sqrt(1 - pow(x, 2))*(346759527252.22327*pow(x, 39) - 3252516578403.7652*pow(x, 37) + 14066078189720.179*pow(x, 35) - 37196962323926.696*pow(x, 33) + 67260260640524.985*pow(x, 31) - 88101468162941.178*pow(x, 29) + 86399024333705.6*pow(x, 27) - 64661103499212.506*pow(x, 25) + 37304482788007.215*pow(x, 23) - 16645562866606.394*pow(x, 21) + 5730439675389.0865*pow(x, 19) - 1509869313546.277*pow(x, 17) + 300207933687.56385*pow(x, 15) - 44086479772.299586*pow(x, 13) + 4634427792.7754282*pow(x, 11) - 333194155.03614189*pow(x, 9) + 15299731.608802434*pow(x, 7) - 402120.6054879238*pow(x, 5) + 4964.4519196039976*pow(x, 3) - 18.229321614212965*x);

Percentage Accurate: 99.9% → 99.9%
Time: 27.6s
Alternatives: 12
Speedup: 1.1×

Specification

?
\[-1 \leq x \land x \leq 1\]
\[\begin{array}{l} \\ \sqrt{1 - {x}^{2}} \cdot \left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(346759527252.22327 \cdot {x}^{39} - 3252516578403.765 \cdot {x}^{37}\right) + 14066078189720.18 \cdot {x}^{35}\right) - 37196962323926.695 \cdot {x}^{33}\right) + 67260260640524.984 \cdot {x}^{31}\right) - 88101468162941.17 \cdot {x}^{29}\right) + 86399024333705.6 \cdot {x}^{27}\right) - 64661103499212.51 \cdot {x}^{25}\right) + 37304482788007.22 \cdot {x}^{23}\right) - 16645562866606.395 \cdot {x}^{21}\right) + 5730439675389.087 \cdot {x}^{19}\right) - 1509869313546.277 \cdot {x}^{17}\right) + 300207933687.56384 \cdot {x}^{15}\right) - 44086479772.29958 \cdot {x}^{13}\right) + 4634427792.775428 \cdot {x}^{11}\right) - 333194155.0361419 \cdot {x}^{9}\right) + 15299731.608802434 \cdot {x}^{7}\right) - 402120.6054879238 \cdot {x}^{5}\right) + 4964.451919603997 \cdot {x}^{3}\right) - 18.229321614212964 \cdot x\right) \end{array} \]
(FPCore (x)
 :precision binary64
 (*
  (sqrt (- 1.0 (pow x 2.0)))
  (-
   (+
    (-
     (+
      (-
       (+
        (-
         (+
          (-
           (+
            (-
             (+
              (-
               (+
                (-
                 (+
                  (-
                   (+
                    (-
                     (* 346759527252.22327 (pow x 39.0))
                     (* 3252516578403.765 (pow x 37.0)))
                    (* 14066078189720.18 (pow x 35.0)))
                   (* 37196962323926.695 (pow x 33.0)))
                  (* 67260260640524.984 (pow x 31.0)))
                 (* 88101468162941.17 (pow x 29.0)))
                (* 86399024333705.6 (pow x 27.0)))
               (* 64661103499212.51 (pow x 25.0)))
              (* 37304482788007.22 (pow x 23.0)))
             (* 16645562866606.395 (pow x 21.0)))
            (* 5730439675389.087 (pow x 19.0)))
           (* 1509869313546.277 (pow x 17.0)))
          (* 300207933687.56384 (pow x 15.0)))
         (* 44086479772.29958 (pow x 13.0)))
        (* 4634427792.775428 (pow x 11.0)))
       (* 333194155.0361419 (pow x 9.0)))
      (* 15299731.608802434 (pow x 7.0)))
     (* 402120.6054879238 (pow x 5.0)))
    (* 4964.451919603997 (pow x 3.0)))
   (* 18.229321614212964 x))))
double code(double x) {
	return sqrt((1.0 - pow(x, 2.0))) * ((((((((((((((((((((346759527252.22327 * pow(x, 39.0)) - (3252516578403.765 * pow(x, 37.0))) + (14066078189720.18 * pow(x, 35.0))) - (37196962323926.695 * pow(x, 33.0))) + (67260260640524.984 * pow(x, 31.0))) - (88101468162941.17 * pow(x, 29.0))) + (86399024333705.6 * pow(x, 27.0))) - (64661103499212.51 * pow(x, 25.0))) + (37304482788007.22 * pow(x, 23.0))) - (16645562866606.395 * pow(x, 21.0))) + (5730439675389.087 * pow(x, 19.0))) - (1509869313546.277 * pow(x, 17.0))) + (300207933687.56384 * pow(x, 15.0))) - (44086479772.29958 * pow(x, 13.0))) + (4634427792.775428 * pow(x, 11.0))) - (333194155.0361419 * pow(x, 9.0))) + (15299731.608802434 * pow(x, 7.0))) - (402120.6054879238 * pow(x, 5.0))) + (4964.451919603997 * pow(x, 3.0))) - (18.229321614212964 * x));
}

real(8) function code(x)
    real(8), intent (in) :: x
    code = sqrt((1.0d0 - (x ** 2.0d0))) * ((((((((((((((((((((346759527252.22327d0 * (x ** 39.0d0)) - (3252516578403.765d0 * (x ** 37.0d0))) + (14066078189720.18d0 * (x ** 35.0d0))) - (37196962323926.695d0 * (x ** 33.0d0))) + (67260260640524.984d0 * (x ** 31.0d0))) - (88101468162941.17d0 * (x ** 29.0d0))) + (86399024333705.6d0 * (x ** 27.0d0))) - (64661103499212.51d0 * (x ** 25.0d0))) + (37304482788007.22d0 * (x ** 23.0d0))) - (16645562866606.395d0 * (x ** 21.0d0))) + (5730439675389.087d0 * (x ** 19.0d0))) - (1509869313546.277d0 * (x ** 17.0d0))) + (300207933687.56384d0 * (x ** 15.0d0))) - (44086479772.29958d0 * (x ** 13.0d0))) + (4634427792.775428d0 * (x ** 11.0d0))) - (333194155.0361419d0 * (x ** 9.0d0))) + (15299731.608802434d0 * (x ** 7.0d0))) - (402120.6054879238d0 * (x ** 5.0d0))) + (4964.451919603997d0 * (x ** 3.0d0))) - (18.229321614212964d0 * x))
end function

public static double code(double x) {
	return Math.sqrt((1.0 - Math.pow(x, 2.0))) * ((((((((((((((((((((346759527252.22327 * Math.pow(x, 39.0)) - (3252516578403.765 * Math.pow(x, 37.0))) + (14066078189720.18 * Math.pow(x, 35.0))) - (37196962323926.695 * Math.pow(x, 33.0))) + (67260260640524.984 * Math.pow(x, 31.0))) - (88101468162941.17 * Math.pow(x, 29.0))) + (86399024333705.6 * Math.pow(x, 27.0))) - (64661103499212.51 * Math.pow(x, 25.0))) + (37304482788007.22 * Math.pow(x, 23.0))) - (16645562866606.395 * Math.pow(x, 21.0))) + (5730439675389.087 * Math.pow(x, 19.0))) - (1509869313546.277 * Math.pow(x, 17.0))) + (300207933687.56384 * Math.pow(x, 15.0))) - (44086479772.29958 * Math.pow(x, 13.0))) + (4634427792.775428 * Math.pow(x, 11.0))) - (333194155.0361419 * Math.pow(x, 9.0))) + (15299731.608802434 * Math.pow(x, 7.0))) - (402120.6054879238 * Math.pow(x, 5.0))) + (4964.451919603997 * Math.pow(x, 3.0))) - (18.229321614212964 * x));
}

def code(x):
	return math.sqrt((1.0 - math.pow(x, 2.0))) * ((((((((((((((((((((346759527252.22327 * math.pow(x, 39.0)) - (3252516578403.765 * math.pow(x, 37.0))) + (14066078189720.18 * math.pow(x, 35.0))) - (37196962323926.695 * math.pow(x, 33.0))) + (67260260640524.984 * math.pow(x, 31.0))) - (88101468162941.17 * math.pow(x, 29.0))) + (86399024333705.6 * math.pow(x, 27.0))) - (64661103499212.51 * math.pow(x, 25.0))) + (37304482788007.22 * math.pow(x, 23.0))) - (16645562866606.395 * math.pow(x, 21.0))) + (5730439675389.087 * math.pow(x, 19.0))) - (1509869313546.277 * math.pow(x, 17.0))) + (300207933687.56384 * math.pow(x, 15.0))) - (44086479772.29958 * math.pow(x, 13.0))) + (4634427792.775428 * math.pow(x, 11.0))) - (333194155.0361419 * math.pow(x, 9.0))) + (15299731.608802434 * math.pow(x, 7.0))) - (402120.6054879238 * math.pow(x, 5.0))) + (4964.451919603997 * math.pow(x, 3.0))) - (18.229321614212964 * x))

function code(x)
	return Float64(sqrt(Float64(1.0 - (x ^ 2.0))) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(346759527252.22327 * (x ^ 39.0)) - Float64(3252516578403.765 * (x ^ 37.0))) + Float64(14066078189720.18 * (x ^ 35.0))) - Float64(37196962323926.695 * (x ^ 33.0))) + Float64(67260260640524.984 * (x ^ 31.0))) - Float64(88101468162941.17 * (x ^ 29.0))) + Float64(86399024333705.6 * (x ^ 27.0))) - Float64(64661103499212.51 * (x ^ 25.0))) + Float64(37304482788007.22 * (x ^ 23.0))) - Float64(16645562866606.395 * (x ^ 21.0))) + Float64(5730439675389.087 * (x ^ 19.0))) - Float64(1509869313546.277 * (x ^ 17.0))) + Float64(300207933687.56384 * (x ^ 15.0))) - Float64(44086479772.29958 * (x ^ 13.0))) + Float64(4634427792.775428 * (x ^ 11.0))) - Float64(333194155.0361419 * (x ^ 9.0))) + Float64(15299731.608802434 * (x ^ 7.0))) - Float64(402120.6054879238 * (x ^ 5.0))) + Float64(4964.451919603997 * (x ^ 3.0))) - Float64(18.229321614212964 * x)))
end

function tmp = code(x)
	tmp = sqrt((1.0 - (x ^ 2.0))) * ((((((((((((((((((((346759527252.22327 * (x ^ 39.0)) - (3252516578403.765 * (x ^ 37.0))) + (14066078189720.18 * (x ^ 35.0))) - (37196962323926.695 * (x ^ 33.0))) + (67260260640524.984 * (x ^ 31.0))) - (88101468162941.17 * (x ^ 29.0))) + (86399024333705.6 * (x ^ 27.0))) - (64661103499212.51 * (x ^ 25.0))) + (37304482788007.22 * (x ^ 23.0))) - (16645562866606.395 * (x ^ 21.0))) + (5730439675389.087 * (x ^ 19.0))) - (1509869313546.277 * (x ^ 17.0))) + (300207933687.56384 * (x ^ 15.0))) - (44086479772.29958 * (x ^ 13.0))) + (4634427792.775428 * (x ^ 11.0))) - (333194155.0361419 * (x ^ 9.0))) + (15299731.608802434 * (x ^ 7.0))) - (402120.6054879238 * (x ^ 5.0))) + (4964.451919603997 * (x ^ 3.0))) - (18.229321614212964 * x));
end

code[x_] := N[(N[Sqrt[N[(1.0 - N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(N[(N[(N[(N[(N[(N[(N[(N[(N[(N[(N[(N[(346759527252.22327 * N[Power[x, 39.0], $MachinePrecision]), $MachinePrecision] - N[(3252516578403.765 * N[Power[x, 37.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(14066078189720.18 * N[Power[x, 35.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(37196962323926.695 * N[Power[x, 33.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(67260260640524.984 * N[Power[x, 31.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(88101468162941.17 * N[Power[x, 29.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(86399024333705.6 * N[Power[x, 27.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(64661103499212.51 * N[Power[x, 25.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(37304482788007.22 * N[Power[x, 23.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(16645562866606.395 * N[Power[x, 21.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(5730439675389.087 * N[Power[x, 19.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(1509869313546.277 * N[Power[x, 17.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(300207933687.56384 * N[Power[x, 15.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(44086479772.29958 * N[Power[x, 13.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(4634427792.775428 * N[Power[x, 11.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(333194155.0361419 * N[Power[x, 9.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(15299731.608802434 * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(402120.6054879238 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(4964.451919603997 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(18.229321614212964 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\begin{array}{l}

\\
\sqrt{1 - {x}^{2}} \cdot \left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(346759527252.22327 \cdot {x}^{39} - 3252516578403.765 \cdot {x}^{37}\right) + 14066078189720.18 \cdot {x}^{35}\right) - 37196962323926.695 \cdot {x}^{33}\right) + 67260260640524.984 \cdot {x}^{31}\right) - 88101468162941.17 \cdot {x}^{29}\right) + 86399024333705.6 \cdot {x}^{27}\right) - 64661103499212.51 \cdot {x}^{25}\right) + 37304482788007.22 \cdot {x}^{23}\right) - 16645562866606.395 \cdot {x}^{21}\right) + 5730439675389.087 \cdot {x}^{19}\right) - 1509869313546.277 \cdot {x}^{17}\right) + 300207933687.56384 \cdot {x}^{15}\right) - 44086479772.29958 \cdot {x}^{13}\right) + 4634427792.775428 \cdot {x}^{11}\right) - 333194155.0361419 \cdot {x}^{9}\right) + 15299731.608802434 \cdot {x}^{7}\right) - 402120.6054879238 \cdot {x}^{5}\right) + 4964.451919603997 \cdot {x}^{3}\right) - 18.229321614212964 \cdot x\right)
\end{array}

Sampling outcomes in binary64 precision:

Local Percentage Accuracy vs ?

The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples.

Accuracy vs Speed?

Herbie found 12 alternatives:

AlternativeAccuracySpeedup
The accuracy (vertical axis) and speed (horizontal axis) of each alternatives. Up and to the right is better. The red square shows the initial program, and each blue circle shows an alternative.The line shows the best available speed-accuracy tradeoffs.

Initial Program: 99.9% accurate, 1.0× speedup?

\[\begin{array}{l} \\ \sqrt{1 - {x}^{2}} \cdot \left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(346759527252.22327 \cdot {x}^{39} - 3252516578403.765 \cdot {x}^{37}\right) + 14066078189720.18 \cdot {x}^{35}\right) - 37196962323926.695 \cdot {x}^{33}\right) + 67260260640524.984 \cdot {x}^{31}\right) - 88101468162941.17 \cdot {x}^{29}\right) + 86399024333705.6 \cdot {x}^{27}\right) - 64661103499212.51 \cdot {x}^{25}\right) + 37304482788007.22 \cdot {x}^{23}\right) - 16645562866606.395 \cdot {x}^{21}\right) + 5730439675389.087 \cdot {x}^{19}\right) - 1509869313546.277 \cdot {x}^{17}\right) + 300207933687.56384 \cdot {x}^{15}\right) - 44086479772.29958 \cdot {x}^{13}\right) + 4634427792.775428 \cdot {x}^{11}\right) - 333194155.0361419 \cdot {x}^{9}\right) + 15299731.608802434 \cdot {x}^{7}\right) - 402120.6054879238 \cdot {x}^{5}\right) + 4964.451919603997 \cdot {x}^{3}\right) - 18.229321614212964 \cdot x\right) \end{array} \]
(FPCore (x)
 :precision binary64
 (*
  (sqrt (- 1.0 (pow x 2.0)))
  (-
   (+
    (-
     (+
      (-
       (+
        (-
         (+
          (-
           (+
            (-
             (+
              (-
               (+
                (-
                 (+
                  (-
                   (+
                    (-
                     (* 346759527252.22327 (pow x 39.0))
                     (* 3252516578403.765 (pow x 37.0)))
                    (* 14066078189720.18 (pow x 35.0)))
                   (* 37196962323926.695 (pow x 33.0)))
                  (* 67260260640524.984 (pow x 31.0)))
                 (* 88101468162941.17 (pow x 29.0)))
                (* 86399024333705.6 (pow x 27.0)))
               (* 64661103499212.51 (pow x 25.0)))
              (* 37304482788007.22 (pow x 23.0)))
             (* 16645562866606.395 (pow x 21.0)))
            (* 5730439675389.087 (pow x 19.0)))
           (* 1509869313546.277 (pow x 17.0)))
          (* 300207933687.56384 (pow x 15.0)))
         (* 44086479772.29958 (pow x 13.0)))
        (* 4634427792.775428 (pow x 11.0)))
       (* 333194155.0361419 (pow x 9.0)))
      (* 15299731.608802434 (pow x 7.0)))
     (* 402120.6054879238 (pow x 5.0)))
    (* 4964.451919603997 (pow x 3.0)))
   (* 18.229321614212964 x))))
double code(double x) {
	return sqrt((1.0 - pow(x, 2.0))) * ((((((((((((((((((((346759527252.22327 * pow(x, 39.0)) - (3252516578403.765 * pow(x, 37.0))) + (14066078189720.18 * pow(x, 35.0))) - (37196962323926.695 * pow(x, 33.0))) + (67260260640524.984 * pow(x, 31.0))) - (88101468162941.17 * pow(x, 29.0))) + (86399024333705.6 * pow(x, 27.0))) - (64661103499212.51 * pow(x, 25.0))) + (37304482788007.22 * pow(x, 23.0))) - (16645562866606.395 * pow(x, 21.0))) + (5730439675389.087 * pow(x, 19.0))) - (1509869313546.277 * pow(x, 17.0))) + (300207933687.56384 * pow(x, 15.0))) - (44086479772.29958 * pow(x, 13.0))) + (4634427792.775428 * pow(x, 11.0))) - (333194155.0361419 * pow(x, 9.0))) + (15299731.608802434 * pow(x, 7.0))) - (402120.6054879238 * pow(x, 5.0))) + (4964.451919603997 * pow(x, 3.0))) - (18.229321614212964 * x));
}

real(8) function code(x)
    real(8), intent (in) :: x
    code = sqrt((1.0d0 - (x ** 2.0d0))) * ((((((((((((((((((((346759527252.22327d0 * (x ** 39.0d0)) - (3252516578403.765d0 * (x ** 37.0d0))) + (14066078189720.18d0 * (x ** 35.0d0))) - (37196962323926.695d0 * (x ** 33.0d0))) + (67260260640524.984d0 * (x ** 31.0d0))) - (88101468162941.17d0 * (x ** 29.0d0))) + (86399024333705.6d0 * (x ** 27.0d0))) - (64661103499212.51d0 * (x ** 25.0d0))) + (37304482788007.22d0 * (x ** 23.0d0))) - (16645562866606.395d0 * (x ** 21.0d0))) + (5730439675389.087d0 * (x ** 19.0d0))) - (1509869313546.277d0 * (x ** 17.0d0))) + (300207933687.56384d0 * (x ** 15.0d0))) - (44086479772.29958d0 * (x ** 13.0d0))) + (4634427792.775428d0 * (x ** 11.0d0))) - (333194155.0361419d0 * (x ** 9.0d0))) + (15299731.608802434d0 * (x ** 7.0d0))) - (402120.6054879238d0 * (x ** 5.0d0))) + (4964.451919603997d0 * (x ** 3.0d0))) - (18.229321614212964d0 * x))
end function

public static double code(double x) {
	return Math.sqrt((1.0 - Math.pow(x, 2.0))) * ((((((((((((((((((((346759527252.22327 * Math.pow(x, 39.0)) - (3252516578403.765 * Math.pow(x, 37.0))) + (14066078189720.18 * Math.pow(x, 35.0))) - (37196962323926.695 * Math.pow(x, 33.0))) + (67260260640524.984 * Math.pow(x, 31.0))) - (88101468162941.17 * Math.pow(x, 29.0))) + (86399024333705.6 * Math.pow(x, 27.0))) - (64661103499212.51 * Math.pow(x, 25.0))) + (37304482788007.22 * Math.pow(x, 23.0))) - (16645562866606.395 * Math.pow(x, 21.0))) + (5730439675389.087 * Math.pow(x, 19.0))) - (1509869313546.277 * Math.pow(x, 17.0))) + (300207933687.56384 * Math.pow(x, 15.0))) - (44086479772.29958 * Math.pow(x, 13.0))) + (4634427792.775428 * Math.pow(x, 11.0))) - (333194155.0361419 * Math.pow(x, 9.0))) + (15299731.608802434 * Math.pow(x, 7.0))) - (402120.6054879238 * Math.pow(x, 5.0))) + (4964.451919603997 * Math.pow(x, 3.0))) - (18.229321614212964 * x));
}

def code(x):
	return math.sqrt((1.0 - math.pow(x, 2.0))) * ((((((((((((((((((((346759527252.22327 * math.pow(x, 39.0)) - (3252516578403.765 * math.pow(x, 37.0))) + (14066078189720.18 * math.pow(x, 35.0))) - (37196962323926.695 * math.pow(x, 33.0))) + (67260260640524.984 * math.pow(x, 31.0))) - (88101468162941.17 * math.pow(x, 29.0))) + (86399024333705.6 * math.pow(x, 27.0))) - (64661103499212.51 * math.pow(x, 25.0))) + (37304482788007.22 * math.pow(x, 23.0))) - (16645562866606.395 * math.pow(x, 21.0))) + (5730439675389.087 * math.pow(x, 19.0))) - (1509869313546.277 * math.pow(x, 17.0))) + (300207933687.56384 * math.pow(x, 15.0))) - (44086479772.29958 * math.pow(x, 13.0))) + (4634427792.775428 * math.pow(x, 11.0))) - (333194155.0361419 * math.pow(x, 9.0))) + (15299731.608802434 * math.pow(x, 7.0))) - (402120.6054879238 * math.pow(x, 5.0))) + (4964.451919603997 * math.pow(x, 3.0))) - (18.229321614212964 * x))

function code(x)
	return Float64(sqrt(Float64(1.0 - (x ^ 2.0))) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(346759527252.22327 * (x ^ 39.0)) - Float64(3252516578403.765 * (x ^ 37.0))) + Float64(14066078189720.18 * (x ^ 35.0))) - Float64(37196962323926.695 * (x ^ 33.0))) + Float64(67260260640524.984 * (x ^ 31.0))) - Float64(88101468162941.17 * (x ^ 29.0))) + Float64(86399024333705.6 * (x ^ 27.0))) - Float64(64661103499212.51 * (x ^ 25.0))) + Float64(37304482788007.22 * (x ^ 23.0))) - Float64(16645562866606.395 * (x ^ 21.0))) + Float64(5730439675389.087 * (x ^ 19.0))) - Float64(1509869313546.277 * (x ^ 17.0))) + Float64(300207933687.56384 * (x ^ 15.0))) - Float64(44086479772.29958 * (x ^ 13.0))) + Float64(4634427792.775428 * (x ^ 11.0))) - Float64(333194155.0361419 * (x ^ 9.0))) + Float64(15299731.608802434 * (x ^ 7.0))) - Float64(402120.6054879238 * (x ^ 5.0))) + Float64(4964.451919603997 * (x ^ 3.0))) - Float64(18.229321614212964 * x)))
end

function tmp = code(x)
	tmp = sqrt((1.0 - (x ^ 2.0))) * ((((((((((((((((((((346759527252.22327 * (x ^ 39.0)) - (3252516578403.765 * (x ^ 37.0))) + (14066078189720.18 * (x ^ 35.0))) - (37196962323926.695 * (x ^ 33.0))) + (67260260640524.984 * (x ^ 31.0))) - (88101468162941.17 * (x ^ 29.0))) + (86399024333705.6 * (x ^ 27.0))) - (64661103499212.51 * (x ^ 25.0))) + (37304482788007.22 * (x ^ 23.0))) - (16645562866606.395 * (x ^ 21.0))) + (5730439675389.087 * (x ^ 19.0))) - (1509869313546.277 * (x ^ 17.0))) + (300207933687.56384 * (x ^ 15.0))) - (44086479772.29958 * (x ^ 13.0))) + (4634427792.775428 * (x ^ 11.0))) - (333194155.0361419 * (x ^ 9.0))) + (15299731.608802434 * (x ^ 7.0))) - (402120.6054879238 * (x ^ 5.0))) + (4964.451919603997 * (x ^ 3.0))) - (18.229321614212964 * x));
end

code[x_] := N[(N[Sqrt[N[(1.0 - N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(N[(N[(N[(N[(N[(N[(N[(N[(N[(N[(N[(N[(346759527252.22327 * N[Power[x, 39.0], $MachinePrecision]), $MachinePrecision] - N[(3252516578403.765 * N[Power[x, 37.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(14066078189720.18 * N[Power[x, 35.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(37196962323926.695 * N[Power[x, 33.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(67260260640524.984 * N[Power[x, 31.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(88101468162941.17 * N[Power[x, 29.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(86399024333705.6 * N[Power[x, 27.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(64661103499212.51 * N[Power[x, 25.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(37304482788007.22 * N[Power[x, 23.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(16645562866606.395 * N[Power[x, 21.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(5730439675389.087 * N[Power[x, 19.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(1509869313546.277 * N[Power[x, 17.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(300207933687.56384 * N[Power[x, 15.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(44086479772.29958 * N[Power[x, 13.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(4634427792.775428 * N[Power[x, 11.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(333194155.0361419 * N[Power[x, 9.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(15299731.608802434 * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(402120.6054879238 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(4964.451919603997 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(18.229321614212964 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\begin{array}{l}

\\
\sqrt{1 - {x}^{2}} \cdot \left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(346759527252.22327 \cdot {x}^{39} - 3252516578403.765 \cdot {x}^{37}\right) + 14066078189720.18 \cdot {x}^{35}\right) - 37196962323926.695 \cdot {x}^{33}\right) + 67260260640524.984 \cdot {x}^{31}\right) - 88101468162941.17 \cdot {x}^{29}\right) + 86399024333705.6 \cdot {x}^{27}\right) - 64661103499212.51 \cdot {x}^{25}\right) + 37304482788007.22 \cdot {x}^{23}\right) - 16645562866606.395 \cdot {x}^{21}\right) + 5730439675389.087 \cdot {x}^{19}\right) - 1509869313546.277 \cdot {x}^{17}\right) + 300207933687.56384 \cdot {x}^{15}\right) - 44086479772.29958 \cdot {x}^{13}\right) + 4634427792.775428 \cdot {x}^{11}\right) - 333194155.0361419 \cdot {x}^{9}\right) + 15299731.608802434 \cdot {x}^{7}\right) - 402120.6054879238 \cdot {x}^{5}\right) + 4964.451919603997 \cdot {x}^{3}\right) - 18.229321614212964 \cdot x\right)
\end{array}

Alternative 1: 99.9% accurate, 1.1× speedup?

\[\begin{array}{l} \\ \sqrt{1 - {x}^{2}} \cdot \mathsf{fma}\left(x \cdot x, x \cdot 4964.451919603997, \mathsf{fma}\left(-402120.6054879238, {x}^{5}, \mathsf{fma}\left(15299731.608802434, {x}^{7}, {x}^{9} \cdot -333194155.0361419\right) + \mathsf{fma}\left(4634427792.775428, {x}^{11}, \mathsf{fma}\left({x}^{13}, -44086479772.29958, \mathsf{fma}\left(300207933687.56384, {x}^{15}, \mathsf{fma}\left({x}^{17}, -1509869313546.277, \mathsf{fma}\left(5730439675389.087, {x}^{19}, \mathsf{fma}\left({x}^{21}, -16645562866606.395, \mathsf{fma}\left(37304482788007.22, {x}^{23}, \mathsf{fma}\left({x}^{25}, -64661103499212.51, \mathsf{fma}\left(86399024333705.6, {x}^{27}, \mathsf{fma}\left({x}^{29}, -88101468162941.17, \mathsf{fma}\left(67260260640524.984, {x}^{31}, \mathsf{fma}\left({x}^{33}, -37196962323926.695, \mathsf{fma}\left(14066078189720.18, {x}^{35}, \mathsf{fma}\left({x}^{37}, -3252516578403.765, 346759527252.22327 \cdot {x}^{39}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) - 18.229321614212964 \cdot x\right) \end{array} \]
(FPCore (x)
 :precision binary64
 (*
  (sqrt (- 1.0 (pow x 2.0)))
  (fma
   (* x x)
   (* x 4964.451919603997)
   (-
    (fma
     -402120.6054879238
     (pow x 5.0)
     (+
      (fma 15299731.608802434 (pow x 7.0) (* (pow x 9.0) -333194155.0361419))
      (fma
       4634427792.775428
       (pow x 11.0)
       (fma
        (pow x 13.0)
        -44086479772.29958
        (fma
         300207933687.56384
         (pow x 15.0)
         (fma
          (pow x 17.0)
          -1509869313546.277
          (fma
           5730439675389.087
           (pow x 19.0)
           (fma
            (pow x 21.0)
            -16645562866606.395
            (fma
             37304482788007.22
             (pow x 23.0)
             (fma
              (pow x 25.0)
              -64661103499212.51
              (fma
               86399024333705.6
               (pow x 27.0)
               (fma
                (pow x 29.0)
                -88101468162941.17
                (fma
                 67260260640524.984
                 (pow x 31.0)
                 (fma
                  (pow x 33.0)
                  -37196962323926.695
                  (fma
                   14066078189720.18
                   (pow x 35.0)
                   (fma
                    (pow x 37.0)
                    -3252516578403.765
                    (* 346759527252.22327 (pow x 39.0))))))))))))))))))
    (* 18.229321614212964 x)))))
double code(double x) {
	return sqrt((1.0 - pow(x, 2.0))) * fma((x * x), (x * 4964.451919603997), (fma(-402120.6054879238, pow(x, 5.0), (fma(15299731.608802434, pow(x, 7.0), (pow(x, 9.0) * -333194155.0361419)) + fma(4634427792.775428, pow(x, 11.0), fma(pow(x, 13.0), -44086479772.29958, fma(300207933687.56384, pow(x, 15.0), fma(pow(x, 17.0), -1509869313546.277, fma(5730439675389.087, pow(x, 19.0), fma(pow(x, 21.0), -16645562866606.395, fma(37304482788007.22, pow(x, 23.0), fma(pow(x, 25.0), -64661103499212.51, fma(86399024333705.6, pow(x, 27.0), fma(pow(x, 29.0), -88101468162941.17, fma(67260260640524.984, pow(x, 31.0), fma(pow(x, 33.0), -37196962323926.695, fma(14066078189720.18, pow(x, 35.0), fma(pow(x, 37.0), -3252516578403.765, (346759527252.22327 * pow(x, 39.0)))))))))))))))))) - (18.229321614212964 * x)));
}

function code(x)
	return Float64(sqrt(Float64(1.0 - (x ^ 2.0))) * fma(Float64(x * x), Float64(x * 4964.451919603997), Float64(fma(-402120.6054879238, (x ^ 5.0), Float64(fma(15299731.608802434, (x ^ 7.0), Float64((x ^ 9.0) * -333194155.0361419)) + fma(4634427792.775428, (x ^ 11.0), fma((x ^ 13.0), -44086479772.29958, fma(300207933687.56384, (x ^ 15.0), fma((x ^ 17.0), -1509869313546.277, fma(5730439675389.087, (x ^ 19.0), fma((x ^ 21.0), -16645562866606.395, fma(37304482788007.22, (x ^ 23.0), fma((x ^ 25.0), -64661103499212.51, fma(86399024333705.6, (x ^ 27.0), fma((x ^ 29.0), -88101468162941.17, fma(67260260640524.984, (x ^ 31.0), fma((x ^ 33.0), -37196962323926.695, fma(14066078189720.18, (x ^ 35.0), fma((x ^ 37.0), -3252516578403.765, Float64(346759527252.22327 * (x ^ 39.0)))))))))))))))))) - Float64(18.229321614212964 * x))))
end

code[x_] := N[(N[Sqrt[N[(1.0 - N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * 4964.451919603997), $MachinePrecision] + N[(N[(-402120.6054879238 * N[Power[x, 5.0], $MachinePrecision] + N[(N[(15299731.608802434 * N[Power[x, 7.0], $MachinePrecision] + N[(N[Power[x, 9.0], $MachinePrecision] * -333194155.0361419), $MachinePrecision]), $MachinePrecision] + N[(4634427792.775428 * N[Power[x, 11.0], $MachinePrecision] + N[(N[Power[x, 13.0], $MachinePrecision] * -44086479772.29958 + N[(300207933687.56384 * N[Power[x, 15.0], $MachinePrecision] + N[(N[Power[x, 17.0], $MachinePrecision] * -1509869313546.277 + N[(5730439675389.087 * N[Power[x, 19.0], $MachinePrecision] + N[(N[Power[x, 21.0], $MachinePrecision] * -16645562866606.395 + N[(37304482788007.22 * N[Power[x, 23.0], $MachinePrecision] + N[(N[Power[x, 25.0], $MachinePrecision] * -64661103499212.51 + N[(86399024333705.6 * N[Power[x, 27.0], $MachinePrecision] + N[(N[Power[x, 29.0], $MachinePrecision] * -88101468162941.17 + N[(67260260640524.984 * N[Power[x, 31.0], $MachinePrecision] + N[(N[Power[x, 33.0], $MachinePrecision] * -37196962323926.695 + N[(14066078189720.18 * N[Power[x, 35.0], $MachinePrecision] + N[(N[Power[x, 37.0], $MachinePrecision] * -3252516578403.765 + N[(346759527252.22327 * N[Power[x, 39.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(18.229321614212964 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\begin{array}{l}

\\
\sqrt{1 - {x}^{2}} \cdot \mathsf{fma}\left(x \cdot x, x \cdot 4964.451919603997, \mathsf{fma}\left(-402120.6054879238, {x}^{5}, \mathsf{fma}\left(15299731.608802434, {x}^{7}, {x}^{9} \cdot -333194155.0361419\right) + \mathsf{fma}\left(4634427792.775428, {x}^{11}, \mathsf{fma}\left({x}^{13}, -44086479772.29958, \mathsf{fma}\left(300207933687.56384, {x}^{15}, \mathsf{fma}\left({x}^{17}, -1509869313546.277, \mathsf{fma}\left(5730439675389.087, {x}^{19}, \mathsf{fma}\left({x}^{21}, -16645562866606.395, \mathsf{fma}\left(37304482788007.22, {x}^{23}, \mathsf{fma}\left({x}^{25}, -64661103499212.51, \mathsf{fma}\left(86399024333705.6, {x}^{27}, \mathsf{fma}\left({x}^{29}, -88101468162941.17, \mathsf{fma}\left(67260260640524.984, {x}^{31}, \mathsf{fma}\left({x}^{33}, -37196962323926.695, \mathsf{fma}\left(14066078189720.18, {x}^{35}, \mathsf{fma}\left({x}^{37}, -3252516578403.765, 346759527252.22327 \cdot {x}^{39}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) - 18.229321614212964 \cdot x\right)
\end{array}
Derivation

  1. Initial program 99.7%

    \[\sqrt{1 - {x}^{2}} \cdot \left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(346759527252.22327 \cdot {x}^{39} - 3252516578403.765 \cdot {x}^{37}\right) + 14066078189720.18 \cdot {x}^{35}\right) - 37196962323926.695 \cdot {x}^{33}\right) + 67260260640524.984 \cdot {x}^{31}\right) - 88101468162941.17 \cdot {x}^{29}\right) + 86399024333705.6 \cdot {x}^{27}\right) - 64661103499212.51 \cdot {x}^{25}\right) + 37304482788007.22 \cdot {x}^{23}\right) - 16645562866606.395 \cdot {x}^{21}\right) + 5730439675389.087 \cdot {x}^{19}\right) - 1509869313546.277 \cdot {x}^{17}\right) + 300207933687.56384 \cdot {x}^{15}\right) - 44086479772.29958 \cdot {x}^{13}\right) + 4634427792.775428 \cdot {x}^{11}\right) - 333194155.0361419 \cdot {x}^{9}\right) + 15299731.608802434 \cdot {x}^{7}\right) - 402120.6054879238 \cdot {x}^{5}\right) + 4964.451919603997 \cdot {x}^{3}\right) - 18.229321614212964 \cdot x\right) \]
  2. Add Preprocessing

  4. Applied rewrites99.7%

    \[\leadsto \sqrt{1 - {x}^{2}} \cdot \color{blue}{\mathsf{fma}\left(x \cdot x, x \cdot 4964.451919603997, \mathsf{fma}\left(-402120.6054879238, {x}^{5}, \mathsf{fma}\left({x}^{7}, 15299731.608802434, \mathsf{fma}\left(-333194155.0361419, {x}^{9}, \mathsf{fma}\left({x}^{11}, 4634427792.775428, \mathsf{fma}\left(-44086479772.29958, {x}^{13}, \mathsf{fma}\left({x}^{15}, 300207933687.56384, \mathsf{fma}\left(-1509869313546.277, {x}^{17}, \mathsf{fma}\left({x}^{19}, 5730439675389.087, \mathsf{fma}\left(-16645562866606.395, {x}^{21}, \mathsf{fma}\left({x}^{23}, 37304482788007.22, \mathsf{fma}\left(-64661103499212.51, {x}^{25}, \mathsf{fma}\left({x}^{27}, 86399024333705.6, \mathsf{fma}\left(-88101468162941.17, {x}^{29}, \mathsf{fma}\left({x}^{31}, 67260260640524.984, \mathsf{fma}\left(-37196962323926.695, {x}^{33}, \mathsf{fma}\left({x}^{35}, 14066078189720.18, \mathsf{fma}\left(-3252516578403.765, {x}^{37}, {x}^{39} \cdot 346759527252.22327\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) - 18.229321614212964 \cdot x\right)} \]

  6. Applied rewrites99.7%

    \[\leadsto \sqrt{1 - {x}^{2}} \cdot \mathsf{fma}\left(x \cdot x, x \cdot 4964.451919603997, \mathsf{fma}\left(-402120.6054879238, {x}^{5}, \color{blue}{\mathsf{fma}\left(15299731.608802434, {x}^{7}, {x}^{9} \cdot -333194155.0361419\right) + \mathsf{fma}\left(4634427792.775428, {x}^{11}, \mathsf{fma}\left({x}^{13}, -44086479772.29958, \mathsf{fma}\left(300207933687.56384, {x}^{15}, \mathsf{fma}\left({x}^{17}, -1509869313546.277, \mathsf{fma}\left(5730439675389.087, {x}^{19}, \mathsf{fma}\left({x}^{21}, -16645562866606.395, \mathsf{fma}\left(37304482788007.22, {x}^{23}, \mathsf{fma}\left({x}^{25}, -64661103499212.51, \mathsf{fma}\left(86399024333705.6, {x}^{27}, \mathsf{fma}\left({x}^{29}, -88101468162941.17, \mathsf{fma}\left(67260260640524.984, {x}^{31}, \mathsf{fma}\left({x}^{33}, -37196962323926.695, \mathsf{fma}\left(14066078189720.18, {x}^{35}, \mathsf{fma}\left({x}^{37}, -3252516578403.765, 346759527252.22327 \cdot {x}^{39}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)}\right) - 18.229321614212964 \cdot x\right) \]
  7. Add Preprocessing

Alternative 2: 99.9% accurate, 1.1× speedup?

\[\begin{array}{l} \\ \sqrt{1 - {x}^{2}} \cdot \mathsf{fma}\left(x, x \cdot \left(4964.451919603997 \cdot x\right), \mathsf{fma}\left({x}^{5}, -402120.6054879238, \mathsf{fma}\left(15299731.608802434, {x}^{7}, \mathsf{fma}\left({x}^{9}, -333194155.0361419, \mathsf{fma}\left(4634427792.775428, {x}^{11}, \mathsf{fma}\left({x}^{13}, -44086479772.29958, \mathsf{fma}\left(300207933687.56384, {x}^{15}, \mathsf{fma}\left({x}^{17}, -1509869313546.277, \mathsf{fma}\left(5730439675389.087, {x}^{19}, \mathsf{fma}\left({x}^{21}, -16645562866606.395, \mathsf{fma}\left(37304482788007.22, {x}^{23}, \mathsf{fma}\left({x}^{25}, -64661103499212.51, \mathsf{fma}\left(86399024333705.6, {x}^{27}, \mathsf{fma}\left({x}^{29}, -88101468162941.17, \mathsf{fma}\left(67260260640524.984, {x}^{31}, \mathsf{fma}\left({x}^{33}, -37196962323926.695, \mathsf{fma}\left(14066078189720.18, {x}^{35}, \mathsf{fma}\left({x}^{37}, -3252516578403.765, 346759527252.22327 \cdot {x}^{39}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) - 18.229321614212964 \cdot x\right) \end{array} \]
(FPCore (x)
 :precision binary64
 (*
  (sqrt (- 1.0 (pow x 2.0)))
  (fma
   x
   (* x (* 4964.451919603997 x))
   (-
    (fma
     (pow x 5.0)
     -402120.6054879238
     (fma
      15299731.608802434
      (pow x 7.0)
      (fma
       (pow x 9.0)
       -333194155.0361419
       (fma
        4634427792.775428
        (pow x 11.0)
        (fma
         (pow x 13.0)
         -44086479772.29958
         (fma
          300207933687.56384
          (pow x 15.0)
          (fma
           (pow x 17.0)
           -1509869313546.277
           (fma
            5730439675389.087
            (pow x 19.0)
            (fma
             (pow x 21.0)
             -16645562866606.395
             (fma
              37304482788007.22
              (pow x 23.0)
              (fma
               (pow x 25.0)
               -64661103499212.51
               (fma
                86399024333705.6
                (pow x 27.0)
                (fma
                 (pow x 29.0)
                 -88101468162941.17
                 (fma
                  67260260640524.984
                  (pow x 31.0)
                  (fma
                   (pow x 33.0)
                   -37196962323926.695
                   (fma
                    14066078189720.18
                    (pow x 35.0)
                    (fma
                     (pow x 37.0)
                     -3252516578403.765
                     (* 346759527252.22327 (pow x 39.0)))))))))))))))))))
    (* 18.229321614212964 x)))))
double code(double x) {
	return sqrt((1.0 - pow(x, 2.0))) * fma(x, (x * (4964.451919603997 * x)), (fma(pow(x, 5.0), -402120.6054879238, fma(15299731.608802434, pow(x, 7.0), fma(pow(x, 9.0), -333194155.0361419, fma(4634427792.775428, pow(x, 11.0), fma(pow(x, 13.0), -44086479772.29958, fma(300207933687.56384, pow(x, 15.0), fma(pow(x, 17.0), -1509869313546.277, fma(5730439675389.087, pow(x, 19.0), fma(pow(x, 21.0), -16645562866606.395, fma(37304482788007.22, pow(x, 23.0), fma(pow(x, 25.0), -64661103499212.51, fma(86399024333705.6, pow(x, 27.0), fma(pow(x, 29.0), -88101468162941.17, fma(67260260640524.984, pow(x, 31.0), fma(pow(x, 33.0), -37196962323926.695, fma(14066078189720.18, pow(x, 35.0), fma(pow(x, 37.0), -3252516578403.765, (346759527252.22327 * pow(x, 39.0))))))))))))))))))) - (18.229321614212964 * x)));
}

function code(x)
	return Float64(sqrt(Float64(1.0 - (x ^ 2.0))) * fma(x, Float64(x * Float64(4964.451919603997 * x)), Float64(fma((x ^ 5.0), -402120.6054879238, fma(15299731.608802434, (x ^ 7.0), fma((x ^ 9.0), -333194155.0361419, fma(4634427792.775428, (x ^ 11.0), fma((x ^ 13.0), -44086479772.29958, fma(300207933687.56384, (x ^ 15.0), fma((x ^ 17.0), -1509869313546.277, fma(5730439675389.087, (x ^ 19.0), fma((x ^ 21.0), -16645562866606.395, fma(37304482788007.22, (x ^ 23.0), fma((x ^ 25.0), -64661103499212.51, fma(86399024333705.6, (x ^ 27.0), fma((x ^ 29.0), -88101468162941.17, fma(67260260640524.984, (x ^ 31.0), fma((x ^ 33.0), -37196962323926.695, fma(14066078189720.18, (x ^ 35.0), fma((x ^ 37.0), -3252516578403.765, Float64(346759527252.22327 * (x ^ 39.0))))))))))))))))))) - Float64(18.229321614212964 * x))))
end

code[x_] := N[(N[Sqrt[N[(1.0 - N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(x * N[(x * N[(4964.451919603997 * x), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[x, 5.0], $MachinePrecision] * -402120.6054879238 + N[(15299731.608802434 * N[Power[x, 7.0], $MachinePrecision] + N[(N[Power[x, 9.0], $MachinePrecision] * -333194155.0361419 + N[(4634427792.775428 * N[Power[x, 11.0], $MachinePrecision] + N[(N[Power[x, 13.0], $MachinePrecision] * -44086479772.29958 + N[(300207933687.56384 * N[Power[x, 15.0], $MachinePrecision] + N[(N[Power[x, 17.0], $MachinePrecision] * -1509869313546.277 + N[(5730439675389.087 * N[Power[x, 19.0], $MachinePrecision] + N[(N[Power[x, 21.0], $MachinePrecision] * -16645562866606.395 + N[(37304482788007.22 * N[Power[x, 23.0], $MachinePrecision] + N[(N[Power[x, 25.0], $MachinePrecision] * -64661103499212.51 + N[(86399024333705.6 * N[Power[x, 27.0], $MachinePrecision] + N[(N[Power[x, 29.0], $MachinePrecision] * -88101468162941.17 + N[(67260260640524.984 * N[Power[x, 31.0], $MachinePrecision] + N[(N[Power[x, 33.0], $MachinePrecision] * -37196962323926.695 + N[(14066078189720.18 * N[Power[x, 35.0], $MachinePrecision] + N[(N[Power[x, 37.0], $MachinePrecision] * -3252516578403.765 + N[(346759527252.22327 * N[Power[x, 39.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(18.229321614212964 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\begin{array}{l}

\\
\sqrt{1 - {x}^{2}} \cdot \mathsf{fma}\left(x, x \cdot \left(4964.451919603997 \cdot x\right), \mathsf{fma}\left({x}^{5}, -402120.6054879238, \mathsf{fma}\left(15299731.608802434, {x}^{7}, \mathsf{fma}\left({x}^{9}, -333194155.0361419, \mathsf{fma}\left(4634427792.775428, {x}^{11}, \mathsf{fma}\left({x}^{13}, -44086479772.29958, \mathsf{fma}\left(300207933687.56384, {x}^{15}, \mathsf{fma}\left({x}^{17}, -1509869313546.277, \mathsf{fma}\left(5730439675389.087, {x}^{19}, \mathsf{fma}\left({x}^{21}, -16645562866606.395, \mathsf{fma}\left(37304482788007.22, {x}^{23}, \mathsf{fma}\left({x}^{25}, -64661103499212.51, \mathsf{fma}\left(86399024333705.6, {x}^{27}, \mathsf{fma}\left({x}^{29}, -88101468162941.17, \mathsf{fma}\left(67260260640524.984, {x}^{31}, \mathsf{fma}\left({x}^{33}, -37196962323926.695, \mathsf{fma}\left(14066078189720.18, {x}^{35}, \mathsf{fma}\left({x}^{37}, -3252516578403.765, 346759527252.22327 \cdot {x}^{39}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) - 18.229321614212964 \cdot x\right)
\end{array}
Derivation

  1. Initial program 99.7%

    \[\sqrt{1 - {x}^{2}} \cdot \left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(346759527252.22327 \cdot {x}^{39} - 3252516578403.765 \cdot {x}^{37}\right) + 14066078189720.18 \cdot {x}^{35}\right) - 37196962323926.695 \cdot {x}^{33}\right) + 67260260640524.984 \cdot {x}^{31}\right) - 88101468162941.17 \cdot {x}^{29}\right) + 86399024333705.6 \cdot {x}^{27}\right) - 64661103499212.51 \cdot {x}^{25}\right) + 37304482788007.22 \cdot {x}^{23}\right) - 16645562866606.395 \cdot {x}^{21}\right) + 5730439675389.087 \cdot {x}^{19}\right) - 1509869313546.277 \cdot {x}^{17}\right) + 300207933687.56384 \cdot {x}^{15}\right) - 44086479772.29958 \cdot {x}^{13}\right) + 4634427792.775428 \cdot {x}^{11}\right) - 333194155.0361419 \cdot {x}^{9}\right) + 15299731.608802434 \cdot {x}^{7}\right) - 402120.6054879238 \cdot {x}^{5}\right) + 4964.451919603997 \cdot {x}^{3}\right) - 18.229321614212964 \cdot x\right) \]
  2. Add Preprocessing

  4. Applied rewrites99.7%

    \[\leadsto \sqrt{1 - {x}^{2}} \cdot \color{blue}{\mathsf{fma}\left(x \cdot x, x \cdot 4964.451919603997, \mathsf{fma}\left(-402120.6054879238, {x}^{5}, \mathsf{fma}\left({x}^{7}, 15299731.608802434, \mathsf{fma}\left(-333194155.0361419, {x}^{9}, \mathsf{fma}\left({x}^{11}, 4634427792.775428, \mathsf{fma}\left(-44086479772.29958, {x}^{13}, \mathsf{fma}\left({x}^{15}, 300207933687.56384, \mathsf{fma}\left(-1509869313546.277, {x}^{17}, \mathsf{fma}\left({x}^{19}, 5730439675389.087, \mathsf{fma}\left(-16645562866606.395, {x}^{21}, \mathsf{fma}\left({x}^{23}, 37304482788007.22, \mathsf{fma}\left(-64661103499212.51, {x}^{25}, \mathsf{fma}\left({x}^{27}, 86399024333705.6, \mathsf{fma}\left(-88101468162941.17, {x}^{29}, \mathsf{fma}\left({x}^{31}, 67260260640524.984, \mathsf{fma}\left(-37196962323926.695, {x}^{33}, \mathsf{fma}\left({x}^{35}, 14066078189720.18, \mathsf{fma}\left(-3252516578403.765, {x}^{37}, {x}^{39} \cdot 346759527252.22327\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) - 18.229321614212964 \cdot x\right)} \]

  6. Applied rewrites99.7%

    \[\leadsto \sqrt{1 - {x}^{2}} \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \left(4964.451919603997 \cdot x\right), \mathsf{fma}\left({x}^{5}, -402120.6054879238, \mathsf{fma}\left(15299731.608802434, {x}^{7}, \mathsf{fma}\left({x}^{9}, -333194155.0361419, \mathsf{fma}\left(4634427792.775428, {x}^{11}, \mathsf{fma}\left({x}^{13}, -44086479772.29958, \mathsf{fma}\left(300207933687.56384, {x}^{15}, \mathsf{fma}\left({x}^{17}, -1509869313546.277, \mathsf{fma}\left(5730439675389.087, {x}^{19}, \mathsf{fma}\left({x}^{21}, -16645562866606.395, \mathsf{fma}\left(37304482788007.22, {x}^{23}, \mathsf{fma}\left({x}^{25}, -64661103499212.51, \mathsf{fma}\left(86399024333705.6, {x}^{27}, \mathsf{fma}\left({x}^{29}, -88101468162941.17, \mathsf{fma}\left(67260260640524.984, {x}^{31}, \mathsf{fma}\left({x}^{33}, -37196962323926.695, \mathsf{fma}\left(14066078189720.18, {x}^{35}, \mathsf{fma}\left({x}^{37}, -3252516578403.765, 346759527252.22327 \cdot {x}^{39}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) - 18.229321614212964 \cdot x\right)} \]
  7. Add Preprocessing

Alternative 3: 99.9% accurate, 1.1× speedup?

\[\begin{array}{l} \\ \sqrt{1 - x \cdot x} \cdot \mathsf{fma}\left(x \cdot x, x \cdot 4964.451919603997, \mathsf{fma}\left(-402120.6054879238, {x}^{5}, \mathsf{fma}\left({x}^{7}, 15299731.608802434, \mathsf{fma}\left(-333194155.0361419, {x}^{9}, \mathsf{fma}\left({x}^{11}, 4634427792.775428, \mathsf{fma}\left(-44086479772.29958, {x}^{13}, \mathsf{fma}\left({x}^{15}, 300207933687.56384, \mathsf{fma}\left(-1509869313546.277, {x}^{17}, \mathsf{fma}\left({x}^{19}, 5730439675389.087, \mathsf{fma}\left(-16645562866606.395, {x}^{21}, \mathsf{fma}\left({x}^{23}, 37304482788007.22, \mathsf{fma}\left(-64661103499212.51, {x}^{25}, \mathsf{fma}\left({x}^{27}, 86399024333705.6, \mathsf{fma}\left(-88101468162941.17, {x}^{29}, \mathsf{fma}\left({x}^{31}, 67260260640524.984, \mathsf{fma}\left(-37196962323926.695, {x}^{33}, \mathsf{fma}\left({x}^{35}, 14066078189720.18, \mathsf{fma}\left(-3252516578403.765, {x}^{37}, {x}^{39} \cdot 346759527252.22327\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) - 18.229321614212964 \cdot x\right) \end{array} \]
(FPCore (x)
 :precision binary64
 (*
  (sqrt (- 1.0 (* x x)))
  (fma
   (* x x)
   (* x 4964.451919603997)
   (-
    (fma
     -402120.6054879238
     (pow x 5.0)
     (fma
      (pow x 7.0)
      15299731.608802434
      (fma
       -333194155.0361419
       (pow x 9.0)
       (fma
        (pow x 11.0)
        4634427792.775428
        (fma
         -44086479772.29958
         (pow x 13.0)
         (fma
          (pow x 15.0)
          300207933687.56384
          (fma
           -1509869313546.277
           (pow x 17.0)
           (fma
            (pow x 19.0)
            5730439675389.087
            (fma
             -16645562866606.395
             (pow x 21.0)
             (fma
              (pow x 23.0)
              37304482788007.22
              (fma
               -64661103499212.51
               (pow x 25.0)
               (fma
                (pow x 27.0)
                86399024333705.6
                (fma
                 -88101468162941.17
                 (pow x 29.0)
                 (fma
                  (pow x 31.0)
                  67260260640524.984
                  (fma
                   -37196962323926.695
                   (pow x 33.0)
                   (fma
                    (pow x 35.0)
                    14066078189720.18
                    (fma
                     -3252516578403.765
                     (pow x 37.0)
                     (* (pow x 39.0) 346759527252.22327))))))))))))))))))
    (* 18.229321614212964 x)))))
double code(double x) {
	return sqrt((1.0 - (x * x))) * fma((x * x), (x * 4964.451919603997), (fma(-402120.6054879238, pow(x, 5.0), fma(pow(x, 7.0), 15299731.608802434, fma(-333194155.0361419, pow(x, 9.0), fma(pow(x, 11.0), 4634427792.775428, fma(-44086479772.29958, pow(x, 13.0), fma(pow(x, 15.0), 300207933687.56384, fma(-1509869313546.277, pow(x, 17.0), fma(pow(x, 19.0), 5730439675389.087, fma(-16645562866606.395, pow(x, 21.0), fma(pow(x, 23.0), 37304482788007.22, fma(-64661103499212.51, pow(x, 25.0), fma(pow(x, 27.0), 86399024333705.6, fma(-88101468162941.17, pow(x, 29.0), fma(pow(x, 31.0), 67260260640524.984, fma(-37196962323926.695, pow(x, 33.0), fma(pow(x, 35.0), 14066078189720.18, fma(-3252516578403.765, pow(x, 37.0), (pow(x, 39.0) * 346759527252.22327)))))))))))))))))) - (18.229321614212964 * x)));
}

function code(x)
	return Float64(sqrt(Float64(1.0 - Float64(x * x))) * fma(Float64(x * x), Float64(x * 4964.451919603997), Float64(fma(-402120.6054879238, (x ^ 5.0), fma((x ^ 7.0), 15299731.608802434, fma(-333194155.0361419, (x ^ 9.0), fma((x ^ 11.0), 4634427792.775428, fma(-44086479772.29958, (x ^ 13.0), fma((x ^ 15.0), 300207933687.56384, fma(-1509869313546.277, (x ^ 17.0), fma((x ^ 19.0), 5730439675389.087, fma(-16645562866606.395, (x ^ 21.0), fma((x ^ 23.0), 37304482788007.22, fma(-64661103499212.51, (x ^ 25.0), fma((x ^ 27.0), 86399024333705.6, fma(-88101468162941.17, (x ^ 29.0), fma((x ^ 31.0), 67260260640524.984, fma(-37196962323926.695, (x ^ 33.0), fma((x ^ 35.0), 14066078189720.18, fma(-3252516578403.765, (x ^ 37.0), Float64((x ^ 39.0) * 346759527252.22327)))))))))))))))))) - Float64(18.229321614212964 * x))))
end

code[x_] := N[(N[Sqrt[N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * 4964.451919603997), $MachinePrecision] + N[(N[(-402120.6054879238 * N[Power[x, 5.0], $MachinePrecision] + N[(N[Power[x, 7.0], $MachinePrecision] * 15299731.608802434 + N[(-333194155.0361419 * N[Power[x, 9.0], $MachinePrecision] + N[(N[Power[x, 11.0], $MachinePrecision] * 4634427792.775428 + N[(-44086479772.29958 * N[Power[x, 13.0], $MachinePrecision] + N[(N[Power[x, 15.0], $MachinePrecision] * 300207933687.56384 + N[(-1509869313546.277 * N[Power[x, 17.0], $MachinePrecision] + N[(N[Power[x, 19.0], $MachinePrecision] * 5730439675389.087 + N[(-16645562866606.395 * N[Power[x, 21.0], $MachinePrecision] + N[(N[Power[x, 23.0], $MachinePrecision] * 37304482788007.22 + N[(-64661103499212.51 * N[Power[x, 25.0], $MachinePrecision] + N[(N[Power[x, 27.0], $MachinePrecision] * 86399024333705.6 + N[(-88101468162941.17 * N[Power[x, 29.0], $MachinePrecision] + N[(N[Power[x, 31.0], $MachinePrecision] * 67260260640524.984 + N[(-37196962323926.695 * N[Power[x, 33.0], $MachinePrecision] + N[(N[Power[x, 35.0], $MachinePrecision] * 14066078189720.18 + N[(-3252516578403.765 * N[Power[x, 37.0], $MachinePrecision] + N[(N[Power[x, 39.0], $MachinePrecision] * 346759527252.22327), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(18.229321614212964 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\begin{array}{l}

\\
\sqrt{1 - x \cdot x} \cdot \mathsf{fma}\left(x \cdot x, x \cdot 4964.451919603997, \mathsf{fma}\left(-402120.6054879238, {x}^{5}, \mathsf{fma}\left({x}^{7}, 15299731.608802434, \mathsf{fma}\left(-333194155.0361419, {x}^{9}, \mathsf{fma}\left({x}^{11}, 4634427792.775428, \mathsf{fma}\left(-44086479772.29958, {x}^{13}, \mathsf{fma}\left({x}^{15}, 300207933687.56384, \mathsf{fma}\left(-1509869313546.277, {x}^{17}, \mathsf{fma}\left({x}^{19}, 5730439675389.087, \mathsf{fma}\left(-16645562866606.395, {x}^{21}, \mathsf{fma}\left({x}^{23}, 37304482788007.22, \mathsf{fma}\left(-64661103499212.51, {x}^{25}, \mathsf{fma}\left({x}^{27}, 86399024333705.6, \mathsf{fma}\left(-88101468162941.17, {x}^{29}, \mathsf{fma}\left({x}^{31}, 67260260640524.984, \mathsf{fma}\left(-37196962323926.695, {x}^{33}, \mathsf{fma}\left({x}^{35}, 14066078189720.18, \mathsf{fma}\left(-3252516578403.765, {x}^{37}, {x}^{39} \cdot 346759527252.22327\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) - 18.229321614212964 \cdot x\right)
\end{array}
Derivation

  1. Initial program 99.7%

    \[\sqrt{1 - {x}^{2}} \cdot \left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(346759527252.22327 \cdot {x}^{39} - 3252516578403.765 \cdot {x}^{37}\right) + 14066078189720.18 \cdot {x}^{35}\right) - 37196962323926.695 \cdot {x}^{33}\right) + 67260260640524.984 \cdot {x}^{31}\right) - 88101468162941.17 \cdot {x}^{29}\right) + 86399024333705.6 \cdot {x}^{27}\right) - 64661103499212.51 \cdot {x}^{25}\right) + 37304482788007.22 \cdot {x}^{23}\right) - 16645562866606.395 \cdot {x}^{21}\right) + 5730439675389.087 \cdot {x}^{19}\right) - 1509869313546.277 \cdot {x}^{17}\right) + 300207933687.56384 \cdot {x}^{15}\right) - 44086479772.29958 \cdot {x}^{13}\right) + 4634427792.775428 \cdot {x}^{11}\right) - 333194155.0361419 \cdot {x}^{9}\right) + 15299731.608802434 \cdot {x}^{7}\right) - 402120.6054879238 \cdot {x}^{5}\right) + 4964.451919603997 \cdot {x}^{3}\right) - 18.229321614212964 \cdot x\right) \]
  2. Add Preprocessing

  4. Applied rewrites99.7%

    \[\leadsto \sqrt{1 - {x}^{2}} \cdot \color{blue}{\mathsf{fma}\left(x \cdot x, x \cdot 4964.451919603997, \mathsf{fma}\left(-402120.6054879238, {x}^{5}, \mathsf{fma}\left({x}^{7}, 15299731.608802434, \mathsf{fma}\left(-333194155.0361419, {x}^{9}, \mathsf{fma}\left({x}^{11}, 4634427792.775428, \mathsf{fma}\left(-44086479772.29958, {x}^{13}, \mathsf{fma}\left({x}^{15}, 300207933687.56384, \mathsf{fma}\left(-1509869313546.277, {x}^{17}, \mathsf{fma}\left({x}^{19}, 5730439675389.087, \mathsf{fma}\left(-16645562866606.395, {x}^{21}, \mathsf{fma}\left({x}^{23}, 37304482788007.22, \mathsf{fma}\left(-64661103499212.51, {x}^{25}, \mathsf{fma}\left({x}^{27}, 86399024333705.6, \mathsf{fma}\left(-88101468162941.17, {x}^{29}, \mathsf{fma}\left({x}^{31}, 67260260640524.984, \mathsf{fma}\left(-37196962323926.695, {x}^{33}, \mathsf{fma}\left({x}^{35}, 14066078189720.18, \mathsf{fma}\left(-3252516578403.765, {x}^{37}, {x}^{39} \cdot 346759527252.22327\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) - 18.229321614212964 \cdot x\right)} \]
  5. Step-by-step derivation

    1. lift-pow.f64N/A

      \[\leadsto \sqrt{1 - \color{blue}{{x}^{2}}} \cdot \mathsf{fma}\left(x \cdot x, x \cdot \frac{5458472611139479}{1099511627776}, \mathsf{fma}\left(\frac{-6908379398473403}{17179869184}, {x}^{5}, \mathsf{fma}\left({x}^{7}, \frac{4106990431086495}{268435456}, \mathsf{fma}\left(\frac{-698758788622355}{2097152}, {x}^{9}, \mathsf{fma}\left({x}^{11}, \frac{4859549757237287}{1048576}, \mathsf{fma}\left(\frac{-5778503076714851}{131072}, {x}^{13}, \mathsf{fma}\left({x}^{15}, \frac{2459303392768523}{8192}, \mathsf{fma}\left(\frac{-6184424708285551}{4096}, {x}^{17}, \mathsf{fma}\left({x}^{19}, \frac{5867970227598425}{1024}, \mathsf{fma}\left(\frac{-4261264093851237}{256}, {x}^{21}, \mathsf{fma}\left({x}^{23}, \frac{1193743449216231}{32}, \mathsf{fma}\left(\frac{-8276621247899201}{128}, {x}^{25}, \mathsf{fma}\left({x}^{27}, \frac{2764768778678579}{32}, \mathsf{fma}\left(\frac{-5638493962428235}{64}, {x}^{29}, \mathsf{fma}\left({x}^{31}, \frac{4304656680993599}{64}, \mathsf{fma}\left(\frac{-4761211177462617}{128}, {x}^{33}, \mathsf{fma}\left({x}^{35}, \frac{1800458008284183}{128}, \mathsf{fma}\left(\frac{-6661153952570911}{2048}, {x}^{37}, {x}^{39} \cdot \frac{2840654047250213}{8192}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) - \frac{1282774469202913}{70368744177664} \cdot x\right) \]

    2. pow2N/A

      \[\leadsto \sqrt{1 - \color{blue}{x \cdot x}} \cdot \mathsf{fma}\left(x \cdot x, x \cdot \frac{5458472611139479}{1099511627776}, \mathsf{fma}\left(\frac{-6908379398473403}{17179869184}, {x}^{5}, \mathsf{fma}\left({x}^{7}, \frac{4106990431086495}{268435456}, \mathsf{fma}\left(\frac{-698758788622355}{2097152}, {x}^{9}, \mathsf{fma}\left({x}^{11}, \frac{4859549757237287}{1048576}, \mathsf{fma}\left(\frac{-5778503076714851}{131072}, {x}^{13}, \mathsf{fma}\left({x}^{15}, \frac{2459303392768523}{8192}, \mathsf{fma}\left(\frac{-6184424708285551}{4096}, {x}^{17}, \mathsf{fma}\left({x}^{19}, \frac{5867970227598425}{1024}, \mathsf{fma}\left(\frac{-4261264093851237}{256}, {x}^{21}, \mathsf{fma}\left({x}^{23}, \frac{1193743449216231}{32}, \mathsf{fma}\left(\frac{-8276621247899201}{128}, {x}^{25}, \mathsf{fma}\left({x}^{27}, \frac{2764768778678579}{32}, \mathsf{fma}\left(\frac{-5638493962428235}{64}, {x}^{29}, \mathsf{fma}\left({x}^{31}, \frac{4304656680993599}{64}, \mathsf{fma}\left(\frac{-4761211177462617}{128}, {x}^{33}, \mathsf{fma}\left({x}^{35}, \frac{1800458008284183}{128}, \mathsf{fma}\left(\frac{-6661153952570911}{2048}, {x}^{37}, {x}^{39} \cdot \frac{2840654047250213}{8192}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) - \frac{1282774469202913}{70368744177664} \cdot x\right) \]

    3. lift-*.f6499.7

      \[\leadsto \sqrt{1 - \color{blue}{x \cdot x}} \cdot \mathsf{fma}\left(x \cdot x, x \cdot 4964.451919603997, \mathsf{fma}\left(-402120.6054879238, {x}^{5}, \mathsf{fma}\left({x}^{7}, 15299731.608802434, \mathsf{fma}\left(-333194155.0361419, {x}^{9}, \mathsf{fma}\left({x}^{11}, 4634427792.775428, \mathsf{fma}\left(-44086479772.29958, {x}^{13}, \mathsf{fma}\left({x}^{15}, 300207933687.56384, \mathsf{fma}\left(-1509869313546.277, {x}^{17}, \mathsf{fma}\left({x}^{19}, 5730439675389.087, \mathsf{fma}\left(-16645562866606.395, {x}^{21}, \mathsf{fma}\left({x}^{23}, 37304482788007.22, \mathsf{fma}\left(-64661103499212.51, {x}^{25}, \mathsf{fma}\left({x}^{27}, 86399024333705.6, \mathsf{fma}\left(-88101468162941.17, {x}^{29}, \mathsf{fma}\left({x}^{31}, 67260260640524.984, \mathsf{fma}\left(-37196962323926.695, {x}^{33}, \mathsf{fma}\left({x}^{35}, 14066078189720.18, \mathsf{fma}\left(-3252516578403.765, {x}^{37}, {x}^{39} \cdot 346759527252.22327\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) - 18.229321614212964 \cdot x\right) \]

  6. Applied rewrites99.7%

    \[\leadsto \color{blue}{\sqrt{1 - x \cdot x}} \cdot \mathsf{fma}\left(x \cdot x, x \cdot 4964.451919603997, \mathsf{fma}\left(-402120.6054879238, {x}^{5}, \mathsf{fma}\left({x}^{7}, 15299731.608802434, \mathsf{fma}\left(-333194155.0361419, {x}^{9}, \mathsf{fma}\left({x}^{11}, 4634427792.775428, \mathsf{fma}\left(-44086479772.29958, {x}^{13}, \mathsf{fma}\left({x}^{15}, 300207933687.56384, \mathsf{fma}\left(-1509869313546.277, {x}^{17}, \mathsf{fma}\left({x}^{19}, 5730439675389.087, \mathsf{fma}\left(-16645562866606.395, {x}^{21}, \mathsf{fma}\left({x}^{23}, 37304482788007.22, \mathsf{fma}\left(-64661103499212.51, {x}^{25}, \mathsf{fma}\left({x}^{27}, 86399024333705.6, \mathsf{fma}\left(-88101468162941.17, {x}^{29}, \mathsf{fma}\left({x}^{31}, 67260260640524.984, \mathsf{fma}\left(-37196962323926.695, {x}^{33}, \mathsf{fma}\left({x}^{35}, 14066078189720.18, \mathsf{fma}\left(-3252516578403.765, {x}^{37}, {x}^{39} \cdot 346759527252.22327\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) - 18.229321614212964 \cdot x\right) \]
  7. Add Preprocessing

Alternative 4: 99.8% accurate, 1.3× speedup?

\[\begin{array}{l} \\ \sqrt{1 - {x}^{2}} \cdot \mathsf{fma}\left(x \cdot x, x \cdot 4964.451919603997, \mathsf{fma}\left(-402120.6054879238, {x}^{5}, \mathsf{fma}\left({x}^{7}, 15299731.608802434, \mathsf{fma}\left(-333194155.0361419, {x}^{9}, \mathsf{fma}\left({x}^{11}, 4634427792.775428, \mathsf{fma}\left(-44086479772.29958, {x}^{13}, \mathsf{fma}\left({x}^{15}, 300207933687.56384, \mathsf{fma}\left(-1509869313546.277, {x}^{17}, \mathsf{fma}\left({x}^{19}, 5730439675389.087, \mathsf{fma}\left(-16645562866606.395, {x}^{21}, \mathsf{fma}\left({x}^{23}, 37304482788007.22, \mathsf{fma}\left(-64661103499212.51, {x}^{25}, \mathsf{fma}\left({x}^{27}, 86399024333705.6, \mathsf{fma}\left(-88101468162941.17, {x}^{29}, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-3252516578403.765, x \cdot x, 14066078189720.18\right), x \cdot x, -37196962323926.695\right), x \cdot x, 67260260640524.984\right) \cdot {x}^{31}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) - 18.229321614212964 \cdot x\right) \end{array} \]
(FPCore (x)
 :precision binary64
 (*
  (sqrt (- 1.0 (pow x 2.0)))
  (fma
   (* x x)
   (* x 4964.451919603997)
   (-
    (fma
     -402120.6054879238
     (pow x 5.0)
     (fma
      (pow x 7.0)
      15299731.608802434
      (fma
       -333194155.0361419
       (pow x 9.0)
       (fma
        (pow x 11.0)
        4634427792.775428
        (fma
         -44086479772.29958
         (pow x 13.0)
         (fma
          (pow x 15.0)
          300207933687.56384
          (fma
           -1509869313546.277
           (pow x 17.0)
           (fma
            (pow x 19.0)
            5730439675389.087
            (fma
             -16645562866606.395
             (pow x 21.0)
             (fma
              (pow x 23.0)
              37304482788007.22
              (fma
               -64661103499212.51
               (pow x 25.0)
               (fma
                (pow x 27.0)
                86399024333705.6
                (fma
                 -88101468162941.17
                 (pow x 29.0)
                 (*
                  (fma
                   (fma
                    (fma -3252516578403.765 (* x x) 14066078189720.18)
                    (* x x)
                    -37196962323926.695)
                   (* x x)
                   67260260640524.984)
                  (pow x 31.0)))))))))))))))
    (* 18.229321614212964 x)))))
double code(double x) {
	return sqrt((1.0 - pow(x, 2.0))) * fma((x * x), (x * 4964.451919603997), (fma(-402120.6054879238, pow(x, 5.0), fma(pow(x, 7.0), 15299731.608802434, fma(-333194155.0361419, pow(x, 9.0), fma(pow(x, 11.0), 4634427792.775428, fma(-44086479772.29958, pow(x, 13.0), fma(pow(x, 15.0), 300207933687.56384, fma(-1509869313546.277, pow(x, 17.0), fma(pow(x, 19.0), 5730439675389.087, fma(-16645562866606.395, pow(x, 21.0), fma(pow(x, 23.0), 37304482788007.22, fma(-64661103499212.51, pow(x, 25.0), fma(pow(x, 27.0), 86399024333705.6, fma(-88101468162941.17, pow(x, 29.0), (fma(fma(fma(-3252516578403.765, (x * x), 14066078189720.18), (x * x), -37196962323926.695), (x * x), 67260260640524.984) * pow(x, 31.0))))))))))))))) - (18.229321614212964 * x)));
}

function code(x)
	return Float64(sqrt(Float64(1.0 - (x ^ 2.0))) * fma(Float64(x * x), Float64(x * 4964.451919603997), Float64(fma(-402120.6054879238, (x ^ 5.0), fma((x ^ 7.0), 15299731.608802434, fma(-333194155.0361419, (x ^ 9.0), fma((x ^ 11.0), 4634427792.775428, fma(-44086479772.29958, (x ^ 13.0), fma((x ^ 15.0), 300207933687.56384, fma(-1509869313546.277, (x ^ 17.0), fma((x ^ 19.0), 5730439675389.087, fma(-16645562866606.395, (x ^ 21.0), fma((x ^ 23.0), 37304482788007.22, fma(-64661103499212.51, (x ^ 25.0), fma((x ^ 27.0), 86399024333705.6, fma(-88101468162941.17, (x ^ 29.0), Float64(fma(fma(fma(-3252516578403.765, Float64(x * x), 14066078189720.18), Float64(x * x), -37196962323926.695), Float64(x * x), 67260260640524.984) * (x ^ 31.0))))))))))))))) - Float64(18.229321614212964 * x))))
end

code[x_] := N[(N[Sqrt[N[(1.0 - N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * 4964.451919603997), $MachinePrecision] + N[(N[(-402120.6054879238 * N[Power[x, 5.0], $MachinePrecision] + N[(N[Power[x, 7.0], $MachinePrecision] * 15299731.608802434 + N[(-333194155.0361419 * N[Power[x, 9.0], $MachinePrecision] + N[(N[Power[x, 11.0], $MachinePrecision] * 4634427792.775428 + N[(-44086479772.29958 * N[Power[x, 13.0], $MachinePrecision] + N[(N[Power[x, 15.0], $MachinePrecision] * 300207933687.56384 + N[(-1509869313546.277 * N[Power[x, 17.0], $MachinePrecision] + N[(N[Power[x, 19.0], $MachinePrecision] * 5730439675389.087 + N[(-16645562866606.395 * N[Power[x, 21.0], $MachinePrecision] + N[(N[Power[x, 23.0], $MachinePrecision] * 37304482788007.22 + N[(-64661103499212.51 * N[Power[x, 25.0], $MachinePrecision] + N[(N[Power[x, 27.0], $MachinePrecision] * 86399024333705.6 + N[(-88101468162941.17 * N[Power[x, 29.0], $MachinePrecision] + N[(N[(N[(N[(-3252516578403.765 * N[(x * x), $MachinePrecision] + 14066078189720.18), $MachinePrecision] * N[(x * x), $MachinePrecision] + -37196962323926.695), $MachinePrecision] * N[(x * x), $MachinePrecision] + 67260260640524.984), $MachinePrecision] * N[Power[x, 31.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(18.229321614212964 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\begin{array}{l}

\\
\sqrt{1 - {x}^{2}} \cdot \mathsf{fma}\left(x \cdot x, x \cdot 4964.451919603997, \mathsf{fma}\left(-402120.6054879238, {x}^{5}, \mathsf{fma}\left({x}^{7}, 15299731.608802434, \mathsf{fma}\left(-333194155.0361419, {x}^{9}, \mathsf{fma}\left({x}^{11}, 4634427792.775428, \mathsf{fma}\left(-44086479772.29958, {x}^{13}, \mathsf{fma}\left({x}^{15}, 300207933687.56384, \mathsf{fma}\left(-1509869313546.277, {x}^{17}, \mathsf{fma}\left({x}^{19}, 5730439675389.087, \mathsf{fma}\left(-16645562866606.395, {x}^{21}, \mathsf{fma}\left({x}^{23}, 37304482788007.22, \mathsf{fma}\left(-64661103499212.51, {x}^{25}, \mathsf{fma}\left({x}^{27}, 86399024333705.6, \mathsf{fma}\left(-88101468162941.17, {x}^{29}, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-3252516578403.765, x \cdot x, 14066078189720.18\right), x \cdot x, -37196962323926.695\right), x \cdot x, 67260260640524.984\right) \cdot {x}^{31}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) - 18.229321614212964 \cdot x\right)
\end{array}
Derivation

  1. Initial program 99.7%

    \[\sqrt{1 - {x}^{2}} \cdot \left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(346759527252.22327 \cdot {x}^{39} - 3252516578403.765 \cdot {x}^{37}\right) + 14066078189720.18 \cdot {x}^{35}\right) - 37196962323926.695 \cdot {x}^{33}\right) + 67260260640524.984 \cdot {x}^{31}\right) - 88101468162941.17 \cdot {x}^{29}\right) + 86399024333705.6 \cdot {x}^{27}\right) - 64661103499212.51 \cdot {x}^{25}\right) + 37304482788007.22 \cdot {x}^{23}\right) - 16645562866606.395 \cdot {x}^{21}\right) + 5730439675389.087 \cdot {x}^{19}\right) - 1509869313546.277 \cdot {x}^{17}\right) + 300207933687.56384 \cdot {x}^{15}\right) - 44086479772.29958 \cdot {x}^{13}\right) + 4634427792.775428 \cdot {x}^{11}\right) - 333194155.0361419 \cdot {x}^{9}\right) + 15299731.608802434 \cdot {x}^{7}\right) - 402120.6054879238 \cdot {x}^{5}\right) + 4964.451919603997 \cdot {x}^{3}\right) - 18.229321614212964 \cdot x\right) \]
  2. Add Preprocessing

  4. Applied rewrites99.7%

    \[\leadsto \sqrt{1 - {x}^{2}} \cdot \color{blue}{\mathsf{fma}\left(x \cdot x, x \cdot 4964.451919603997, \mathsf{fma}\left(-402120.6054879238, {x}^{5}, \mathsf{fma}\left({x}^{7}, 15299731.608802434, \mathsf{fma}\left(-333194155.0361419, {x}^{9}, \mathsf{fma}\left({x}^{11}, 4634427792.775428, \mathsf{fma}\left(-44086479772.29958, {x}^{13}, \mathsf{fma}\left({x}^{15}, 300207933687.56384, \mathsf{fma}\left(-1509869313546.277, {x}^{17}, \mathsf{fma}\left({x}^{19}, 5730439675389.087, \mathsf{fma}\left(-16645562866606.395, {x}^{21}, \mathsf{fma}\left({x}^{23}, 37304482788007.22, \mathsf{fma}\left(-64661103499212.51, {x}^{25}, \mathsf{fma}\left({x}^{27}, 86399024333705.6, \mathsf{fma}\left(-88101468162941.17, {x}^{29}, \mathsf{fma}\left({x}^{31}, 67260260640524.984, \mathsf{fma}\left(-37196962323926.695, {x}^{33}, \mathsf{fma}\left({x}^{35}, 14066078189720.18, \mathsf{fma}\left(-3252516578403.765, {x}^{37}, {x}^{39} \cdot 346759527252.22327\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) - 18.229321614212964 \cdot x\right)} \]

  5. Taylor expanded in x around 0

    \[\leadsto \sqrt{1 - {x}^{2}} \cdot \mathsf{fma}\left(x \cdot x, x \cdot \frac{5458472611139479}{1099511627776}, \mathsf{fma}\left(\frac{-6908379398473403}{17179869184}, {x}^{5}, \mathsf{fma}\left({x}^{7}, \frac{4106990431086495}{268435456}, \mathsf{fma}\left(\frac{-698758788622355}{2097152}, {x}^{9}, \mathsf{fma}\left({x}^{11}, \frac{4859549757237287}{1048576}, \mathsf{fma}\left(\frac{-5778503076714851}{131072}, {x}^{13}, \mathsf{fma}\left({x}^{15}, \frac{2459303392768523}{8192}, \mathsf{fma}\left(\frac{-6184424708285551}{4096}, {x}^{17}, \mathsf{fma}\left({x}^{19}, \frac{5867970227598425}{1024}, \mathsf{fma}\left(\frac{-4261264093851237}{256}, {x}^{21}, \mathsf{fma}\left({x}^{23}, \frac{1193743449216231}{32}, \mathsf{fma}\left(\frac{-8276621247899201}{128}, {x}^{25}, \mathsf{fma}\left({x}^{27}, \frac{2764768778678579}{32}, \mathsf{fma}\left(\frac{-5638493962428235}{64}, {x}^{29}, \color{blue}{{x}^{31} \cdot \left(\frac{4304656680993599}{64} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1800458008284183}{128} + \frac{-6661153952570911}{2048} \cdot {x}^{2}\right) - \frac{4761211177462617}{128}\right)\right)}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) - \frac{1282774469202913}{70368744177664} \cdot x\right) \]
  6. Step-by-step derivation

    1. *-commutativeN/A

      \[\leadsto \sqrt{1 - {x}^{2}} \cdot \mathsf{fma}\left(x \cdot x, x \cdot \frac{5458472611139479}{1099511627776}, \mathsf{fma}\left(\frac{-6908379398473403}{17179869184}, {x}^{5}, \mathsf{fma}\left({x}^{7}, \frac{4106990431086495}{268435456}, \mathsf{fma}\left(\frac{-698758788622355}{2097152}, {x}^{9}, \mathsf{fma}\left({x}^{11}, \frac{4859549757237287}{1048576}, \mathsf{fma}\left(\frac{-5778503076714851}{131072}, {x}^{13}, \mathsf{fma}\left({x}^{15}, \frac{2459303392768523}{8192}, \mathsf{fma}\left(\frac{-6184424708285551}{4096}, {x}^{17}, \mathsf{fma}\left({x}^{19}, \frac{5867970227598425}{1024}, \mathsf{fma}\left(\frac{-4261264093851237}{256}, {x}^{21}, \mathsf{fma}\left({x}^{23}, \frac{1193743449216231}{32}, \mathsf{fma}\left(\frac{-8276621247899201}{128}, {x}^{25}, \mathsf{fma}\left({x}^{27}, \frac{2764768778678579}{32}, \mathsf{fma}\left(\frac{-5638493962428235}{64}, {x}^{29}, \color{blue}{\left(\frac{4304656680993599}{64} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1800458008284183}{128} + \frac{-6661153952570911}{2048} \cdot {x}^{2}\right) - \frac{4761211177462617}{128}\right)\right) \cdot {x}^{31}}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) - \frac{1282774469202913}{70368744177664} \cdot x\right) \]

    2. lower-*.f64N/A

      \[\leadsto \sqrt{1 - {x}^{2}} \cdot \mathsf{fma}\left(x \cdot x, x \cdot \frac{5458472611139479}{1099511627776}, \mathsf{fma}\left(\frac{-6908379398473403}{17179869184}, {x}^{5}, \mathsf{fma}\left({x}^{7}, \frac{4106990431086495}{268435456}, \mathsf{fma}\left(\frac{-698758788622355}{2097152}, {x}^{9}, \mathsf{fma}\left({x}^{11}, \frac{4859549757237287}{1048576}, \mathsf{fma}\left(\frac{-5778503076714851}{131072}, {x}^{13}, \mathsf{fma}\left({x}^{15}, \frac{2459303392768523}{8192}, \mathsf{fma}\left(\frac{-6184424708285551}{4096}, {x}^{17}, \mathsf{fma}\left({x}^{19}, \frac{5867970227598425}{1024}, \mathsf{fma}\left(\frac{-4261264093851237}{256}, {x}^{21}, \mathsf{fma}\left({x}^{23}, \frac{1193743449216231}{32}, \mathsf{fma}\left(\frac{-8276621247899201}{128}, {x}^{25}, \mathsf{fma}\left({x}^{27}, \frac{2764768778678579}{32}, \mathsf{fma}\left(\frac{-5638493962428235}{64}, {x}^{29}, \color{blue}{\left(\frac{4304656680993599}{64} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1800458008284183}{128} + \frac{-6661153952570911}{2048} \cdot {x}^{2}\right) - \frac{4761211177462617}{128}\right)\right) \cdot {x}^{31}}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) - \frac{1282774469202913}{70368744177664} \cdot x\right) \]

    3. +-commutativeN/A

      \[\leadsto \sqrt{1 - {x}^{2}} \cdot \mathsf{fma}\left(x \cdot x, x \cdot \frac{5458472611139479}{1099511627776}, \mathsf{fma}\left(\frac{-6908379398473403}{17179869184}, {x}^{5}, \mathsf{fma}\left({x}^{7}, \frac{4106990431086495}{268435456}, \mathsf{fma}\left(\frac{-698758788622355}{2097152}, {x}^{9}, \mathsf{fma}\left({x}^{11}, \frac{4859549757237287}{1048576}, \mathsf{fma}\left(\frac{-5778503076714851}{131072}, {x}^{13}, \mathsf{fma}\left({x}^{15}, \frac{2459303392768523}{8192}, \mathsf{fma}\left(\frac{-6184424708285551}{4096}, {x}^{17}, \mathsf{fma}\left({x}^{19}, \frac{5867970227598425}{1024}, \mathsf{fma}\left(\frac{-4261264093851237}{256}, {x}^{21}, \mathsf{fma}\left({x}^{23}, \frac{1193743449216231}{32}, \mathsf{fma}\left(\frac{-8276621247899201}{128}, {x}^{25}, \mathsf{fma}\left({x}^{27}, \frac{2764768778678579}{32}, \mathsf{fma}\left(\frac{-5638493962428235}{64}, {x}^{29}, \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1800458008284183}{128} + \frac{-6661153952570911}{2048} \cdot {x}^{2}\right) - \frac{4761211177462617}{128}\right) + \frac{4304656680993599}{64}\right)} \cdot {x}^{31}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) - \frac{1282774469202913}{70368744177664} \cdot x\right) \]

    4. *-commutativeN/A

      \[\leadsto \sqrt{1 - {x}^{2}} \cdot \mathsf{fma}\left(x \cdot x, x \cdot \frac{5458472611139479}{1099511627776}, \mathsf{fma}\left(\frac{-6908379398473403}{17179869184}, {x}^{5}, \mathsf{fma}\left({x}^{7}, \frac{4106990431086495}{268435456}, \mathsf{fma}\left(\frac{-698758788622355}{2097152}, {x}^{9}, \mathsf{fma}\left({x}^{11}, \frac{4859549757237287}{1048576}, \mathsf{fma}\left(\frac{-5778503076714851}{131072}, {x}^{13}, \mathsf{fma}\left({x}^{15}, \frac{2459303392768523}{8192}, \mathsf{fma}\left(\frac{-6184424708285551}{4096}, {x}^{17}, \mathsf{fma}\left({x}^{19}, \frac{5867970227598425}{1024}, \mathsf{fma}\left(\frac{-4261264093851237}{256}, {x}^{21}, \mathsf{fma}\left({x}^{23}, \frac{1193743449216231}{32}, \mathsf{fma}\left(\frac{-8276621247899201}{128}, {x}^{25}, \mathsf{fma}\left({x}^{27}, \frac{2764768778678579}{32}, \mathsf{fma}\left(\frac{-5638493962428235}{64}, {x}^{29}, \left(\color{blue}{\left({x}^{2} \cdot \left(\frac{1800458008284183}{128} + \frac{-6661153952570911}{2048} \cdot {x}^{2}\right) - \frac{4761211177462617}{128}\right) \cdot {x}^{2}} + \frac{4304656680993599}{64}\right) \cdot {x}^{31}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) - \frac{1282774469202913}{70368744177664} \cdot x\right) \]

    5. lower-fma.f64N/A

      \[\leadsto \sqrt{1 - {x}^{2}} \cdot \mathsf{fma}\left(x \cdot x, x \cdot \frac{5458472611139479}{1099511627776}, \mathsf{fma}\left(\frac{-6908379398473403}{17179869184}, {x}^{5}, \mathsf{fma}\left({x}^{7}, \frac{4106990431086495}{268435456}, \mathsf{fma}\left(\frac{-698758788622355}{2097152}, {x}^{9}, \mathsf{fma}\left({x}^{11}, \frac{4859549757237287}{1048576}, \mathsf{fma}\left(\frac{-5778503076714851}{131072}, {x}^{13}, \mathsf{fma}\left({x}^{15}, \frac{2459303392768523}{8192}, \mathsf{fma}\left(\frac{-6184424708285551}{4096}, {x}^{17}, \mathsf{fma}\left({x}^{19}, \frac{5867970227598425}{1024}, \mathsf{fma}\left(\frac{-4261264093851237}{256}, {x}^{21}, \mathsf{fma}\left({x}^{23}, \frac{1193743449216231}{32}, \mathsf{fma}\left(\frac{-8276621247899201}{128}, {x}^{25}, \mathsf{fma}\left({x}^{27}, \frac{2764768778678579}{32}, \mathsf{fma}\left(\frac{-5638493962428235}{64}, {x}^{29}, \color{blue}{\mathsf{fma}\left({x}^{2} \cdot \left(\frac{1800458008284183}{128} + \frac{-6661153952570911}{2048} \cdot {x}^{2}\right) - \frac{4761211177462617}{128}, {x}^{2}, \frac{4304656680993599}{64}\right)} \cdot {x}^{31}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) - \frac{1282774469202913}{70368744177664} \cdot x\right) \]

    6. sub-negN/A

      \[\leadsto \sqrt{1 - {x}^{2}} \cdot \mathsf{fma}\left(x \cdot x, x \cdot \frac{5458472611139479}{1099511627776}, \mathsf{fma}\left(\frac{-6908379398473403}{17179869184}, {x}^{5}, \mathsf{fma}\left({x}^{7}, \frac{4106990431086495}{268435456}, \mathsf{fma}\left(\frac{-698758788622355}{2097152}, {x}^{9}, \mathsf{fma}\left({x}^{11}, \frac{4859549757237287}{1048576}, \mathsf{fma}\left(\frac{-5778503076714851}{131072}, {x}^{13}, \mathsf{fma}\left({x}^{15}, \frac{2459303392768523}{8192}, \mathsf{fma}\left(\frac{-6184424708285551}{4096}, {x}^{17}, \mathsf{fma}\left({x}^{19}, \frac{5867970227598425}{1024}, \mathsf{fma}\left(\frac{-4261264093851237}{256}, {x}^{21}, \mathsf{fma}\left({x}^{23}, \frac{1193743449216231}{32}, \mathsf{fma}\left(\frac{-8276621247899201}{128}, {x}^{25}, \mathsf{fma}\left({x}^{27}, \frac{2764768778678579}{32}, \mathsf{fma}\left(\frac{-5638493962428235}{64}, {x}^{29}, \mathsf{fma}\left(\color{blue}{{x}^{2} \cdot \left(\frac{1800458008284183}{128} + \frac{-6661153952570911}{2048} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{4761211177462617}{128}\right)\right)}, {x}^{2}, \frac{4304656680993599}{64}\right) \cdot {x}^{31}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) - \frac{1282774469202913}{70368744177664} \cdot x\right) \]

    7. *-commutativeN/A

      \[\leadsto \sqrt{1 - {x}^{2}} \cdot \mathsf{fma}\left(x \cdot x, x \cdot \frac{5458472611139479}{1099511627776}, \mathsf{fma}\left(\frac{-6908379398473403}{17179869184}, {x}^{5}, \mathsf{fma}\left({x}^{7}, \frac{4106990431086495}{268435456}, \mathsf{fma}\left(\frac{-698758788622355}{2097152}, {x}^{9}, \mathsf{fma}\left({x}^{11}, \frac{4859549757237287}{1048576}, \mathsf{fma}\left(\frac{-5778503076714851}{131072}, {x}^{13}, \mathsf{fma}\left({x}^{15}, \frac{2459303392768523}{8192}, \mathsf{fma}\left(\frac{-6184424708285551}{4096}, {x}^{17}, \mathsf{fma}\left({x}^{19}, \frac{5867970227598425}{1024}, \mathsf{fma}\left(\frac{-4261264093851237}{256}, {x}^{21}, \mathsf{fma}\left({x}^{23}, \frac{1193743449216231}{32}, \mathsf{fma}\left(\frac{-8276621247899201}{128}, {x}^{25}, \mathsf{fma}\left({x}^{27}, \frac{2764768778678579}{32}, \mathsf{fma}\left(\frac{-5638493962428235}{64}, {x}^{29}, \mathsf{fma}\left(\color{blue}{\left(\frac{1800458008284183}{128} + \frac{-6661153952570911}{2048} \cdot {x}^{2}\right) \cdot {x}^{2}} + \left(\mathsf{neg}\left(\frac{4761211177462617}{128}\right)\right), {x}^{2}, \frac{4304656680993599}{64}\right) \cdot {x}^{31}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) - \frac{1282774469202913}{70368744177664} \cdot x\right) \]

    8. metadata-evalN/A

      \[\leadsto \sqrt{1 - {x}^{2}} \cdot \mathsf{fma}\left(x \cdot x, x \cdot \frac{5458472611139479}{1099511627776}, \mathsf{fma}\left(\frac{-6908379398473403}{17179869184}, {x}^{5}, \mathsf{fma}\left({x}^{7}, \frac{4106990431086495}{268435456}, \mathsf{fma}\left(\frac{-698758788622355}{2097152}, {x}^{9}, \mathsf{fma}\left({x}^{11}, \frac{4859549757237287}{1048576}, \mathsf{fma}\left(\frac{-5778503076714851}{131072}, {x}^{13}, \mathsf{fma}\left({x}^{15}, \frac{2459303392768523}{8192}, \mathsf{fma}\left(\frac{-6184424708285551}{4096}, {x}^{17}, \mathsf{fma}\left({x}^{19}, \frac{5867970227598425}{1024}, \mathsf{fma}\left(\frac{-4261264093851237}{256}, {x}^{21}, \mathsf{fma}\left({x}^{23}, \frac{1193743449216231}{32}, \mathsf{fma}\left(\frac{-8276621247899201}{128}, {x}^{25}, \mathsf{fma}\left({x}^{27}, \frac{2764768778678579}{32}, \mathsf{fma}\left(\frac{-5638493962428235}{64}, {x}^{29}, \mathsf{fma}\left(\left(\frac{1800458008284183}{128} + \frac{-6661153952570911}{2048} \cdot {x}^{2}\right) \cdot {x}^{2} + \color{blue}{\frac{-4761211177462617}{128}}, {x}^{2}, \frac{4304656680993599}{64}\right) \cdot {x}^{31}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) - \frac{1282774469202913}{70368744177664} \cdot x\right) \]

    9. lower-fma.f64N/A

      \[\leadsto \sqrt{1 - {x}^{2}} \cdot \mathsf{fma}\left(x \cdot x, x \cdot \frac{5458472611139479}{1099511627776}, \mathsf{fma}\left(\frac{-6908379398473403}{17179869184}, {x}^{5}, \mathsf{fma}\left({x}^{7}, \frac{4106990431086495}{268435456}, \mathsf{fma}\left(\frac{-698758788622355}{2097152}, {x}^{9}, \mathsf{fma}\left({x}^{11}, \frac{4859549757237287}{1048576}, \mathsf{fma}\left(\frac{-5778503076714851}{131072}, {x}^{13}, \mathsf{fma}\left({x}^{15}, \frac{2459303392768523}{8192}, \mathsf{fma}\left(\frac{-6184424708285551}{4096}, {x}^{17}, \mathsf{fma}\left({x}^{19}, \frac{5867970227598425}{1024}, \mathsf{fma}\left(\frac{-4261264093851237}{256}, {x}^{21}, \mathsf{fma}\left({x}^{23}, \frac{1193743449216231}{32}, \mathsf{fma}\left(\frac{-8276621247899201}{128}, {x}^{25}, \mathsf{fma}\left({x}^{27}, \frac{2764768778678579}{32}, \mathsf{fma}\left(\frac{-5638493962428235}{64}, {x}^{29}, \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1800458008284183}{128} + \frac{-6661153952570911}{2048} \cdot {x}^{2}, {x}^{2}, \frac{-4761211177462617}{128}\right)}, {x}^{2}, \frac{4304656680993599}{64}\right) \cdot {x}^{31}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) - \frac{1282774469202913}{70368744177664} \cdot x\right) \]

    10. +-commutativeN/A

      \[\leadsto \sqrt{1 - {x}^{2}} \cdot \mathsf{fma}\left(x \cdot x, x \cdot \frac{5458472611139479}{1099511627776}, \mathsf{fma}\left(\frac{-6908379398473403}{17179869184}, {x}^{5}, \mathsf{fma}\left({x}^{7}, \frac{4106990431086495}{268435456}, \mathsf{fma}\left(\frac{-698758788622355}{2097152}, {x}^{9}, \mathsf{fma}\left({x}^{11}, \frac{4859549757237287}{1048576}, \mathsf{fma}\left(\frac{-5778503076714851}{131072}, {x}^{13}, \mathsf{fma}\left({x}^{15}, \frac{2459303392768523}{8192}, \mathsf{fma}\left(\frac{-6184424708285551}{4096}, {x}^{17}, \mathsf{fma}\left({x}^{19}, \frac{5867970227598425}{1024}, \mathsf{fma}\left(\frac{-4261264093851237}{256}, {x}^{21}, \mathsf{fma}\left({x}^{23}, \frac{1193743449216231}{32}, \mathsf{fma}\left(\frac{-8276621247899201}{128}, {x}^{25}, \mathsf{fma}\left({x}^{27}, \frac{2764768778678579}{32}, \mathsf{fma}\left(\frac{-5638493962428235}{64}, {x}^{29}, \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{-6661153952570911}{2048} \cdot {x}^{2} + \frac{1800458008284183}{128}}, {x}^{2}, \frac{-4761211177462617}{128}\right), {x}^{2}, \frac{4304656680993599}{64}\right) \cdot {x}^{31}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) - \frac{1282774469202913}{70368744177664} \cdot x\right) \]

    11. lower-fma.f64N/A

      \[\leadsto \sqrt{1 - {x}^{2}} \cdot \mathsf{fma}\left(x \cdot x, x \cdot \frac{5458472611139479}{1099511627776}, \mathsf{fma}\left(\frac{-6908379398473403}{17179869184}, {x}^{5}, \mathsf{fma}\left({x}^{7}, \frac{4106990431086495}{268435456}, \mathsf{fma}\left(\frac{-698758788622355}{2097152}, {x}^{9}, \mathsf{fma}\left({x}^{11}, \frac{4859549757237287}{1048576}, \mathsf{fma}\left(\frac{-5778503076714851}{131072}, {x}^{13}, \mathsf{fma}\left({x}^{15}, \frac{2459303392768523}{8192}, \mathsf{fma}\left(\frac{-6184424708285551}{4096}, {x}^{17}, \mathsf{fma}\left({x}^{19}, \frac{5867970227598425}{1024}, \mathsf{fma}\left(\frac{-4261264093851237}{256}, {x}^{21}, \mathsf{fma}\left({x}^{23}, \frac{1193743449216231}{32}, \mathsf{fma}\left(\frac{-8276621247899201}{128}, {x}^{25}, \mathsf{fma}\left({x}^{27}, \frac{2764768778678579}{32}, \mathsf{fma}\left(\frac{-5638493962428235}{64}, {x}^{29}, \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-6661153952570911}{2048}, {x}^{2}, \frac{1800458008284183}{128}\right)}, {x}^{2}, \frac{-4761211177462617}{128}\right), {x}^{2}, \frac{4304656680993599}{64}\right) \cdot {x}^{31}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) - \frac{1282774469202913}{70368744177664} \cdot x\right) \]

    12. unpow2N/A

      \[\leadsto \sqrt{1 - {x}^{2}} \cdot \mathsf{fma}\left(x \cdot x, x \cdot \frac{5458472611139479}{1099511627776}, \mathsf{fma}\left(\frac{-6908379398473403}{17179869184}, {x}^{5}, \mathsf{fma}\left({x}^{7}, \frac{4106990431086495}{268435456}, \mathsf{fma}\left(\frac{-698758788622355}{2097152}, {x}^{9}, \mathsf{fma}\left({x}^{11}, \frac{4859549757237287}{1048576}, \mathsf{fma}\left(\frac{-5778503076714851}{131072}, {x}^{13}, \mathsf{fma}\left({x}^{15}, \frac{2459303392768523}{8192}, \mathsf{fma}\left(\frac{-6184424708285551}{4096}, {x}^{17}, \mathsf{fma}\left({x}^{19}, \frac{5867970227598425}{1024}, \mathsf{fma}\left(\frac{-4261264093851237}{256}, {x}^{21}, \mathsf{fma}\left({x}^{23}, \frac{1193743449216231}{32}, \mathsf{fma}\left(\frac{-8276621247899201}{128}, {x}^{25}, \mathsf{fma}\left({x}^{27}, \frac{2764768778678579}{32}, \mathsf{fma}\left(\frac{-5638493962428235}{64}, {x}^{29}, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-6661153952570911}{2048}, \color{blue}{x \cdot x}, \frac{1800458008284183}{128}\right), {x}^{2}, \frac{-4761211177462617}{128}\right), {x}^{2}, \frac{4304656680993599}{64}\right) \cdot {x}^{31}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) - \frac{1282774469202913}{70368744177664} \cdot x\right) \]

    13. lower-*.f64N/A

      \[\leadsto \sqrt{1 - {x}^{2}} \cdot \mathsf{fma}\left(x \cdot x, x \cdot \frac{5458472611139479}{1099511627776}, \mathsf{fma}\left(\frac{-6908379398473403}{17179869184}, {x}^{5}, \mathsf{fma}\left({x}^{7}, \frac{4106990431086495}{268435456}, \mathsf{fma}\left(\frac{-698758788622355}{2097152}, {x}^{9}, \mathsf{fma}\left({x}^{11}, \frac{4859549757237287}{1048576}, \mathsf{fma}\left(\frac{-5778503076714851}{131072}, {x}^{13}, \mathsf{fma}\left({x}^{15}, \frac{2459303392768523}{8192}, \mathsf{fma}\left(\frac{-6184424708285551}{4096}, {x}^{17}, \mathsf{fma}\left({x}^{19}, \frac{5867970227598425}{1024}, \mathsf{fma}\left(\frac{-4261264093851237}{256}, {x}^{21}, \mathsf{fma}\left({x}^{23}, \frac{1193743449216231}{32}, \mathsf{fma}\left(\frac{-8276621247899201}{128}, {x}^{25}, \mathsf{fma}\left({x}^{27}, \frac{2764768778678579}{32}, \mathsf{fma}\left(\frac{-5638493962428235}{64}, {x}^{29}, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-6661153952570911}{2048}, \color{blue}{x \cdot x}, \frac{1800458008284183}{128}\right), {x}^{2}, \frac{-4761211177462617}{128}\right), {x}^{2}, \frac{4304656680993599}{64}\right) \cdot {x}^{31}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) - \frac{1282774469202913}{70368744177664} \cdot x\right) \]

    14. unpow2N/A

      \[\leadsto \sqrt{1 - {x}^{2}} \cdot \mathsf{fma}\left(x \cdot x, x \cdot \frac{5458472611139479}{1099511627776}, \mathsf{fma}\left(\frac{-6908379398473403}{17179869184}, {x}^{5}, \mathsf{fma}\left({x}^{7}, \frac{4106990431086495}{268435456}, \mathsf{fma}\left(\frac{-698758788622355}{2097152}, {x}^{9}, \mathsf{fma}\left({x}^{11}, \frac{4859549757237287}{1048576}, \mathsf{fma}\left(\frac{-5778503076714851}{131072}, {x}^{13}, \mathsf{fma}\left({x}^{15}, \frac{2459303392768523}{8192}, \mathsf{fma}\left(\frac{-6184424708285551}{4096}, {x}^{17}, \mathsf{fma}\left({x}^{19}, \frac{5867970227598425}{1024}, \mathsf{fma}\left(\frac{-4261264093851237}{256}, {x}^{21}, \mathsf{fma}\left({x}^{23}, \frac{1193743449216231}{32}, \mathsf{fma}\left(\frac{-8276621247899201}{128}, {x}^{25}, \mathsf{fma}\left({x}^{27}, \frac{2764768778678579}{32}, \mathsf{fma}\left(\frac{-5638493962428235}{64}, {x}^{29}, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-6661153952570911}{2048}, x \cdot x, \frac{1800458008284183}{128}\right), \color{blue}{x \cdot x}, \frac{-4761211177462617}{128}\right), {x}^{2}, \frac{4304656680993599}{64}\right) \cdot {x}^{31}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) - \frac{1282774469202913}{70368744177664} \cdot x\right) \]

    15. lower-*.f64N/A

      \[\leadsto \sqrt{1 - {x}^{2}} \cdot \mathsf{fma}\left(x \cdot x, x \cdot \frac{5458472611139479}{1099511627776}, \mathsf{fma}\left(\frac{-6908379398473403}{17179869184}, {x}^{5}, \mathsf{fma}\left({x}^{7}, \frac{4106990431086495}{268435456}, \mathsf{fma}\left(\frac{-698758788622355}{2097152}, {x}^{9}, \mathsf{fma}\left({x}^{11}, \frac{4859549757237287}{1048576}, \mathsf{fma}\left(\frac{-5778503076714851}{131072}, {x}^{13}, \mathsf{fma}\left({x}^{15}, \frac{2459303392768523}{8192}, \mathsf{fma}\left(\frac{-6184424708285551}{4096}, {x}^{17}, \mathsf{fma}\left({x}^{19}, \frac{5867970227598425}{1024}, \mathsf{fma}\left(\frac{-4261264093851237}{256}, {x}^{21}, \mathsf{fma}\left({x}^{23}, \frac{1193743449216231}{32}, \mathsf{fma}\left(\frac{-8276621247899201}{128}, {x}^{25}, \mathsf{fma}\left({x}^{27}, \frac{2764768778678579}{32}, \mathsf{fma}\left(\frac{-5638493962428235}{64}, {x}^{29}, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-6661153952570911}{2048}, x \cdot x, \frac{1800458008284183}{128}\right), \color{blue}{x \cdot x}, \frac{-4761211177462617}{128}\right), {x}^{2}, \frac{4304656680993599}{64}\right) \cdot {x}^{31}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) - \frac{1282774469202913}{70368744177664} \cdot x\right) \]

    16. unpow2N/A

      \[\leadsto \sqrt{1 - {x}^{2}} \cdot \mathsf{fma}\left(x \cdot x, x \cdot \frac{5458472611139479}{1099511627776}, \mathsf{fma}\left(\frac{-6908379398473403}{17179869184}, {x}^{5}, \mathsf{fma}\left({x}^{7}, \frac{4106990431086495}{268435456}, \mathsf{fma}\left(\frac{-698758788622355}{2097152}, {x}^{9}, \mathsf{fma}\left({x}^{11}, \frac{4859549757237287}{1048576}, \mathsf{fma}\left(\frac{-5778503076714851}{131072}, {x}^{13}, \mathsf{fma}\left({x}^{15}, \frac{2459303392768523}{8192}, \mathsf{fma}\left(\frac{-6184424708285551}{4096}, {x}^{17}, \mathsf{fma}\left({x}^{19}, \frac{5867970227598425}{1024}, \mathsf{fma}\left(\frac{-4261264093851237}{256}, {x}^{21}, \mathsf{fma}\left({x}^{23}, \frac{1193743449216231}{32}, \mathsf{fma}\left(\frac{-8276621247899201}{128}, {x}^{25}, \mathsf{fma}\left({x}^{27}, \frac{2764768778678579}{32}, \mathsf{fma}\left(\frac{-5638493962428235}{64}, {x}^{29}, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-6661153952570911}{2048}, x \cdot x, \frac{1800458008284183}{128}\right), x \cdot x, \frac{-4761211177462617}{128}\right), \color{blue}{x \cdot x}, \frac{4304656680993599}{64}\right) \cdot {x}^{31}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) - \frac{1282774469202913}{70368744177664} \cdot x\right) \]

    17. lower-*.f64N/A

      \[\leadsto \sqrt{1 - {x}^{2}} \cdot \mathsf{fma}\left(x \cdot x, x \cdot \frac{5458472611139479}{1099511627776}, \mathsf{fma}\left(\frac{-6908379398473403}{17179869184}, {x}^{5}, \mathsf{fma}\left({x}^{7}, \frac{4106990431086495}{268435456}, \mathsf{fma}\left(\frac{-698758788622355}{2097152}, {x}^{9}, \mathsf{fma}\left({x}^{11}, \frac{4859549757237287}{1048576}, \mathsf{fma}\left(\frac{-5778503076714851}{131072}, {x}^{13}, \mathsf{fma}\left({x}^{15}, \frac{2459303392768523}{8192}, \mathsf{fma}\left(\frac{-6184424708285551}{4096}, {x}^{17}, \mathsf{fma}\left({x}^{19}, \frac{5867970227598425}{1024}, \mathsf{fma}\left(\frac{-4261264093851237}{256}, {x}^{21}, \mathsf{fma}\left({x}^{23}, \frac{1193743449216231}{32}, \mathsf{fma}\left(\frac{-8276621247899201}{128}, {x}^{25}, \mathsf{fma}\left({x}^{27}, \frac{2764768778678579}{32}, \mathsf{fma}\left(\frac{-5638493962428235}{64}, {x}^{29}, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-6661153952570911}{2048}, x \cdot x, \frac{1800458008284183}{128}\right), x \cdot x, \frac{-4761211177462617}{128}\right), \color{blue}{x \cdot x}, \frac{4304656680993599}{64}\right) \cdot {x}^{31}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) - \frac{1282774469202913}{70368744177664} \cdot x\right) \]

    18. lower-pow.f6499.4

      \[\leadsto \sqrt{1 - {x}^{2}} \cdot \mathsf{fma}\left(x \cdot x, x \cdot 4964.451919603997, \mathsf{fma}\left(-402120.6054879238, {x}^{5}, \mathsf{fma}\left({x}^{7}, 15299731.608802434, \mathsf{fma}\left(-333194155.0361419, {x}^{9}, \mathsf{fma}\left({x}^{11}, 4634427792.775428, \mathsf{fma}\left(-44086479772.29958, {x}^{13}, \mathsf{fma}\left({x}^{15}, 300207933687.56384, \mathsf{fma}\left(-1509869313546.277, {x}^{17}, \mathsf{fma}\left({x}^{19}, 5730439675389.087, \mathsf{fma}\left(-16645562866606.395, {x}^{21}, \mathsf{fma}\left({x}^{23}, 37304482788007.22, \mathsf{fma}\left(-64661103499212.51, {x}^{25}, \mathsf{fma}\left({x}^{27}, 86399024333705.6, \mathsf{fma}\left(-88101468162941.17, {x}^{29}, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-3252516578403.765, x \cdot x, 14066078189720.18\right), x \cdot x, -37196962323926.695\right), x \cdot x, 67260260640524.984\right) \cdot \color{blue}{{x}^{31}}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) - 18.229321614212964 \cdot x\right) \]

  7. Applied rewrites99.4%

    \[\leadsto \sqrt{1 - {x}^{2}} \cdot \mathsf{fma}\left(x \cdot x, x \cdot 4964.451919603997, \mathsf{fma}\left(-402120.6054879238, {x}^{5}, \mathsf{fma}\left({x}^{7}, 15299731.608802434, \mathsf{fma}\left(-333194155.0361419, {x}^{9}, \mathsf{fma}\left({x}^{11}, 4634427792.775428, \mathsf{fma}\left(-44086479772.29958, {x}^{13}, \mathsf{fma}\left({x}^{15}, 300207933687.56384, \mathsf{fma}\left(-1509869313546.277, {x}^{17}, \mathsf{fma}\left({x}^{19}, 5730439675389.087, \mathsf{fma}\left(-16645562866606.395, {x}^{21}, \mathsf{fma}\left({x}^{23}, 37304482788007.22, \mathsf{fma}\left(-64661103499212.51, {x}^{25}, \mathsf{fma}\left({x}^{27}, 86399024333705.6, \mathsf{fma}\left(-88101468162941.17, {x}^{29}, \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-3252516578403.765, x \cdot x, 14066078189720.18\right), x \cdot x, -37196962323926.695\right), x \cdot x, 67260260640524.984\right) \cdot {x}^{31}}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) - 18.229321614212964 \cdot x\right) \]
  8. Add Preprocessing

Alternative 5: 99.8% accurate, 1.4× speedup?

\[\begin{array}{l} \\ \sqrt{1 - {x}^{2}} \cdot \mathsf{fma}\left(x \cdot x, x \cdot 4964.451919603997, \mathsf{fma}\left(-402120.6054879238, {x}^{5}, \mathsf{fma}\left({x}^{7}, 15299731.608802434, \mathsf{fma}\left(-333194155.0361419, {x}^{9}, \mathsf{fma}\left({x}^{11}, 4634427792.775428, \mathsf{fma}\left(-44086479772.29958, {x}^{13}, \mathsf{fma}\left({x}^{15}, 300207933687.56384, \mathsf{fma}\left(-1509869313546.277, {x}^{17}, \mathsf{fma}\left({x}^{19}, 5730439675389.087, \mathsf{fma}\left(-16645562866606.395, {x}^{21}, \mathsf{fma}\left({x}^{23}, 37304482788007.22, \mathsf{fma}\left(-64661103499212.51, {x}^{25}, \mathsf{fma}\left({x}^{27}, 86399024333705.6, \mathsf{fma}\left(\mathsf{fma}\left(-37196962323926.695, x \cdot x, 67260260640524.984\right), x \cdot x, -88101468162941.17\right) \cdot {x}^{29}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) - 18.229321614212964 \cdot x\right) \end{array} \]
(FPCore (x)
 :precision binary64
 (*
  (sqrt (- 1.0 (pow x 2.0)))
  (fma
   (* x x)
   (* x 4964.451919603997)
   (-
    (fma
     -402120.6054879238
     (pow x 5.0)
     (fma
      (pow x 7.0)
      15299731.608802434
      (fma
       -333194155.0361419
       (pow x 9.0)
       (fma
        (pow x 11.0)
        4634427792.775428
        (fma
         -44086479772.29958
         (pow x 13.0)
         (fma
          (pow x 15.0)
          300207933687.56384
          (fma
           -1509869313546.277
           (pow x 17.0)
           (fma
            (pow x 19.0)
            5730439675389.087
            (fma
             -16645562866606.395
             (pow x 21.0)
             (fma
              (pow x 23.0)
              37304482788007.22
              (fma
               -64661103499212.51
               (pow x 25.0)
               (fma
                (pow x 27.0)
                86399024333705.6
                (*
                 (fma
                  (fma -37196962323926.695 (* x x) 67260260640524.984)
                  (* x x)
                  -88101468162941.17)
                 (pow x 29.0))))))))))))))
    (* 18.229321614212964 x)))))
double code(double x) {
	return sqrt((1.0 - pow(x, 2.0))) * fma((x * x), (x * 4964.451919603997), (fma(-402120.6054879238, pow(x, 5.0), fma(pow(x, 7.0), 15299731.608802434, fma(-333194155.0361419, pow(x, 9.0), fma(pow(x, 11.0), 4634427792.775428, fma(-44086479772.29958, pow(x, 13.0), fma(pow(x, 15.0), 300207933687.56384, fma(-1509869313546.277, pow(x, 17.0), fma(pow(x, 19.0), 5730439675389.087, fma(-16645562866606.395, pow(x, 21.0), fma(pow(x, 23.0), 37304482788007.22, fma(-64661103499212.51, pow(x, 25.0), fma(pow(x, 27.0), 86399024333705.6, (fma(fma(-37196962323926.695, (x * x), 67260260640524.984), (x * x), -88101468162941.17) * pow(x, 29.0)))))))))))))) - (18.229321614212964 * x)));
}

function code(x)
	return Float64(sqrt(Float64(1.0 - (x ^ 2.0))) * fma(Float64(x * x), Float64(x * 4964.451919603997), Float64(fma(-402120.6054879238, (x ^ 5.0), fma((x ^ 7.0), 15299731.608802434, fma(-333194155.0361419, (x ^ 9.0), fma((x ^ 11.0), 4634427792.775428, fma(-44086479772.29958, (x ^ 13.0), fma((x ^ 15.0), 300207933687.56384, fma(-1509869313546.277, (x ^ 17.0), fma((x ^ 19.0), 5730439675389.087, fma(-16645562866606.395, (x ^ 21.0), fma((x ^ 23.0), 37304482788007.22, fma(-64661103499212.51, (x ^ 25.0), fma((x ^ 27.0), 86399024333705.6, Float64(fma(fma(-37196962323926.695, Float64(x * x), 67260260640524.984), Float64(x * x), -88101468162941.17) * (x ^ 29.0)))))))))))))) - Float64(18.229321614212964 * x))))
end

code[x_] := N[(N[Sqrt[N[(1.0 - N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * 4964.451919603997), $MachinePrecision] + N[(N[(-402120.6054879238 * N[Power[x, 5.0], $MachinePrecision] + N[(N[Power[x, 7.0], $MachinePrecision] * 15299731.608802434 + N[(-333194155.0361419 * N[Power[x, 9.0], $MachinePrecision] + N[(N[Power[x, 11.0], $MachinePrecision] * 4634427792.775428 + N[(-44086479772.29958 * N[Power[x, 13.0], $MachinePrecision] + N[(N[Power[x, 15.0], $MachinePrecision] * 300207933687.56384 + N[(-1509869313546.277 * N[Power[x, 17.0], $MachinePrecision] + N[(N[Power[x, 19.0], $MachinePrecision] * 5730439675389.087 + N[(-16645562866606.395 * N[Power[x, 21.0], $MachinePrecision] + N[(N[Power[x, 23.0], $MachinePrecision] * 37304482788007.22 + N[(-64661103499212.51 * N[Power[x, 25.0], $MachinePrecision] + N[(N[Power[x, 27.0], $MachinePrecision] * 86399024333705.6 + N[(N[(N[(-37196962323926.695 * N[(x * x), $MachinePrecision] + 67260260640524.984), $MachinePrecision] * N[(x * x), $MachinePrecision] + -88101468162941.17), $MachinePrecision] * N[Power[x, 29.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(18.229321614212964 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\begin{array}{l}

\\
\sqrt{1 - {x}^{2}} \cdot \mathsf{fma}\left(x \cdot x, x \cdot 4964.451919603997, \mathsf{fma}\left(-402120.6054879238, {x}^{5}, \mathsf{fma}\left({x}^{7}, 15299731.608802434, \mathsf{fma}\left(-333194155.0361419, {x}^{9}, \mathsf{fma}\left({x}^{11}, 4634427792.775428, \mathsf{fma}\left(-44086479772.29958, {x}^{13}, \mathsf{fma}\left({x}^{15}, 300207933687.56384, \mathsf{fma}\left(-1509869313546.277, {x}^{17}, \mathsf{fma}\left({x}^{19}, 5730439675389.087, \mathsf{fma}\left(-16645562866606.395, {x}^{21}, \mathsf{fma}\left({x}^{23}, 37304482788007.22, \mathsf{fma}\left(-64661103499212.51, {x}^{25}, \mathsf{fma}\left({x}^{27}, 86399024333705.6, \mathsf{fma}\left(\mathsf{fma}\left(-37196962323926.695, x \cdot x, 67260260640524.984\right), x \cdot x, -88101468162941.17\right) \cdot {x}^{29}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) - 18.229321614212964 \cdot x\right)
\end{array}
Derivation

  1. Initial program 99.7%

    \[\sqrt{1 - {x}^{2}} \cdot \left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(346759527252.22327 \cdot {x}^{39} - 3252516578403.765 \cdot {x}^{37}\right) + 14066078189720.18 \cdot {x}^{35}\right) - 37196962323926.695 \cdot {x}^{33}\right) + 67260260640524.984 \cdot {x}^{31}\right) - 88101468162941.17 \cdot {x}^{29}\right) + 86399024333705.6 \cdot {x}^{27}\right) - 64661103499212.51 \cdot {x}^{25}\right) + 37304482788007.22 \cdot {x}^{23}\right) - 16645562866606.395 \cdot {x}^{21}\right) + 5730439675389.087 \cdot {x}^{19}\right) - 1509869313546.277 \cdot {x}^{17}\right) + 300207933687.56384 \cdot {x}^{15}\right) - 44086479772.29958 \cdot {x}^{13}\right) + 4634427792.775428 \cdot {x}^{11}\right) - 333194155.0361419 \cdot {x}^{9}\right) + 15299731.608802434 \cdot {x}^{7}\right) - 402120.6054879238 \cdot {x}^{5}\right) + 4964.451919603997 \cdot {x}^{3}\right) - 18.229321614212964 \cdot x\right) \]
  2. Add Preprocessing

  4. Applied rewrites99.7%

    \[\leadsto \sqrt{1 - {x}^{2}} \cdot \color{blue}{\mathsf{fma}\left(x \cdot x, x \cdot 4964.451919603997, \mathsf{fma}\left(-402120.6054879238, {x}^{5}, \mathsf{fma}\left({x}^{7}, 15299731.608802434, \mathsf{fma}\left(-333194155.0361419, {x}^{9}, \mathsf{fma}\left({x}^{11}, 4634427792.775428, \mathsf{fma}\left(-44086479772.29958, {x}^{13}, \mathsf{fma}\left({x}^{15}, 300207933687.56384, \mathsf{fma}\left(-1509869313546.277, {x}^{17}, \mathsf{fma}\left({x}^{19}, 5730439675389.087, \mathsf{fma}\left(-16645562866606.395, {x}^{21}, \mathsf{fma}\left({x}^{23}, 37304482788007.22, \mathsf{fma}\left(-64661103499212.51, {x}^{25}, \mathsf{fma}\left({x}^{27}, 86399024333705.6, \mathsf{fma}\left(-88101468162941.17, {x}^{29}, \mathsf{fma}\left({x}^{31}, 67260260640524.984, \mathsf{fma}\left(-37196962323926.695, {x}^{33}, \mathsf{fma}\left({x}^{35}, 14066078189720.18, \mathsf{fma}\left(-3252516578403.765, {x}^{37}, {x}^{39} \cdot 346759527252.22327\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) - 18.229321614212964 \cdot x\right)} \]

  5. Taylor expanded in x around 0

    \[\leadsto \sqrt{1 - {x}^{2}} \cdot \mathsf{fma}\left(x \cdot x, x \cdot \frac{5458472611139479}{1099511627776}, \mathsf{fma}\left(\frac{-6908379398473403}{17179869184}, {x}^{5}, \mathsf{fma}\left({x}^{7}, \frac{4106990431086495}{268435456}, \mathsf{fma}\left(\frac{-698758788622355}{2097152}, {x}^{9}, \mathsf{fma}\left({x}^{11}, \frac{4859549757237287}{1048576}, \mathsf{fma}\left(\frac{-5778503076714851}{131072}, {x}^{13}, \mathsf{fma}\left({x}^{15}, \frac{2459303392768523}{8192}, \mathsf{fma}\left(\frac{-6184424708285551}{4096}, {x}^{17}, \mathsf{fma}\left({x}^{19}, \frac{5867970227598425}{1024}, \mathsf{fma}\left(\frac{-4261264093851237}{256}, {x}^{21}, \mathsf{fma}\left({x}^{23}, \frac{1193743449216231}{32}, \mathsf{fma}\left(\frac{-8276621247899201}{128}, {x}^{25}, \mathsf{fma}\left({x}^{27}, \frac{2764768778678579}{32}, \color{blue}{{x}^{29} \cdot \left({x}^{2} \cdot \left(\frac{4304656680993599}{64} + \frac{-4761211177462617}{128} \cdot {x}^{2}\right) - \frac{5638493962428235}{64}\right)}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) - \frac{1282774469202913}{70368744177664} \cdot x\right) \]
  6. Step-by-step derivation

    1. *-commutativeN/A

      \[\leadsto \sqrt{1 - {x}^{2}} \cdot \mathsf{fma}\left(x \cdot x, x \cdot \frac{5458472611139479}{1099511627776}, \mathsf{fma}\left(\frac{-6908379398473403}{17179869184}, {x}^{5}, \mathsf{fma}\left({x}^{7}, \frac{4106990431086495}{268435456}, \mathsf{fma}\left(\frac{-698758788622355}{2097152}, {x}^{9}, \mathsf{fma}\left({x}^{11}, \frac{4859549757237287}{1048576}, \mathsf{fma}\left(\frac{-5778503076714851}{131072}, {x}^{13}, \mathsf{fma}\left({x}^{15}, \frac{2459303392768523}{8192}, \mathsf{fma}\left(\frac{-6184424708285551}{4096}, {x}^{17}, \mathsf{fma}\left({x}^{19}, \frac{5867970227598425}{1024}, \mathsf{fma}\left(\frac{-4261264093851237}{256}, {x}^{21}, \mathsf{fma}\left({x}^{23}, \frac{1193743449216231}{32}, \mathsf{fma}\left(\frac{-8276621247899201}{128}, {x}^{25}, \mathsf{fma}\left({x}^{27}, \frac{2764768778678579}{32}, \color{blue}{\left({x}^{2} \cdot \left(\frac{4304656680993599}{64} + \frac{-4761211177462617}{128} \cdot {x}^{2}\right) - \frac{5638493962428235}{64}\right) \cdot {x}^{29}}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) - \frac{1282774469202913}{70368744177664} \cdot x\right) \]

    2. lower-*.f64N/A

      \[\leadsto \sqrt{1 - {x}^{2}} \cdot \mathsf{fma}\left(x \cdot x, x \cdot \frac{5458472611139479}{1099511627776}, \mathsf{fma}\left(\frac{-6908379398473403}{17179869184}, {x}^{5}, \mathsf{fma}\left({x}^{7}, \frac{4106990431086495}{268435456}, \mathsf{fma}\left(\frac{-698758788622355}{2097152}, {x}^{9}, \mathsf{fma}\left({x}^{11}, \frac{4859549757237287}{1048576}, \mathsf{fma}\left(\frac{-5778503076714851}{131072}, {x}^{13}, \mathsf{fma}\left({x}^{15}, \frac{2459303392768523}{8192}, \mathsf{fma}\left(\frac{-6184424708285551}{4096}, {x}^{17}, \mathsf{fma}\left({x}^{19}, \frac{5867970227598425}{1024}, \mathsf{fma}\left(\frac{-4261264093851237}{256}, {x}^{21}, \mathsf{fma}\left({x}^{23}, \frac{1193743449216231}{32}, \mathsf{fma}\left(\frac{-8276621247899201}{128}, {x}^{25}, \mathsf{fma}\left({x}^{27}, \frac{2764768778678579}{32}, \color{blue}{\left({x}^{2} \cdot \left(\frac{4304656680993599}{64} + \frac{-4761211177462617}{128} \cdot {x}^{2}\right) - \frac{5638493962428235}{64}\right) \cdot {x}^{29}}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) - \frac{1282774469202913}{70368744177664} \cdot x\right) \]

    3. sub-negN/A

      \[\leadsto \sqrt{1 - {x}^{2}} \cdot \mathsf{fma}\left(x \cdot x, x \cdot \frac{5458472611139479}{1099511627776}, \mathsf{fma}\left(\frac{-6908379398473403}{17179869184}, {x}^{5}, \mathsf{fma}\left({x}^{7}, \frac{4106990431086495}{268435456}, \mathsf{fma}\left(\frac{-698758788622355}{2097152}, {x}^{9}, \mathsf{fma}\left({x}^{11}, \frac{4859549757237287}{1048576}, \mathsf{fma}\left(\frac{-5778503076714851}{131072}, {x}^{13}, \mathsf{fma}\left({x}^{15}, \frac{2459303392768523}{8192}, \mathsf{fma}\left(\frac{-6184424708285551}{4096}, {x}^{17}, \mathsf{fma}\left({x}^{19}, \frac{5867970227598425}{1024}, \mathsf{fma}\left(\frac{-4261264093851237}{256}, {x}^{21}, \mathsf{fma}\left({x}^{23}, \frac{1193743449216231}{32}, \mathsf{fma}\left(\frac{-8276621247899201}{128}, {x}^{25}, \mathsf{fma}\left({x}^{27}, \frac{2764768778678579}{32}, \color{blue}{\left({x}^{2} \cdot \left(\frac{4304656680993599}{64} + \frac{-4761211177462617}{128} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{5638493962428235}{64}\right)\right)\right)} \cdot {x}^{29}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) - \frac{1282774469202913}{70368744177664} \cdot x\right) \]

    4. *-commutativeN/A

      \[\leadsto \sqrt{1 - {x}^{2}} \cdot \mathsf{fma}\left(x \cdot x, x \cdot \frac{5458472611139479}{1099511627776}, \mathsf{fma}\left(\frac{-6908379398473403}{17179869184}, {x}^{5}, \mathsf{fma}\left({x}^{7}, \frac{4106990431086495}{268435456}, \mathsf{fma}\left(\frac{-698758788622355}{2097152}, {x}^{9}, \mathsf{fma}\left({x}^{11}, \frac{4859549757237287}{1048576}, \mathsf{fma}\left(\frac{-5778503076714851}{131072}, {x}^{13}, \mathsf{fma}\left({x}^{15}, \frac{2459303392768523}{8192}, \mathsf{fma}\left(\frac{-6184424708285551}{4096}, {x}^{17}, \mathsf{fma}\left({x}^{19}, \frac{5867970227598425}{1024}, \mathsf{fma}\left(\frac{-4261264093851237}{256}, {x}^{21}, \mathsf{fma}\left({x}^{23}, \frac{1193743449216231}{32}, \mathsf{fma}\left(\frac{-8276621247899201}{128}, {x}^{25}, \mathsf{fma}\left({x}^{27}, \frac{2764768778678579}{32}, \left(\color{blue}{\left(\frac{4304656680993599}{64} + \frac{-4761211177462617}{128} \cdot {x}^{2}\right) \cdot {x}^{2}} + \left(\mathsf{neg}\left(\frac{5638493962428235}{64}\right)\right)\right) \cdot {x}^{29}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) - \frac{1282774469202913}{70368744177664} \cdot x\right) \]

    5. metadata-evalN/A

      \[\leadsto \sqrt{1 - {x}^{2}} \cdot \mathsf{fma}\left(x \cdot x, x \cdot \frac{5458472611139479}{1099511627776}, \mathsf{fma}\left(\frac{-6908379398473403}{17179869184}, {x}^{5}, \mathsf{fma}\left({x}^{7}, \frac{4106990431086495}{268435456}, \mathsf{fma}\left(\frac{-698758788622355}{2097152}, {x}^{9}, \mathsf{fma}\left({x}^{11}, \frac{4859549757237287}{1048576}, \mathsf{fma}\left(\frac{-5778503076714851}{131072}, {x}^{13}, \mathsf{fma}\left({x}^{15}, \frac{2459303392768523}{8192}, \mathsf{fma}\left(\frac{-6184424708285551}{4096}, {x}^{17}, \mathsf{fma}\left({x}^{19}, \frac{5867970227598425}{1024}, \mathsf{fma}\left(\frac{-4261264093851237}{256}, {x}^{21}, \mathsf{fma}\left({x}^{23}, \frac{1193743449216231}{32}, \mathsf{fma}\left(\frac{-8276621247899201}{128}, {x}^{25}, \mathsf{fma}\left({x}^{27}, \frac{2764768778678579}{32}, \left(\left(\frac{4304656680993599}{64} + \frac{-4761211177462617}{128} \cdot {x}^{2}\right) \cdot {x}^{2} + \color{blue}{\frac{-5638493962428235}{64}}\right) \cdot {x}^{29}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) - \frac{1282774469202913}{70368744177664} \cdot x\right) \]

    6. lower-fma.f64N/A

      \[\leadsto \sqrt{1 - {x}^{2}} \cdot \mathsf{fma}\left(x \cdot x, x \cdot \frac{5458472611139479}{1099511627776}, \mathsf{fma}\left(\frac{-6908379398473403}{17179869184}, {x}^{5}, \mathsf{fma}\left({x}^{7}, \frac{4106990431086495}{268435456}, \mathsf{fma}\left(\frac{-698758788622355}{2097152}, {x}^{9}, \mathsf{fma}\left({x}^{11}, \frac{4859549757237287}{1048576}, \mathsf{fma}\left(\frac{-5778503076714851}{131072}, {x}^{13}, \mathsf{fma}\left({x}^{15}, \frac{2459303392768523}{8192}, \mathsf{fma}\left(\frac{-6184424708285551}{4096}, {x}^{17}, \mathsf{fma}\left({x}^{19}, \frac{5867970227598425}{1024}, \mathsf{fma}\left(\frac{-4261264093851237}{256}, {x}^{21}, \mathsf{fma}\left({x}^{23}, \frac{1193743449216231}{32}, \mathsf{fma}\left(\frac{-8276621247899201}{128}, {x}^{25}, \mathsf{fma}\left({x}^{27}, \frac{2764768778678579}{32}, \color{blue}{\mathsf{fma}\left(\frac{4304656680993599}{64} + \frac{-4761211177462617}{128} \cdot {x}^{2}, {x}^{2}, \frac{-5638493962428235}{64}\right)} \cdot {x}^{29}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) - \frac{1282774469202913}{70368744177664} \cdot x\right) \]

    7. +-commutativeN/A

      \[\leadsto \sqrt{1 - {x}^{2}} \cdot \mathsf{fma}\left(x \cdot x, x \cdot \frac{5458472611139479}{1099511627776}, \mathsf{fma}\left(\frac{-6908379398473403}{17179869184}, {x}^{5}, \mathsf{fma}\left({x}^{7}, \frac{4106990431086495}{268435456}, \mathsf{fma}\left(\frac{-698758788622355}{2097152}, {x}^{9}, \mathsf{fma}\left({x}^{11}, \frac{4859549757237287}{1048576}, \mathsf{fma}\left(\frac{-5778503076714851}{131072}, {x}^{13}, \mathsf{fma}\left({x}^{15}, \frac{2459303392768523}{8192}, \mathsf{fma}\left(\frac{-6184424708285551}{4096}, {x}^{17}, \mathsf{fma}\left({x}^{19}, \frac{5867970227598425}{1024}, \mathsf{fma}\left(\frac{-4261264093851237}{256}, {x}^{21}, \mathsf{fma}\left({x}^{23}, \frac{1193743449216231}{32}, \mathsf{fma}\left(\frac{-8276621247899201}{128}, {x}^{25}, \mathsf{fma}\left({x}^{27}, \frac{2764768778678579}{32}, \mathsf{fma}\left(\color{blue}{\frac{-4761211177462617}{128} \cdot {x}^{2} + \frac{4304656680993599}{64}}, {x}^{2}, \frac{-5638493962428235}{64}\right) \cdot {x}^{29}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) - \frac{1282774469202913}{70368744177664} \cdot x\right) \]

    8. lower-fma.f64N/A

      \[\leadsto \sqrt{1 - {x}^{2}} \cdot \mathsf{fma}\left(x \cdot x, x \cdot \frac{5458472611139479}{1099511627776}, \mathsf{fma}\left(\frac{-6908379398473403}{17179869184}, {x}^{5}, \mathsf{fma}\left({x}^{7}, \frac{4106990431086495}{268435456}, \mathsf{fma}\left(\frac{-698758788622355}{2097152}, {x}^{9}, \mathsf{fma}\left({x}^{11}, \frac{4859549757237287}{1048576}, \mathsf{fma}\left(\frac{-5778503076714851}{131072}, {x}^{13}, \mathsf{fma}\left({x}^{15}, \frac{2459303392768523}{8192}, \mathsf{fma}\left(\frac{-6184424708285551}{4096}, {x}^{17}, \mathsf{fma}\left({x}^{19}, \frac{5867970227598425}{1024}, \mathsf{fma}\left(\frac{-4261264093851237}{256}, {x}^{21}, \mathsf{fma}\left({x}^{23}, \frac{1193743449216231}{32}, \mathsf{fma}\left(\frac{-8276621247899201}{128}, {x}^{25}, \mathsf{fma}\left({x}^{27}, \frac{2764768778678579}{32}, \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-4761211177462617}{128}, {x}^{2}, \frac{4304656680993599}{64}\right)}, {x}^{2}, \frac{-5638493962428235}{64}\right) \cdot {x}^{29}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) - \frac{1282774469202913}{70368744177664} \cdot x\right) \]

    9. unpow2N/A

      \[\leadsto \sqrt{1 - {x}^{2}} \cdot \mathsf{fma}\left(x \cdot x, x \cdot \frac{5458472611139479}{1099511627776}, \mathsf{fma}\left(\frac{-6908379398473403}{17179869184}, {x}^{5}, \mathsf{fma}\left({x}^{7}, \frac{4106990431086495}{268435456}, \mathsf{fma}\left(\frac{-698758788622355}{2097152}, {x}^{9}, \mathsf{fma}\left({x}^{11}, \frac{4859549757237287}{1048576}, \mathsf{fma}\left(\frac{-5778503076714851}{131072}, {x}^{13}, \mathsf{fma}\left({x}^{15}, \frac{2459303392768523}{8192}, \mathsf{fma}\left(\frac{-6184424708285551}{4096}, {x}^{17}, \mathsf{fma}\left({x}^{19}, \frac{5867970227598425}{1024}, \mathsf{fma}\left(\frac{-4261264093851237}{256}, {x}^{21}, \mathsf{fma}\left({x}^{23}, \frac{1193743449216231}{32}, \mathsf{fma}\left(\frac{-8276621247899201}{128}, {x}^{25}, \mathsf{fma}\left({x}^{27}, \frac{2764768778678579}{32}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{-4761211177462617}{128}, \color{blue}{x \cdot x}, \frac{4304656680993599}{64}\right), {x}^{2}, \frac{-5638493962428235}{64}\right) \cdot {x}^{29}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) - \frac{1282774469202913}{70368744177664} \cdot x\right) \]

    10. lower-*.f64N/A

      \[\leadsto \sqrt{1 - {x}^{2}} \cdot \mathsf{fma}\left(x \cdot x, x \cdot \frac{5458472611139479}{1099511627776}, \mathsf{fma}\left(\frac{-6908379398473403}{17179869184}, {x}^{5}, \mathsf{fma}\left({x}^{7}, \frac{4106990431086495}{268435456}, \mathsf{fma}\left(\frac{-698758788622355}{2097152}, {x}^{9}, \mathsf{fma}\left({x}^{11}, \frac{4859549757237287}{1048576}, \mathsf{fma}\left(\frac{-5778503076714851}{131072}, {x}^{13}, \mathsf{fma}\left({x}^{15}, \frac{2459303392768523}{8192}, \mathsf{fma}\left(\frac{-6184424708285551}{4096}, {x}^{17}, \mathsf{fma}\left({x}^{19}, \frac{5867970227598425}{1024}, \mathsf{fma}\left(\frac{-4261264093851237}{256}, {x}^{21}, \mathsf{fma}\left({x}^{23}, \frac{1193743449216231}{32}, \mathsf{fma}\left(\frac{-8276621247899201}{128}, {x}^{25}, \mathsf{fma}\left({x}^{27}, \frac{2764768778678579}{32}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{-4761211177462617}{128}, \color{blue}{x \cdot x}, \frac{4304656680993599}{64}\right), {x}^{2}, \frac{-5638493962428235}{64}\right) \cdot {x}^{29}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) - \frac{1282774469202913}{70368744177664} \cdot x\right) \]

    11. unpow2N/A

      \[\leadsto \sqrt{1 - {x}^{2}} \cdot \mathsf{fma}\left(x \cdot x, x \cdot \frac{5458472611139479}{1099511627776}, \mathsf{fma}\left(\frac{-6908379398473403}{17179869184}, {x}^{5}, \mathsf{fma}\left({x}^{7}, \frac{4106990431086495}{268435456}, \mathsf{fma}\left(\frac{-698758788622355}{2097152}, {x}^{9}, \mathsf{fma}\left({x}^{11}, \frac{4859549757237287}{1048576}, \mathsf{fma}\left(\frac{-5778503076714851}{131072}, {x}^{13}, \mathsf{fma}\left({x}^{15}, \frac{2459303392768523}{8192}, \mathsf{fma}\left(\frac{-6184424708285551}{4096}, {x}^{17}, \mathsf{fma}\left({x}^{19}, \frac{5867970227598425}{1024}, \mathsf{fma}\left(\frac{-4261264093851237}{256}, {x}^{21}, \mathsf{fma}\left({x}^{23}, \frac{1193743449216231}{32}, \mathsf{fma}\left(\frac{-8276621247899201}{128}, {x}^{25}, \mathsf{fma}\left({x}^{27}, \frac{2764768778678579}{32}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{-4761211177462617}{128}, x \cdot x, \frac{4304656680993599}{64}\right), \color{blue}{x \cdot x}, \frac{-5638493962428235}{64}\right) \cdot {x}^{29}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) - \frac{1282774469202913}{70368744177664} \cdot x\right) \]

    12. lower-*.f64N/A

      \[\leadsto \sqrt{1 - {x}^{2}} \cdot \mathsf{fma}\left(x \cdot x, x \cdot \frac{5458472611139479}{1099511627776}, \mathsf{fma}\left(\frac{-6908379398473403}{17179869184}, {x}^{5}, \mathsf{fma}\left({x}^{7}, \frac{4106990431086495}{268435456}, \mathsf{fma}\left(\frac{-698758788622355}{2097152}, {x}^{9}, \mathsf{fma}\left({x}^{11}, \frac{4859549757237287}{1048576}, \mathsf{fma}\left(\frac{-5778503076714851}{131072}, {x}^{13}, \mathsf{fma}\left({x}^{15}, \frac{2459303392768523}{8192}, \mathsf{fma}\left(\frac{-6184424708285551}{4096}, {x}^{17}, \mathsf{fma}\left({x}^{19}, \frac{5867970227598425}{1024}, \mathsf{fma}\left(\frac{-4261264093851237}{256}, {x}^{21}, \mathsf{fma}\left({x}^{23}, \frac{1193743449216231}{32}, \mathsf{fma}\left(\frac{-8276621247899201}{128}, {x}^{25}, \mathsf{fma}\left({x}^{27}, \frac{2764768778678579}{32}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{-4761211177462617}{128}, x \cdot x, \frac{4304656680993599}{64}\right), \color{blue}{x \cdot x}, \frac{-5638493962428235}{64}\right) \cdot {x}^{29}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) - \frac{1282774469202913}{70368744177664} \cdot x\right) \]

    13. lower-pow.f6499.4

      \[\leadsto \sqrt{1 - {x}^{2}} \cdot \mathsf{fma}\left(x \cdot x, x \cdot 4964.451919603997, \mathsf{fma}\left(-402120.6054879238, {x}^{5}, \mathsf{fma}\left({x}^{7}, 15299731.608802434, \mathsf{fma}\left(-333194155.0361419, {x}^{9}, \mathsf{fma}\left({x}^{11}, 4634427792.775428, \mathsf{fma}\left(-44086479772.29958, {x}^{13}, \mathsf{fma}\left({x}^{15}, 300207933687.56384, \mathsf{fma}\left(-1509869313546.277, {x}^{17}, \mathsf{fma}\left({x}^{19}, 5730439675389.087, \mathsf{fma}\left(-16645562866606.395, {x}^{21}, \mathsf{fma}\left({x}^{23}, 37304482788007.22, \mathsf{fma}\left(-64661103499212.51, {x}^{25}, \mathsf{fma}\left({x}^{27}, 86399024333705.6, \mathsf{fma}\left(\mathsf{fma}\left(-37196962323926.695, x \cdot x, 67260260640524.984\right), x \cdot x, -88101468162941.17\right) \cdot \color{blue}{{x}^{29}}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) - 18.229321614212964 \cdot x\right) \]

  7. Applied rewrites99.4%

    \[\leadsto \sqrt{1 - {x}^{2}} \cdot \mathsf{fma}\left(x \cdot x, x \cdot 4964.451919603997, \mathsf{fma}\left(-402120.6054879238, {x}^{5}, \mathsf{fma}\left({x}^{7}, 15299731.608802434, \mathsf{fma}\left(-333194155.0361419, {x}^{9}, \mathsf{fma}\left({x}^{11}, 4634427792.775428, \mathsf{fma}\left(-44086479772.29958, {x}^{13}, \mathsf{fma}\left({x}^{15}, 300207933687.56384, \mathsf{fma}\left(-1509869313546.277, {x}^{17}, \mathsf{fma}\left({x}^{19}, 5730439675389.087, \mathsf{fma}\left(-16645562866606.395, {x}^{21}, \mathsf{fma}\left({x}^{23}, 37304482788007.22, \mathsf{fma}\left(-64661103499212.51, {x}^{25}, \mathsf{fma}\left({x}^{27}, 86399024333705.6, \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-37196962323926.695, x \cdot x, 67260260640524.984\right), x \cdot x, -88101468162941.17\right) \cdot {x}^{29}}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) - 18.229321614212964 \cdot x\right) \]
  8. Add Preprocessing

Alternative 6: 99.8% accurate, 1.4× speedup?

\[\begin{array}{l} \\ \sqrt{1 - {x}^{2}} \cdot \mathsf{fma}\left(x \cdot x, x \cdot 4964.451919603997, \mathsf{fma}\left(-402120.6054879238, {x}^{5}, \mathsf{fma}\left({x}^{7}, 15299731.608802434, \mathsf{fma}\left(-333194155.0361419, {x}^{9}, \mathsf{fma}\left({x}^{11}, 4634427792.775428, \mathsf{fma}\left(-44086479772.29958, {x}^{13}, \mathsf{fma}\left({x}^{15}, 300207933687.56384, \mathsf{fma}\left(-1509869313546.277, {x}^{17}, \mathsf{fma}\left({x}^{19}, 5730439675389.087, \mathsf{fma}\left(-16645562866606.395, {x}^{21}, \mathsf{fma}\left({x}^{23}, 37304482788007.22, \mathsf{fma}\left(-64661103499212.51, {x}^{25}, \mathsf{fma}\left({x}^{27}, 86399024333705.6, -88101468162941.17 \cdot {x}^{29}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) - 18.229321614212964 \cdot x\right) \end{array} \]
(FPCore (x)
 :precision binary64
 (*
  (sqrt (- 1.0 (pow x 2.0)))
  (fma
   (* x x)
   (* x 4964.451919603997)
   (-
    (fma
     -402120.6054879238
     (pow x 5.0)
     (fma
      (pow x 7.0)
      15299731.608802434
      (fma
       -333194155.0361419
       (pow x 9.0)
       (fma
        (pow x 11.0)
        4634427792.775428
        (fma
         -44086479772.29958
         (pow x 13.0)
         (fma
          (pow x 15.0)
          300207933687.56384
          (fma
           -1509869313546.277
           (pow x 17.0)
           (fma
            (pow x 19.0)
            5730439675389.087
            (fma
             -16645562866606.395
             (pow x 21.0)
             (fma
              (pow x 23.0)
              37304482788007.22
              (fma
               -64661103499212.51
               (pow x 25.0)
               (fma
                (pow x 27.0)
                86399024333705.6
                (* -88101468162941.17 (pow x 29.0))))))))))))))
    (* 18.229321614212964 x)))))
double code(double x) {
	return sqrt((1.0 - pow(x, 2.0))) * fma((x * x), (x * 4964.451919603997), (fma(-402120.6054879238, pow(x, 5.0), fma(pow(x, 7.0), 15299731.608802434, fma(-333194155.0361419, pow(x, 9.0), fma(pow(x, 11.0), 4634427792.775428, fma(-44086479772.29958, pow(x, 13.0), fma(pow(x, 15.0), 300207933687.56384, fma(-1509869313546.277, pow(x, 17.0), fma(pow(x, 19.0), 5730439675389.087, fma(-16645562866606.395, pow(x, 21.0), fma(pow(x, 23.0), 37304482788007.22, fma(-64661103499212.51, pow(x, 25.0), fma(pow(x, 27.0), 86399024333705.6, (-88101468162941.17 * pow(x, 29.0)))))))))))))) - (18.229321614212964 * x)));
}

function code(x)
	return Float64(sqrt(Float64(1.0 - (x ^ 2.0))) * fma(Float64(x * x), Float64(x * 4964.451919603997), Float64(fma(-402120.6054879238, (x ^ 5.0), fma((x ^ 7.0), 15299731.608802434, fma(-333194155.0361419, (x ^ 9.0), fma((x ^ 11.0), 4634427792.775428, fma(-44086479772.29958, (x ^ 13.0), fma((x ^ 15.0), 300207933687.56384, fma(-1509869313546.277, (x ^ 17.0), fma((x ^ 19.0), 5730439675389.087, fma(-16645562866606.395, (x ^ 21.0), fma((x ^ 23.0), 37304482788007.22, fma(-64661103499212.51, (x ^ 25.0), fma((x ^ 27.0), 86399024333705.6, Float64(-88101468162941.17 * (x ^ 29.0)))))))))))))) - Float64(18.229321614212964 * x))))
end

code[x_] := N[(N[Sqrt[N[(1.0 - N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * 4964.451919603997), $MachinePrecision] + N[(N[(-402120.6054879238 * N[Power[x, 5.0], $MachinePrecision] + N[(N[Power[x, 7.0], $MachinePrecision] * 15299731.608802434 + N[(-333194155.0361419 * N[Power[x, 9.0], $MachinePrecision] + N[(N[Power[x, 11.0], $MachinePrecision] * 4634427792.775428 + N[(-44086479772.29958 * N[Power[x, 13.0], $MachinePrecision] + N[(N[Power[x, 15.0], $MachinePrecision] * 300207933687.56384 + N[(-1509869313546.277 * N[Power[x, 17.0], $MachinePrecision] + N[(N[Power[x, 19.0], $MachinePrecision] * 5730439675389.087 + N[(-16645562866606.395 * N[Power[x, 21.0], $MachinePrecision] + N[(N[Power[x, 23.0], $MachinePrecision] * 37304482788007.22 + N[(-64661103499212.51 * N[Power[x, 25.0], $MachinePrecision] + N[(N[Power[x, 27.0], $MachinePrecision] * 86399024333705.6 + N[(-88101468162941.17 * N[Power[x, 29.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(18.229321614212964 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\begin{array}{l}

\\
\sqrt{1 - {x}^{2}} \cdot \mathsf{fma}\left(x \cdot x, x \cdot 4964.451919603997, \mathsf{fma}\left(-402120.6054879238, {x}^{5}, \mathsf{fma}\left({x}^{7}, 15299731.608802434, \mathsf{fma}\left(-333194155.0361419, {x}^{9}, \mathsf{fma}\left({x}^{11}, 4634427792.775428, \mathsf{fma}\left(-44086479772.29958, {x}^{13}, \mathsf{fma}\left({x}^{15}, 300207933687.56384, \mathsf{fma}\left(-1509869313546.277, {x}^{17}, \mathsf{fma}\left({x}^{19}, 5730439675389.087, \mathsf{fma}\left(-16645562866606.395, {x}^{21}, \mathsf{fma}\left({x}^{23}, 37304482788007.22, \mathsf{fma}\left(-64661103499212.51, {x}^{25}, \mathsf{fma}\left({x}^{27}, 86399024333705.6, -88101468162941.17 \cdot {x}^{29}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) - 18.229321614212964 \cdot x\right)
\end{array}
Derivation

  1. Initial program 99.7%

    \[\sqrt{1 - {x}^{2}} \cdot \left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(346759527252.22327 \cdot {x}^{39} - 3252516578403.765 \cdot {x}^{37}\right) + 14066078189720.18 \cdot {x}^{35}\right) - 37196962323926.695 \cdot {x}^{33}\right) + 67260260640524.984 \cdot {x}^{31}\right) - 88101468162941.17 \cdot {x}^{29}\right) + 86399024333705.6 \cdot {x}^{27}\right) - 64661103499212.51 \cdot {x}^{25}\right) + 37304482788007.22 \cdot {x}^{23}\right) - 16645562866606.395 \cdot {x}^{21}\right) + 5730439675389.087 \cdot {x}^{19}\right) - 1509869313546.277 \cdot {x}^{17}\right) + 300207933687.56384 \cdot {x}^{15}\right) - 44086479772.29958 \cdot {x}^{13}\right) + 4634427792.775428 \cdot {x}^{11}\right) - 333194155.0361419 \cdot {x}^{9}\right) + 15299731.608802434 \cdot {x}^{7}\right) - 402120.6054879238 \cdot {x}^{5}\right) + 4964.451919603997 \cdot {x}^{3}\right) - 18.229321614212964 \cdot x\right) \]
  2. Add Preprocessing

  4. Applied rewrites99.7%

    \[\leadsto \sqrt{1 - {x}^{2}} \cdot \color{blue}{\mathsf{fma}\left(x \cdot x, x \cdot 4964.451919603997, \mathsf{fma}\left(-402120.6054879238, {x}^{5}, \mathsf{fma}\left({x}^{7}, 15299731.608802434, \mathsf{fma}\left(-333194155.0361419, {x}^{9}, \mathsf{fma}\left({x}^{11}, 4634427792.775428, \mathsf{fma}\left(-44086479772.29958, {x}^{13}, \mathsf{fma}\left({x}^{15}, 300207933687.56384, \mathsf{fma}\left(-1509869313546.277, {x}^{17}, \mathsf{fma}\left({x}^{19}, 5730439675389.087, \mathsf{fma}\left(-16645562866606.395, {x}^{21}, \mathsf{fma}\left({x}^{23}, 37304482788007.22, \mathsf{fma}\left(-64661103499212.51, {x}^{25}, \mathsf{fma}\left({x}^{27}, 86399024333705.6, \mathsf{fma}\left(-88101468162941.17, {x}^{29}, \mathsf{fma}\left({x}^{31}, 67260260640524.984, \mathsf{fma}\left(-37196962323926.695, {x}^{33}, \mathsf{fma}\left({x}^{35}, 14066078189720.18, \mathsf{fma}\left(-3252516578403.765, {x}^{37}, {x}^{39} \cdot 346759527252.22327\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) - 18.229321614212964 \cdot x\right)} \]

  5. Taylor expanded in x around 0

    \[\leadsto \sqrt{1 - {x}^{2}} \cdot \mathsf{fma}\left(x \cdot x, x \cdot \frac{5458472611139479}{1099511627776}, \mathsf{fma}\left(\frac{-6908379398473403}{17179869184}, {x}^{5}, \mathsf{fma}\left({x}^{7}, \frac{4106990431086495}{268435456}, \mathsf{fma}\left(\frac{-698758788622355}{2097152}, {x}^{9}, \mathsf{fma}\left({x}^{11}, \frac{4859549757237287}{1048576}, \mathsf{fma}\left(\frac{-5778503076714851}{131072}, {x}^{13}, \mathsf{fma}\left({x}^{15}, \frac{2459303392768523}{8192}, \mathsf{fma}\left(\frac{-6184424708285551}{4096}, {x}^{17}, \mathsf{fma}\left({x}^{19}, \frac{5867970227598425}{1024}, \mathsf{fma}\left(\frac{-4261264093851237}{256}, {x}^{21}, \mathsf{fma}\left({x}^{23}, \frac{1193743449216231}{32}, \mathsf{fma}\left(\frac{-8276621247899201}{128}, {x}^{25}, \mathsf{fma}\left({x}^{27}, \frac{2764768778678579}{32}, \color{blue}{\frac{-5638493962428235}{64} \cdot {x}^{29}}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) - \frac{1282774469202913}{70368744177664} \cdot x\right) \]
  6. Step-by-step derivation

    1. lower-*.f64N/A

      \[\leadsto \sqrt{1 - {x}^{2}} \cdot \mathsf{fma}\left(x \cdot x, x \cdot \frac{5458472611139479}{1099511627776}, \mathsf{fma}\left(\frac{-6908379398473403}{17179869184}, {x}^{5}, \mathsf{fma}\left({x}^{7}, \frac{4106990431086495}{268435456}, \mathsf{fma}\left(\frac{-698758788622355}{2097152}, {x}^{9}, \mathsf{fma}\left({x}^{11}, \frac{4859549757237287}{1048576}, \mathsf{fma}\left(\frac{-5778503076714851}{131072}, {x}^{13}, \mathsf{fma}\left({x}^{15}, \frac{2459303392768523}{8192}, \mathsf{fma}\left(\frac{-6184424708285551}{4096}, {x}^{17}, \mathsf{fma}\left({x}^{19}, \frac{5867970227598425}{1024}, \mathsf{fma}\left(\frac{-4261264093851237}{256}, {x}^{21}, \mathsf{fma}\left({x}^{23}, \frac{1193743449216231}{32}, \mathsf{fma}\left(\frac{-8276621247899201}{128}, {x}^{25}, \mathsf{fma}\left({x}^{27}, \frac{2764768778678579}{32}, \color{blue}{\frac{-5638493962428235}{64} \cdot {x}^{29}}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) - \frac{1282774469202913}{70368744177664} \cdot x\right) \]

    2. lower-pow.f6499.3

      \[\leadsto \sqrt{1 - {x}^{2}} \cdot \mathsf{fma}\left(x \cdot x, x \cdot 4964.451919603997, \mathsf{fma}\left(-402120.6054879238, {x}^{5}, \mathsf{fma}\left({x}^{7}, 15299731.608802434, \mathsf{fma}\left(-333194155.0361419, {x}^{9}, \mathsf{fma}\left({x}^{11}, 4634427792.775428, \mathsf{fma}\left(-44086479772.29958, {x}^{13}, \mathsf{fma}\left({x}^{15}, 300207933687.56384, \mathsf{fma}\left(-1509869313546.277, {x}^{17}, \mathsf{fma}\left({x}^{19}, 5730439675389.087, \mathsf{fma}\left(-16645562866606.395, {x}^{21}, \mathsf{fma}\left({x}^{23}, 37304482788007.22, \mathsf{fma}\left(-64661103499212.51, {x}^{25}, \mathsf{fma}\left({x}^{27}, 86399024333705.6, -88101468162941.17 \cdot \color{blue}{{x}^{29}}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) - 18.229321614212964 \cdot x\right) \]

  7. Applied rewrites99.3%

    \[\leadsto \sqrt{1 - {x}^{2}} \cdot \mathsf{fma}\left(x \cdot x, x \cdot 4964.451919603997, \mathsf{fma}\left(-402120.6054879238, {x}^{5}, \mathsf{fma}\left({x}^{7}, 15299731.608802434, \mathsf{fma}\left(-333194155.0361419, {x}^{9}, \mathsf{fma}\left({x}^{11}, 4634427792.775428, \mathsf{fma}\left(-44086479772.29958, {x}^{13}, \mathsf{fma}\left({x}^{15}, 300207933687.56384, \mathsf{fma}\left(-1509869313546.277, {x}^{17}, \mathsf{fma}\left({x}^{19}, 5730439675389.087, \mathsf{fma}\left(-16645562866606.395, {x}^{21}, \mathsf{fma}\left({x}^{23}, 37304482788007.22, \mathsf{fma}\left(-64661103499212.51, {x}^{25}, \mathsf{fma}\left({x}^{27}, 86399024333705.6, \color{blue}{-88101468162941.17 \cdot {x}^{29}}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) - 18.229321614212964 \cdot x\right) \]
  8. Add Preprocessing

Alternative 7: 99.7% accurate, 5.6× speedup?

\[\begin{array}{l} \\ \sqrt{1 - {x}^{2}} \cdot \mathsf{fma}\left(x \cdot x, x \cdot 4964.451919603997, \mathsf{fma}\left(-402120.6054879238, {x}^{5}, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-44086479772.29958, x \cdot x, 4634427792.775428\right), x \cdot x, -333194155.0361419\right), x \cdot x, 15299731.608802434\right) \cdot {x}^{7}\right) - 18.229321614212964 \cdot x\right) \end{array} \]
(FPCore (x)
 :precision binary64
 (*
  (sqrt (- 1.0 (pow x 2.0)))
  (fma
   (* x x)
   (* x 4964.451919603997)
   (-
    (fma
     -402120.6054879238
     (pow x 5.0)
     (*
      (fma
       (fma
        (fma -44086479772.29958 (* x x) 4634427792.775428)
        (* x x)
        -333194155.0361419)
       (* x x)
       15299731.608802434)
      (pow x 7.0)))
    (* 18.229321614212964 x)))))
double code(double x) {
	return sqrt((1.0 - pow(x, 2.0))) * fma((x * x), (x * 4964.451919603997), (fma(-402120.6054879238, pow(x, 5.0), (fma(fma(fma(-44086479772.29958, (x * x), 4634427792.775428), (x * x), -333194155.0361419), (x * x), 15299731.608802434) * pow(x, 7.0))) - (18.229321614212964 * x)));
}

function code(x)
	return Float64(sqrt(Float64(1.0 - (x ^ 2.0))) * fma(Float64(x * x), Float64(x * 4964.451919603997), Float64(fma(-402120.6054879238, (x ^ 5.0), Float64(fma(fma(fma(-44086479772.29958, Float64(x * x), 4634427792.775428), Float64(x * x), -333194155.0361419), Float64(x * x), 15299731.608802434) * (x ^ 7.0))) - Float64(18.229321614212964 * x))))
end

code[x_] := N[(N[Sqrt[N[(1.0 - N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * 4964.451919603997), $MachinePrecision] + N[(N[(-402120.6054879238 * N[Power[x, 5.0], $MachinePrecision] + N[(N[(N[(N[(-44086479772.29958 * N[(x * x), $MachinePrecision] + 4634427792.775428), $MachinePrecision] * N[(x * x), $MachinePrecision] + -333194155.0361419), $MachinePrecision] * N[(x * x), $MachinePrecision] + 15299731.608802434), $MachinePrecision] * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(18.229321614212964 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\begin{array}{l}

\\
\sqrt{1 - {x}^{2}} \cdot \mathsf{fma}\left(x \cdot x, x \cdot 4964.451919603997, \mathsf{fma}\left(-402120.6054879238, {x}^{5}, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-44086479772.29958, x \cdot x, 4634427792.775428\right), x \cdot x, -333194155.0361419\right), x \cdot x, 15299731.608802434\right) \cdot {x}^{7}\right) - 18.229321614212964 \cdot x\right)
\end{array}
Derivation

  1. Initial program 99.7%

    \[\sqrt{1 - {x}^{2}} \cdot \left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(346759527252.22327 \cdot {x}^{39} - 3252516578403.765 \cdot {x}^{37}\right) + 14066078189720.18 \cdot {x}^{35}\right) - 37196962323926.695 \cdot {x}^{33}\right) + 67260260640524.984 \cdot {x}^{31}\right) - 88101468162941.17 \cdot {x}^{29}\right) + 86399024333705.6 \cdot {x}^{27}\right) - 64661103499212.51 \cdot {x}^{25}\right) + 37304482788007.22 \cdot {x}^{23}\right) - 16645562866606.395 \cdot {x}^{21}\right) + 5730439675389.087 \cdot {x}^{19}\right) - 1509869313546.277 \cdot {x}^{17}\right) + 300207933687.56384 \cdot {x}^{15}\right) - 44086479772.29958 \cdot {x}^{13}\right) + 4634427792.775428 \cdot {x}^{11}\right) - 333194155.0361419 \cdot {x}^{9}\right) + 15299731.608802434 \cdot {x}^{7}\right) - 402120.6054879238 \cdot {x}^{5}\right) + 4964.451919603997 \cdot {x}^{3}\right) - 18.229321614212964 \cdot x\right) \]
  2. Add Preprocessing

  4. Applied rewrites99.7%

    \[\leadsto \sqrt{1 - {x}^{2}} \cdot \color{blue}{\mathsf{fma}\left(x \cdot x, x \cdot 4964.451919603997, \mathsf{fma}\left(-402120.6054879238, {x}^{5}, \mathsf{fma}\left({x}^{7}, 15299731.608802434, \mathsf{fma}\left(-333194155.0361419, {x}^{9}, \mathsf{fma}\left({x}^{11}, 4634427792.775428, \mathsf{fma}\left(-44086479772.29958, {x}^{13}, \mathsf{fma}\left({x}^{15}, 300207933687.56384, \mathsf{fma}\left(-1509869313546.277, {x}^{17}, \mathsf{fma}\left({x}^{19}, 5730439675389.087, \mathsf{fma}\left(-16645562866606.395, {x}^{21}, \mathsf{fma}\left({x}^{23}, 37304482788007.22, \mathsf{fma}\left(-64661103499212.51, {x}^{25}, \mathsf{fma}\left({x}^{27}, 86399024333705.6, \mathsf{fma}\left(-88101468162941.17, {x}^{29}, \mathsf{fma}\left({x}^{31}, 67260260640524.984, \mathsf{fma}\left(-37196962323926.695, {x}^{33}, \mathsf{fma}\left({x}^{35}, 14066078189720.18, \mathsf{fma}\left(-3252516578403.765, {x}^{37}, {x}^{39} \cdot 346759527252.22327\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) - 18.229321614212964 \cdot x\right)} \]

  5. Taylor expanded in x around 0

    \[\leadsto \sqrt{1 - {x}^{2}} \cdot \mathsf{fma}\left(x \cdot x, x \cdot \frac{5458472611139479}{1099511627776}, \mathsf{fma}\left(\frac{-6908379398473403}{17179869184}, {x}^{5}, \color{blue}{{x}^{7} \cdot \left(\frac{4106990431086495}{268435456} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{4859549757237287}{1048576} + \frac{-5778503076714851}{131072} \cdot {x}^{2}\right) - \frac{698758788622355}{2097152}\right)\right)}\right) - \frac{1282774469202913}{70368744177664} \cdot x\right) \]
  6. Step-by-step derivation

    1. *-commutativeN/A

      \[\leadsto \sqrt{1 - {x}^{2}} \cdot \mathsf{fma}\left(x \cdot x, x \cdot \frac{5458472611139479}{1099511627776}, \mathsf{fma}\left(\frac{-6908379398473403}{17179869184}, {x}^{5}, \color{blue}{\left(\frac{4106990431086495}{268435456} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{4859549757237287}{1048576} + \frac{-5778503076714851}{131072} \cdot {x}^{2}\right) - \frac{698758788622355}{2097152}\right)\right) \cdot {x}^{7}}\right) - \frac{1282774469202913}{70368744177664} \cdot x\right) \]

    2. lower-*.f64N/A

      \[\leadsto \sqrt{1 - {x}^{2}} \cdot \mathsf{fma}\left(x \cdot x, x \cdot \frac{5458472611139479}{1099511627776}, \mathsf{fma}\left(\frac{-6908379398473403}{17179869184}, {x}^{5}, \color{blue}{\left(\frac{4106990431086495}{268435456} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{4859549757237287}{1048576} + \frac{-5778503076714851}{131072} \cdot {x}^{2}\right) - \frac{698758788622355}{2097152}\right)\right) \cdot {x}^{7}}\right) - \frac{1282774469202913}{70368744177664} \cdot x\right) \]

    3. +-commutativeN/A

      \[\leadsto \sqrt{1 - {x}^{2}} \cdot \mathsf{fma}\left(x \cdot x, x \cdot \frac{5458472611139479}{1099511627776}, \mathsf{fma}\left(\frac{-6908379398473403}{17179869184}, {x}^{5}, \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{4859549757237287}{1048576} + \frac{-5778503076714851}{131072} \cdot {x}^{2}\right) - \frac{698758788622355}{2097152}\right) + \frac{4106990431086495}{268435456}\right)} \cdot {x}^{7}\right) - \frac{1282774469202913}{70368744177664} \cdot x\right) \]

    4. *-commutativeN/A

      \[\leadsto \sqrt{1 - {x}^{2}} \cdot \mathsf{fma}\left(x \cdot x, x \cdot \frac{5458472611139479}{1099511627776}, \mathsf{fma}\left(\frac{-6908379398473403}{17179869184}, {x}^{5}, \left(\color{blue}{\left({x}^{2} \cdot \left(\frac{4859549757237287}{1048576} + \frac{-5778503076714851}{131072} \cdot {x}^{2}\right) - \frac{698758788622355}{2097152}\right) \cdot {x}^{2}} + \frac{4106990431086495}{268435456}\right) \cdot {x}^{7}\right) - \frac{1282774469202913}{70368744177664} \cdot x\right) \]

    5. lower-fma.f64N/A

      \[\leadsto \sqrt{1 - {x}^{2}} \cdot \mathsf{fma}\left(x \cdot x, x \cdot \frac{5458472611139479}{1099511627776}, \mathsf{fma}\left(\frac{-6908379398473403}{17179869184}, {x}^{5}, \color{blue}{\mathsf{fma}\left({x}^{2} \cdot \left(\frac{4859549757237287}{1048576} + \frac{-5778503076714851}{131072} \cdot {x}^{2}\right) - \frac{698758788622355}{2097152}, {x}^{2}, \frac{4106990431086495}{268435456}\right)} \cdot {x}^{7}\right) - \frac{1282774469202913}{70368744177664} \cdot x\right) \]

    6. sub-negN/A

      \[\leadsto \sqrt{1 - {x}^{2}} \cdot \mathsf{fma}\left(x \cdot x, x \cdot \frac{5458472611139479}{1099511627776}, \mathsf{fma}\left(\frac{-6908379398473403}{17179869184}, {x}^{5}, \mathsf{fma}\left(\color{blue}{{x}^{2} \cdot \left(\frac{4859549757237287}{1048576} + \frac{-5778503076714851}{131072} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{698758788622355}{2097152}\right)\right)}, {x}^{2}, \frac{4106990431086495}{268435456}\right) \cdot {x}^{7}\right) - \frac{1282774469202913}{70368744177664} \cdot x\right) \]

    7. *-commutativeN/A

      \[\leadsto \sqrt{1 - {x}^{2}} \cdot \mathsf{fma}\left(x \cdot x, x \cdot \frac{5458472611139479}{1099511627776}, \mathsf{fma}\left(\frac{-6908379398473403}{17179869184}, {x}^{5}, \mathsf{fma}\left(\color{blue}{\left(\frac{4859549757237287}{1048576} + \frac{-5778503076714851}{131072} \cdot {x}^{2}\right) \cdot {x}^{2}} + \left(\mathsf{neg}\left(\frac{698758788622355}{2097152}\right)\right), {x}^{2}, \frac{4106990431086495}{268435456}\right) \cdot {x}^{7}\right) - \frac{1282774469202913}{70368744177664} \cdot x\right) \]

    8. metadata-evalN/A

      \[\leadsto \sqrt{1 - {x}^{2}} \cdot \mathsf{fma}\left(x \cdot x, x \cdot \frac{5458472611139479}{1099511627776}, \mathsf{fma}\left(\frac{-6908379398473403}{17179869184}, {x}^{5}, \mathsf{fma}\left(\left(\frac{4859549757237287}{1048576} + \frac{-5778503076714851}{131072} \cdot {x}^{2}\right) \cdot {x}^{2} + \color{blue}{\frac{-698758788622355}{2097152}}, {x}^{2}, \frac{4106990431086495}{268435456}\right) \cdot {x}^{7}\right) - \frac{1282774469202913}{70368744177664} \cdot x\right) \]

    9. lower-fma.f64N/A

      \[\leadsto \sqrt{1 - {x}^{2}} \cdot \mathsf{fma}\left(x \cdot x, x \cdot \frac{5458472611139479}{1099511627776}, \mathsf{fma}\left(\frac{-6908379398473403}{17179869184}, {x}^{5}, \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{4859549757237287}{1048576} + \frac{-5778503076714851}{131072} \cdot {x}^{2}, {x}^{2}, \frac{-698758788622355}{2097152}\right)}, {x}^{2}, \frac{4106990431086495}{268435456}\right) \cdot {x}^{7}\right) - \frac{1282774469202913}{70368744177664} \cdot x\right) \]

    10. +-commutativeN/A

      \[\leadsto \sqrt{1 - {x}^{2}} \cdot \mathsf{fma}\left(x \cdot x, x \cdot \frac{5458472611139479}{1099511627776}, \mathsf{fma}\left(\frac{-6908379398473403}{17179869184}, {x}^{5}, \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{-5778503076714851}{131072} \cdot {x}^{2} + \frac{4859549757237287}{1048576}}, {x}^{2}, \frac{-698758788622355}{2097152}\right), {x}^{2}, \frac{4106990431086495}{268435456}\right) \cdot {x}^{7}\right) - \frac{1282774469202913}{70368744177664} \cdot x\right) \]

    11. lower-fma.f64N/A

      \[\leadsto \sqrt{1 - {x}^{2}} \cdot \mathsf{fma}\left(x \cdot x, x \cdot \frac{5458472611139479}{1099511627776}, \mathsf{fma}\left(\frac{-6908379398473403}{17179869184}, {x}^{5}, \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-5778503076714851}{131072}, {x}^{2}, \frac{4859549757237287}{1048576}\right)}, {x}^{2}, \frac{-698758788622355}{2097152}\right), {x}^{2}, \frac{4106990431086495}{268435456}\right) \cdot {x}^{7}\right) - \frac{1282774469202913}{70368744177664} \cdot x\right) \]

    12. unpow2N/A

      \[\leadsto \sqrt{1 - {x}^{2}} \cdot \mathsf{fma}\left(x \cdot x, x \cdot \frac{5458472611139479}{1099511627776}, \mathsf{fma}\left(\frac{-6908379398473403}{17179869184}, {x}^{5}, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-5778503076714851}{131072}, \color{blue}{x \cdot x}, \frac{4859549757237287}{1048576}\right), {x}^{2}, \frac{-698758788622355}{2097152}\right), {x}^{2}, \frac{4106990431086495}{268435456}\right) \cdot {x}^{7}\right) - \frac{1282774469202913}{70368744177664} \cdot x\right) \]

    13. lower-*.f64N/A

      \[\leadsto \sqrt{1 - {x}^{2}} \cdot \mathsf{fma}\left(x \cdot x, x \cdot \frac{5458472611139479}{1099511627776}, \mathsf{fma}\left(\frac{-6908379398473403}{17179869184}, {x}^{5}, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-5778503076714851}{131072}, \color{blue}{x \cdot x}, \frac{4859549757237287}{1048576}\right), {x}^{2}, \frac{-698758788622355}{2097152}\right), {x}^{2}, \frac{4106990431086495}{268435456}\right) \cdot {x}^{7}\right) - \frac{1282774469202913}{70368744177664} \cdot x\right) \]

    14. unpow2N/A

      \[\leadsto \sqrt{1 - {x}^{2}} \cdot \mathsf{fma}\left(x \cdot x, x \cdot \frac{5458472611139479}{1099511627776}, \mathsf{fma}\left(\frac{-6908379398473403}{17179869184}, {x}^{5}, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-5778503076714851}{131072}, x \cdot x, \frac{4859549757237287}{1048576}\right), \color{blue}{x \cdot x}, \frac{-698758788622355}{2097152}\right), {x}^{2}, \frac{4106990431086495}{268435456}\right) \cdot {x}^{7}\right) - \frac{1282774469202913}{70368744177664} \cdot x\right) \]

    15. lower-*.f64N/A

      \[\leadsto \sqrt{1 - {x}^{2}} \cdot \mathsf{fma}\left(x \cdot x, x \cdot \frac{5458472611139479}{1099511627776}, \mathsf{fma}\left(\frac{-6908379398473403}{17179869184}, {x}^{5}, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-5778503076714851}{131072}, x \cdot x, \frac{4859549757237287}{1048576}\right), \color{blue}{x \cdot x}, \frac{-698758788622355}{2097152}\right), {x}^{2}, \frac{4106990431086495}{268435456}\right) \cdot {x}^{7}\right) - \frac{1282774469202913}{70368744177664} \cdot x\right) \]

    16. unpow2N/A

      \[\leadsto \sqrt{1 - {x}^{2}} \cdot \mathsf{fma}\left(x \cdot x, x \cdot \frac{5458472611139479}{1099511627776}, \mathsf{fma}\left(\frac{-6908379398473403}{17179869184}, {x}^{5}, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-5778503076714851}{131072}, x \cdot x, \frac{4859549757237287}{1048576}\right), x \cdot x, \frac{-698758788622355}{2097152}\right), \color{blue}{x \cdot x}, \frac{4106990431086495}{268435456}\right) \cdot {x}^{7}\right) - \frac{1282774469202913}{70368744177664} \cdot x\right) \]

    17. lower-*.f64N/A

      \[\leadsto \sqrt{1 - {x}^{2}} \cdot \mathsf{fma}\left(x \cdot x, x \cdot \frac{5458472611139479}{1099511627776}, \mathsf{fma}\left(\frac{-6908379398473403}{17179869184}, {x}^{5}, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-5778503076714851}{131072}, x \cdot x, \frac{4859549757237287}{1048576}\right), x \cdot x, \frac{-698758788622355}{2097152}\right), \color{blue}{x \cdot x}, \frac{4106990431086495}{268435456}\right) \cdot {x}^{7}\right) - \frac{1282774469202913}{70368744177664} \cdot x\right) \]

    18. lower-pow.f6499.3

      \[\leadsto \sqrt{1 - {x}^{2}} \cdot \mathsf{fma}\left(x \cdot x, x \cdot 4964.451919603997, \mathsf{fma}\left(-402120.6054879238, {x}^{5}, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-44086479772.29958, x \cdot x, 4634427792.775428\right), x \cdot x, -333194155.0361419\right), x \cdot x, 15299731.608802434\right) \cdot \color{blue}{{x}^{7}}\right) - 18.229321614212964 \cdot x\right) \]

  7. Applied rewrites99.3%

    \[\leadsto \sqrt{1 - {x}^{2}} \cdot \mathsf{fma}\left(x \cdot x, x \cdot 4964.451919603997, \mathsf{fma}\left(-402120.6054879238, {x}^{5}, \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-44086479772.29958, x \cdot x, 4634427792.775428\right), x \cdot x, -333194155.0361419\right), x \cdot x, 15299731.608802434\right) \cdot {x}^{7}}\right) - 18.229321614212964 \cdot x\right) \]
  8. Add Preprocessing

Alternative 8: 99.6% accurate, 8.2× speedup?

\[\begin{array}{l} \\ \left(\mathsf{fma}\left({x}^{3}, 4964.451919603997, \mathsf{fma}\left(\mathsf{fma}\left(-333194155.0361419, x \cdot x, 15299731.608802434\right), x \cdot x, -402120.6054879238\right) \cdot {x}^{5}\right) - 18.229321614212964 \cdot x\right) \cdot \sqrt{1 - x \cdot x} \end{array} \]
(FPCore (x)
 :precision binary64
 (*
  (-
   (fma
    (pow x 3.0)
    4964.451919603997
    (*
     (fma
      (fma -333194155.0361419 (* x x) 15299731.608802434)
      (* x x)
      -402120.6054879238)
     (pow x 5.0)))
   (* 18.229321614212964 x))
  (sqrt (- 1.0 (* x x)))))
double code(double x) {
	return (fma(pow(x, 3.0), 4964.451919603997, (fma(fma(-333194155.0361419, (x * x), 15299731.608802434), (x * x), -402120.6054879238) * pow(x, 5.0))) - (18.229321614212964 * x)) * sqrt((1.0 - (x * x)));
}

function code(x)
	return Float64(Float64(fma((x ^ 3.0), 4964.451919603997, Float64(fma(fma(-333194155.0361419, Float64(x * x), 15299731.608802434), Float64(x * x), -402120.6054879238) * (x ^ 5.0))) - Float64(18.229321614212964 * x)) * sqrt(Float64(1.0 - Float64(x * x))))
end

code[x_] := N[(N[(N[(N[Power[x, 3.0], $MachinePrecision] * 4964.451919603997 + N[(N[(N[(-333194155.0361419 * N[(x * x), $MachinePrecision] + 15299731.608802434), $MachinePrecision] * N[(x * x), $MachinePrecision] + -402120.6054879238), $MachinePrecision] * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(18.229321614212964 * x), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

\begin{array}{l}

\\
\left(\mathsf{fma}\left({x}^{3}, 4964.451919603997, \mathsf{fma}\left(\mathsf{fma}\left(-333194155.0361419, x \cdot x, 15299731.608802434\right), x \cdot x, -402120.6054879238\right) \cdot {x}^{5}\right) - 18.229321614212964 \cdot x\right) \cdot \sqrt{1 - x \cdot x}
\end{array}
Derivation

  1. Initial program 99.7%

    \[\sqrt{1 - {x}^{2}} \cdot \left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(346759527252.22327 \cdot {x}^{39} - 3252516578403.765 \cdot {x}^{37}\right) + 14066078189720.18 \cdot {x}^{35}\right) - 37196962323926.695 \cdot {x}^{33}\right) + 67260260640524.984 \cdot {x}^{31}\right) - 88101468162941.17 \cdot {x}^{29}\right) + 86399024333705.6 \cdot {x}^{27}\right) - 64661103499212.51 \cdot {x}^{25}\right) + 37304482788007.22 \cdot {x}^{23}\right) - 16645562866606.395 \cdot {x}^{21}\right) + 5730439675389.087 \cdot {x}^{19}\right) - 1509869313546.277 \cdot {x}^{17}\right) + 300207933687.56384 \cdot {x}^{15}\right) - 44086479772.29958 \cdot {x}^{13}\right) + 4634427792.775428 \cdot {x}^{11}\right) - 333194155.0361419 \cdot {x}^{9}\right) + 15299731.608802434 \cdot {x}^{7}\right) - 402120.6054879238 \cdot {x}^{5}\right) + 4964.451919603997 \cdot {x}^{3}\right) - 18.229321614212964 \cdot x\right) \]
  2. Add Preprocessing

  3. Taylor expanded in x around 0

    \[\leadsto \sqrt{1 - {x}^{2}} \cdot \left(\left(\color{blue}{{x}^{5} \cdot \left({x}^{2} \cdot \left(\frac{4106990431086495}{268435456} + \frac{-698758788622355}{2097152} \cdot {x}^{2}\right) - \frac{6908379398473403}{17179869184}\right)} + \frac{5458472611139479}{1099511627776} \cdot {x}^{3}\right) - \frac{1282774469202913}{70368744177664} \cdot x\right) \]
  4. Step-by-step derivation

    1. lower-*.f64N/A

      \[\leadsto \sqrt{1 - {x}^{2}} \cdot \left(\left(\color{blue}{{x}^{5} \cdot \left({x}^{2} \cdot \left(\frac{4106990431086495}{268435456} + \frac{-698758788622355}{2097152} \cdot {x}^{2}\right) - \frac{6908379398473403}{17179869184}\right)} + \frac{5458472611139479}{1099511627776} \cdot {x}^{3}\right) - \frac{1282774469202913}{70368744177664} \cdot x\right) \]

    2. lower-pow.f64N/A

      \[\leadsto \sqrt{1 - {x}^{2}} \cdot \left(\left(\color{blue}{{x}^{5}} \cdot \left({x}^{2} \cdot \left(\frac{4106990431086495}{268435456} + \frac{-698758788622355}{2097152} \cdot {x}^{2}\right) - \frac{6908379398473403}{17179869184}\right) + \frac{5458472611139479}{1099511627776} \cdot {x}^{3}\right) - \frac{1282774469202913}{70368744177664} \cdot x\right) \]

    3. sub-negN/A

      \[\leadsto \sqrt{1 - {x}^{2}} \cdot \left(\left({x}^{5} \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{4106990431086495}{268435456} + \frac{-698758788622355}{2097152} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{6908379398473403}{17179869184}\right)\right)\right)} + \frac{5458472611139479}{1099511627776} \cdot {x}^{3}\right) - \frac{1282774469202913}{70368744177664} \cdot x\right) \]

    4. *-commutativeN/A

      \[\leadsto \sqrt{1 - {x}^{2}} \cdot \left(\left({x}^{5} \cdot \left(\color{blue}{\left(\frac{4106990431086495}{268435456} + \frac{-698758788622355}{2097152} \cdot {x}^{2}\right) \cdot {x}^{2}} + \left(\mathsf{neg}\left(\frac{6908379398473403}{17179869184}\right)\right)\right) + \frac{5458472611139479}{1099511627776} \cdot {x}^{3}\right) - \frac{1282774469202913}{70368744177664} \cdot x\right) \]

    5. metadata-evalN/A

      \[\leadsto \sqrt{1 - {x}^{2}} \cdot \left(\left({x}^{5} \cdot \left(\left(\frac{4106990431086495}{268435456} + \frac{-698758788622355}{2097152} \cdot {x}^{2}\right) \cdot {x}^{2} + \color{blue}{\frac{-6908379398473403}{17179869184}}\right) + \frac{5458472611139479}{1099511627776} \cdot {x}^{3}\right) - \frac{1282774469202913}{70368744177664} \cdot x\right) \]

    6. lower-fma.f64N/A

      \[\leadsto \sqrt{1 - {x}^{2}} \cdot \left(\left({x}^{5} \cdot \color{blue}{\mathsf{fma}\left(\frac{4106990431086495}{268435456} + \frac{-698758788622355}{2097152} \cdot {x}^{2}, {x}^{2}, \frac{-6908379398473403}{17179869184}\right)} + \frac{5458472611139479}{1099511627776} \cdot {x}^{3}\right) - \frac{1282774469202913}{70368744177664} \cdot x\right) \]

    7. +-commutativeN/A

      \[\leadsto \sqrt{1 - {x}^{2}} \cdot \left(\left({x}^{5} \cdot \mathsf{fma}\left(\color{blue}{\frac{-698758788622355}{2097152} \cdot {x}^{2} + \frac{4106990431086495}{268435456}}, {x}^{2}, \frac{-6908379398473403}{17179869184}\right) + \frac{5458472611139479}{1099511627776} \cdot {x}^{3}\right) - \frac{1282774469202913}{70368744177664} \cdot x\right) \]

    8. lower-fma.f64N/A

      \[\leadsto \sqrt{1 - {x}^{2}} \cdot \left(\left({x}^{5} \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-698758788622355}{2097152}, {x}^{2}, \frac{4106990431086495}{268435456}\right)}, {x}^{2}, \frac{-6908379398473403}{17179869184}\right) + \frac{5458472611139479}{1099511627776} \cdot {x}^{3}\right) - \frac{1282774469202913}{70368744177664} \cdot x\right) \]

    9. unpow2N/A

      \[\leadsto \sqrt{1 - {x}^{2}} \cdot \left(\left({x}^{5} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-698758788622355}{2097152}, \color{blue}{x \cdot x}, \frac{4106990431086495}{268435456}\right), {x}^{2}, \frac{-6908379398473403}{17179869184}\right) + \frac{5458472611139479}{1099511627776} \cdot {x}^{3}\right) - \frac{1282774469202913}{70368744177664} \cdot x\right) \]

    10. lower-*.f64N/A

      \[\leadsto \sqrt{1 - {x}^{2}} \cdot \left(\left({x}^{5} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-698758788622355}{2097152}, \color{blue}{x \cdot x}, \frac{4106990431086495}{268435456}\right), {x}^{2}, \frac{-6908379398473403}{17179869184}\right) + \frac{5458472611139479}{1099511627776} \cdot {x}^{3}\right) - \frac{1282774469202913}{70368744177664} \cdot x\right) \]

    11. unpow2N/A

      \[\leadsto \sqrt{1 - {x}^{2}} \cdot \left(\left({x}^{5} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-698758788622355}{2097152}, x \cdot x, \frac{4106990431086495}{268435456}\right), \color{blue}{x \cdot x}, \frac{-6908379398473403}{17179869184}\right) + \frac{5458472611139479}{1099511627776} \cdot {x}^{3}\right) - \frac{1282774469202913}{70368744177664} \cdot x\right) \]

    12. lower-*.f6499.2

      \[\leadsto \sqrt{1 - {x}^{2}} \cdot \left(\left({x}^{5} \cdot \mathsf{fma}\left(\mathsf{fma}\left(-333194155.0361419, x \cdot x, 15299731.608802434\right), \color{blue}{x \cdot x}, -402120.6054879238\right) + 4964.451919603997 \cdot {x}^{3}\right) - 18.229321614212964 \cdot x\right) \]

  5. Applied rewrites99.2%

    \[\leadsto \sqrt{1 - {x}^{2}} \cdot \left(\left(\color{blue}{{x}^{5} \cdot \mathsf{fma}\left(\mathsf{fma}\left(-333194155.0361419, x \cdot x, 15299731.608802434\right), x \cdot x, -402120.6054879238\right)} + 4964.451919603997 \cdot {x}^{3}\right) - 18.229321614212964 \cdot x\right) \]

  7. Applied rewrites99.2%

    \[\leadsto \color{blue}{\left(\mathsf{fma}\left({x}^{3}, 4964.451919603997, \mathsf{fma}\left(\mathsf{fma}\left(-333194155.0361419, x \cdot x, 15299731.608802434\right), x \cdot x, -402120.6054879238\right) \cdot {x}^{5}\right) - 18.229321614212964 \cdot x\right) \cdot \sqrt{1 - x \cdot x}} \]
  8. Add Preprocessing

Alternative 9: 99.5% accurate, 56.3× speedup?

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(15500172.494389046, x \cdot x, -404600.552782524\right), x \cdot x, 4973.566580411104\right), x \cdot x, -18.229321614212964\right) \cdot x \end{array} \]
(FPCore (x)
 :precision binary64
 (*
  (fma
   (fma
    (fma 15500172.494389046 (* x x) -404600.552782524)
    (* x x)
    4973.566580411104)
   (* x x)
   -18.229321614212964)
  x))
double code(double x) {
	return fma(fma(fma(15500172.494389046, (x * x), -404600.552782524), (x * x), 4973.566580411104), (x * x), -18.229321614212964) * x;
}

function code(x)
	return Float64(fma(fma(fma(15500172.494389046, Float64(x * x), -404600.552782524), Float64(x * x), 4973.566580411104), Float64(x * x), -18.229321614212964) * x)
end

code[x_] := N[(N[(N[(N[(15500172.494389046 * N[(x * x), $MachinePrecision] + -404600.552782524), $MachinePrecision] * N[(x * x), $MachinePrecision] + 4973.566580411104), $MachinePrecision] * N[(x * x), $MachinePrecision] + -18.229321614212964), $MachinePrecision] * x), $MachinePrecision]

\begin{array}{l}

\\
\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(15500172.494389046, x \cdot x, -404600.552782524\right), x \cdot x, 4973.566580411104\right), x \cdot x, -18.229321614212964\right) \cdot x
\end{array}
Derivation

  1. Initial program 99.7%

    \[\sqrt{1 - {x}^{2}} \cdot \left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(346759527252.22327 \cdot {x}^{39} - 3252516578403.765 \cdot {x}^{37}\right) + 14066078189720.18 \cdot {x}^{35}\right) - 37196962323926.695 \cdot {x}^{33}\right) + 67260260640524.984 \cdot {x}^{31}\right) - 88101468162941.17 \cdot {x}^{29}\right) + 86399024333705.6 \cdot {x}^{27}\right) - 64661103499212.51 \cdot {x}^{25}\right) + 37304482788007.22 \cdot {x}^{23}\right) - 16645562866606.395 \cdot {x}^{21}\right) + 5730439675389.087 \cdot {x}^{19}\right) - 1509869313546.277 \cdot {x}^{17}\right) + 300207933687.56384 \cdot {x}^{15}\right) - 44086479772.29958 \cdot {x}^{13}\right) + 4634427792.775428 \cdot {x}^{11}\right) - 333194155.0361419 \cdot {x}^{9}\right) + 15299731.608802434 \cdot {x}^{7}\right) - 402120.6054879238 \cdot {x}^{5}\right) + 4964.451919603997 \cdot {x}^{3}\right) - 18.229321614212964 \cdot x\right) \]
  2. Add Preprocessing

  4. Applied rewrites99.7%

    \[\leadsto \sqrt{1 - {x}^{2}} \cdot \color{blue}{\mathsf{fma}\left(x \cdot x, x \cdot 4964.451919603997, \mathsf{fma}\left(-402120.6054879238, {x}^{5}, \mathsf{fma}\left({x}^{7}, 15299731.608802434, \mathsf{fma}\left(-333194155.0361419, {x}^{9}, \mathsf{fma}\left({x}^{11}, 4634427792.775428, \mathsf{fma}\left(-44086479772.29958, {x}^{13}, \mathsf{fma}\left({x}^{15}, 300207933687.56384, \mathsf{fma}\left(-1509869313546.277, {x}^{17}, \mathsf{fma}\left({x}^{19}, 5730439675389.087, \mathsf{fma}\left(-16645562866606.395, {x}^{21}, \mathsf{fma}\left({x}^{23}, 37304482788007.22, \mathsf{fma}\left(-64661103499212.51, {x}^{25}, \mathsf{fma}\left({x}^{27}, 86399024333705.6, \mathsf{fma}\left(-88101468162941.17, {x}^{29}, \mathsf{fma}\left({x}^{31}, 67260260640524.984, \mathsf{fma}\left(-37196962323926.695, {x}^{33}, \mathsf{fma}\left({x}^{35}, 14066078189720.18, \mathsf{fma}\left(-3252516578403.765, {x}^{37}, {x}^{39} \cdot 346759527252.22327\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) - 18.229321614212964 \cdot x\right)} \]

  5. Taylor expanded in x around 0

    \[\leadsto \color{blue}{x \cdot \left({x}^{2} \cdot \left(\frac{699967268695056225}{140737488355328} + {x}^{2} \cdot \left(\frac{17451642767477230143585}{1125899906842624} \cdot {x}^{2} - \frac{227769862343158973215}{562949953421312}\right)\right) - \frac{1282774469202913}{70368744177664}\right)} \]
  6. Step-by-step derivation

    1. *-commutativeN/A

      \[\leadsto \color{blue}{\left({x}^{2} \cdot \left(\frac{699967268695056225}{140737488355328} + {x}^{2} \cdot \left(\frac{17451642767477230143585}{1125899906842624} \cdot {x}^{2} - \frac{227769862343158973215}{562949953421312}\right)\right) - \frac{1282774469202913}{70368744177664}\right) \cdot x} \]

    2. lower-*.f64N/A

      \[\leadsto \color{blue}{\left({x}^{2} \cdot \left(\frac{699967268695056225}{140737488355328} + {x}^{2} \cdot \left(\frac{17451642767477230143585}{1125899906842624} \cdot {x}^{2} - \frac{227769862343158973215}{562949953421312}\right)\right) - \frac{1282774469202913}{70368744177664}\right) \cdot x} \]

  7. Applied rewrites99.1%

    \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(15500172.494389046, x \cdot x, -404600.552782524\right), x \cdot x, 4973.566580411104\right), x \cdot x, -18.229321614212964\right) \cdot x} \]
  8. Add Preprocessing

Alternative 10: 99.4% accurate, 78.4× speedup?

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(-404600.552782524, x \cdot x, 4973.566580411104\right), x \cdot x, -18.229321614212964\right) \cdot x \end{array} \]
(FPCore (x)
 :precision binary64
 (*
  (fma
   (fma -404600.552782524 (* x x) 4973.566580411104)
   (* x x)
   -18.229321614212964)
  x))
double code(double x) {
	return fma(fma(-404600.552782524, (x * x), 4973.566580411104), (x * x), -18.229321614212964) * x;
}

function code(x)
	return Float64(fma(fma(-404600.552782524, Float64(x * x), 4973.566580411104), Float64(x * x), -18.229321614212964) * x)
end

code[x_] := N[(N[(N[(-404600.552782524 * N[(x * x), $MachinePrecision] + 4973.566580411104), $MachinePrecision] * N[(x * x), $MachinePrecision] + -18.229321614212964), $MachinePrecision] * x), $MachinePrecision]

\begin{array}{l}

\\
\mathsf{fma}\left(\mathsf{fma}\left(-404600.552782524, x \cdot x, 4973.566580411104\right), x \cdot x, -18.229321614212964\right) \cdot x
\end{array}
Derivation

  1. Initial program 99.7%

    \[\sqrt{1 - {x}^{2}} \cdot \left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(346759527252.22327 \cdot {x}^{39} - 3252516578403.765 \cdot {x}^{37}\right) + 14066078189720.18 \cdot {x}^{35}\right) - 37196962323926.695 \cdot {x}^{33}\right) + 67260260640524.984 \cdot {x}^{31}\right) - 88101468162941.17 \cdot {x}^{29}\right) + 86399024333705.6 \cdot {x}^{27}\right) - 64661103499212.51 \cdot {x}^{25}\right) + 37304482788007.22 \cdot {x}^{23}\right) - 16645562866606.395 \cdot {x}^{21}\right) + 5730439675389.087 \cdot {x}^{19}\right) - 1509869313546.277 \cdot {x}^{17}\right) + 300207933687.56384 \cdot {x}^{15}\right) - 44086479772.29958 \cdot {x}^{13}\right) + 4634427792.775428 \cdot {x}^{11}\right) - 333194155.0361419 \cdot {x}^{9}\right) + 15299731.608802434 \cdot {x}^{7}\right) - 402120.6054879238 \cdot {x}^{5}\right) + 4964.451919603997 \cdot {x}^{3}\right) - 18.229321614212964 \cdot x\right) \]
  2. Add Preprocessing

  4. Applied rewrites99.7%

    \[\leadsto \sqrt{1 - {x}^{2}} \cdot \color{blue}{\mathsf{fma}\left(x \cdot x, x \cdot 4964.451919603997, \mathsf{fma}\left(-402120.6054879238, {x}^{5}, \mathsf{fma}\left({x}^{7}, 15299731.608802434, \mathsf{fma}\left(-333194155.0361419, {x}^{9}, \mathsf{fma}\left({x}^{11}, 4634427792.775428, \mathsf{fma}\left(-44086479772.29958, {x}^{13}, \mathsf{fma}\left({x}^{15}, 300207933687.56384, \mathsf{fma}\left(-1509869313546.277, {x}^{17}, \mathsf{fma}\left({x}^{19}, 5730439675389.087, \mathsf{fma}\left(-16645562866606.395, {x}^{21}, \mathsf{fma}\left({x}^{23}, 37304482788007.22, \mathsf{fma}\left(-64661103499212.51, {x}^{25}, \mathsf{fma}\left({x}^{27}, 86399024333705.6, \mathsf{fma}\left(-88101468162941.17, {x}^{29}, \mathsf{fma}\left({x}^{31}, 67260260640524.984, \mathsf{fma}\left(-37196962323926.695, {x}^{33}, \mathsf{fma}\left({x}^{35}, 14066078189720.18, \mathsf{fma}\left(-3252516578403.765, {x}^{37}, {x}^{39} \cdot 346759527252.22327\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) - 18.229321614212964 \cdot x\right)} \]

  5. Taylor expanded in x around 0

    \[\leadsto \color{blue}{x \cdot \left({x}^{2} \cdot \left(\frac{699967268695056225}{140737488355328} + \frac{-227769862343158973215}{562949953421312} \cdot {x}^{2}\right) - \frac{1282774469202913}{70368744177664}\right)} \]
  6. Step-by-step derivation

    1. *-commutativeN/A

      \[\leadsto \color{blue}{\left({x}^{2} \cdot \left(\frac{699967268695056225}{140737488355328} + \frac{-227769862343158973215}{562949953421312} \cdot {x}^{2}\right) - \frac{1282774469202913}{70368744177664}\right) \cdot x} \]

    2. lower-*.f64N/A

      \[\leadsto \color{blue}{\left({x}^{2} \cdot \left(\frac{699967268695056225}{140737488355328} + \frac{-227769862343158973215}{562949953421312} \cdot {x}^{2}\right) - \frac{1282774469202913}{70368744177664}\right) \cdot x} \]

    3. sub-negN/A

      \[\leadsto \color{blue}{\left({x}^{2} \cdot \left(\frac{699967268695056225}{140737488355328} + \frac{-227769862343158973215}{562949953421312} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1282774469202913}{70368744177664}\right)\right)\right)} \cdot x \]

    4. *-commutativeN/A

      \[\leadsto \left(\color{blue}{\left(\frac{699967268695056225}{140737488355328} + \frac{-227769862343158973215}{562949953421312} \cdot {x}^{2}\right) \cdot {x}^{2}} + \left(\mathsf{neg}\left(\frac{1282774469202913}{70368744177664}\right)\right)\right) \cdot x \]

    5. metadata-evalN/A

      \[\leadsto \left(\left(\frac{699967268695056225}{140737488355328} + \frac{-227769862343158973215}{562949953421312} \cdot {x}^{2}\right) \cdot {x}^{2} + \color{blue}{\frac{-1282774469202913}{70368744177664}}\right) \cdot x \]

    6. lower-fma.f64N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{699967268695056225}{140737488355328} + \frac{-227769862343158973215}{562949953421312} \cdot {x}^{2}, {x}^{2}, \frac{-1282774469202913}{70368744177664}\right)} \cdot x \]

    7. +-commutativeN/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-227769862343158973215}{562949953421312} \cdot {x}^{2} + \frac{699967268695056225}{140737488355328}}, {x}^{2}, \frac{-1282774469202913}{70368744177664}\right) \cdot x \]

    8. lower-fma.f64N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-227769862343158973215}{562949953421312}, {x}^{2}, \frac{699967268695056225}{140737488355328}\right)}, {x}^{2}, \frac{-1282774469202913}{70368744177664}\right) \cdot x \]

    9. unpow2N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-227769862343158973215}{562949953421312}, \color{blue}{x \cdot x}, \frac{699967268695056225}{140737488355328}\right), {x}^{2}, \frac{-1282774469202913}{70368744177664}\right) \cdot x \]

    10. lower-*.f64N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-227769862343158973215}{562949953421312}, \color{blue}{x \cdot x}, \frac{699967268695056225}{140737488355328}\right), {x}^{2}, \frac{-1282774469202913}{70368744177664}\right) \cdot x \]

    11. unpow2N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-227769862343158973215}{562949953421312}, x \cdot x, \frac{699967268695056225}{140737488355328}\right), \color{blue}{x \cdot x}, \frac{-1282774469202913}{70368744177664}\right) \cdot x \]

    12. lower-*.f6499.0

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-404600.552782524, x \cdot x, 4973.566580411104\right), \color{blue}{x \cdot x}, -18.229321614212964\right) \cdot x \]

  7. Applied rewrites99.0%

    \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-404600.552782524, x \cdot x, 4973.566580411104\right), x \cdot x, -18.229321614212964\right) \cdot x} \]
  8. Add Preprocessing

Alternative 11: 99.2% accurate, 129.2× speedup?

\[\begin{array}{l} \\ \mathsf{fma}\left(4973.566580411104, x \cdot x, -18.229321614212964\right) \cdot x \end{array} \]
(FPCore (x)
 :precision binary64
 (* (fma 4973.566580411104 (* x x) -18.229321614212964) x))
double code(double x) {
	return fma(4973.566580411104, (x * x), -18.229321614212964) * x;
}

function code(x)
	return Float64(fma(4973.566580411104, Float64(x * x), -18.229321614212964) * x)
end

code[x_] := N[(N[(4973.566580411104 * N[(x * x), $MachinePrecision] + -18.229321614212964), $MachinePrecision] * x), $MachinePrecision]

\begin{array}{l}

\\
\mathsf{fma}\left(4973.566580411104, x \cdot x, -18.229321614212964\right) \cdot x
\end{array}
Derivation

  1. Initial program 99.7%

    \[\sqrt{1 - {x}^{2}} \cdot \left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(346759527252.22327 \cdot {x}^{39} - 3252516578403.765 \cdot {x}^{37}\right) + 14066078189720.18 \cdot {x}^{35}\right) - 37196962323926.695 \cdot {x}^{33}\right) + 67260260640524.984 \cdot {x}^{31}\right) - 88101468162941.17 \cdot {x}^{29}\right) + 86399024333705.6 \cdot {x}^{27}\right) - 64661103499212.51 \cdot {x}^{25}\right) + 37304482788007.22 \cdot {x}^{23}\right) - 16645562866606.395 \cdot {x}^{21}\right) + 5730439675389.087 \cdot {x}^{19}\right) - 1509869313546.277 \cdot {x}^{17}\right) + 300207933687.56384 \cdot {x}^{15}\right) - 44086479772.29958 \cdot {x}^{13}\right) + 4634427792.775428 \cdot {x}^{11}\right) - 333194155.0361419 \cdot {x}^{9}\right) + 15299731.608802434 \cdot {x}^{7}\right) - 402120.6054879238 \cdot {x}^{5}\right) + 4964.451919603997 \cdot {x}^{3}\right) - 18.229321614212964 \cdot x\right) \]
  2. Add Preprocessing

  4. Applied rewrites99.7%

    \[\leadsto \sqrt{1 - {x}^{2}} \cdot \color{blue}{\mathsf{fma}\left(x \cdot x, x \cdot 4964.451919603997, \mathsf{fma}\left(-402120.6054879238, {x}^{5}, \mathsf{fma}\left({x}^{7}, 15299731.608802434, \mathsf{fma}\left(-333194155.0361419, {x}^{9}, \mathsf{fma}\left({x}^{11}, 4634427792.775428, \mathsf{fma}\left(-44086479772.29958, {x}^{13}, \mathsf{fma}\left({x}^{15}, 300207933687.56384, \mathsf{fma}\left(-1509869313546.277, {x}^{17}, \mathsf{fma}\left({x}^{19}, 5730439675389.087, \mathsf{fma}\left(-16645562866606.395, {x}^{21}, \mathsf{fma}\left({x}^{23}, 37304482788007.22, \mathsf{fma}\left(-64661103499212.51, {x}^{25}, \mathsf{fma}\left({x}^{27}, 86399024333705.6, \mathsf{fma}\left(-88101468162941.17, {x}^{29}, \mathsf{fma}\left({x}^{31}, 67260260640524.984, \mathsf{fma}\left(-37196962323926.695, {x}^{33}, \mathsf{fma}\left({x}^{35}, 14066078189720.18, \mathsf{fma}\left(-3252516578403.765, {x}^{37}, {x}^{39} \cdot 346759527252.22327\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) - 18.229321614212964 \cdot x\right)} \]

  5. Taylor expanded in x around 0

    \[\leadsto \color{blue}{x \cdot \left(\frac{699967268695056225}{140737488355328} \cdot {x}^{2} - \frac{1282774469202913}{70368744177664}\right)} \]
  6. Step-by-step derivation

    1. *-commutativeN/A

      \[\leadsto \color{blue}{\left(\frac{699967268695056225}{140737488355328} \cdot {x}^{2} - \frac{1282774469202913}{70368744177664}\right) \cdot x} \]

    2. lower-*.f64N/A

      \[\leadsto \color{blue}{\left(\frac{699967268695056225}{140737488355328} \cdot {x}^{2} - \frac{1282774469202913}{70368744177664}\right) \cdot x} \]

    3. sub-negN/A

      \[\leadsto \color{blue}{\left(\frac{699967268695056225}{140737488355328} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1282774469202913}{70368744177664}\right)\right)\right)} \cdot x \]

    4. metadata-evalN/A

      \[\leadsto \left(\frac{699967268695056225}{140737488355328} \cdot {x}^{2} + \color{blue}{\frac{-1282774469202913}{70368744177664}}\right) \cdot x \]

    5. lower-fma.f64N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{699967268695056225}{140737488355328}, {x}^{2}, \frac{-1282774469202913}{70368744177664}\right)} \cdot x \]

    6. unpow2N/A

      \[\leadsto \mathsf{fma}\left(\frac{699967268695056225}{140737488355328}, \color{blue}{x \cdot x}, \frac{-1282774469202913}{70368744177664}\right) \cdot x \]

    7. lower-*.f6498.8

      \[\leadsto \mathsf{fma}\left(4973.566580411104, \color{blue}{x \cdot x}, -18.229321614212964\right) \cdot x \]

  7. Applied rewrites98.8%

    \[\leadsto \color{blue}{\mathsf{fma}\left(4973.566580411104, x \cdot x, -18.229321614212964\right) \cdot x} \]
  8. Add Preprocessing

Alternative 12: 98.5% accurate, 199.6× speedup?

\[\begin{array}{l} \\ 1 \cdot \left(-18.229321614212964 \cdot x\right) \end{array} \]
(FPCore (x) :precision binary64 (* 1.0 (* -18.229321614212964 x)))
double code(double x) {
	return 1.0 * (-18.229321614212964 * x);
}

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 * ((-18.229321614212964d0) * x)
end function

public static double code(double x) {
	return 1.0 * (-18.229321614212964 * x);
}

def code(x):
	return 1.0 * (-18.229321614212964 * x)

function code(x)
	return Float64(1.0 * Float64(-18.229321614212964 * x))
end

function tmp = code(x)
	tmp = 1.0 * (-18.229321614212964 * x);
end

code[x_] := N[(1.0 * N[(-18.229321614212964 * x), $MachinePrecision]), $MachinePrecision]

\begin{array}{l}

\\
1 \cdot \left(-18.229321614212964 \cdot x\right)
\end{array}
Derivation

  1. Initial program 99.7%

    \[\sqrt{1 - {x}^{2}} \cdot \left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(\left(346759527252.22327 \cdot {x}^{39} - 3252516578403.765 \cdot {x}^{37}\right) + 14066078189720.18 \cdot {x}^{35}\right) - 37196962323926.695 \cdot {x}^{33}\right) + 67260260640524.984 \cdot {x}^{31}\right) - 88101468162941.17 \cdot {x}^{29}\right) + 86399024333705.6 \cdot {x}^{27}\right) - 64661103499212.51 \cdot {x}^{25}\right) + 37304482788007.22 \cdot {x}^{23}\right) - 16645562866606.395 \cdot {x}^{21}\right) + 5730439675389.087 \cdot {x}^{19}\right) - 1509869313546.277 \cdot {x}^{17}\right) + 300207933687.56384 \cdot {x}^{15}\right) - 44086479772.29958 \cdot {x}^{13}\right) + 4634427792.775428 \cdot {x}^{11}\right) - 333194155.0361419 \cdot {x}^{9}\right) + 15299731.608802434 \cdot {x}^{7}\right) - 402120.6054879238 \cdot {x}^{5}\right) + 4964.451919603997 \cdot {x}^{3}\right) - 18.229321614212964 \cdot x\right) \]
  2. Add Preprocessing

  3. Taylor expanded in x around 0

    \[\leadsto \sqrt{1 - {x}^{2}} \cdot \color{blue}{\left(\frac{-1282774469202913}{70368744177664} \cdot x\right)} \]
  4. Step-by-step derivation

    1. lower-*.f6498.2

      \[\leadsto \sqrt{1 - {x}^{2}} \cdot \color{blue}{\left(-18.229321614212964 \cdot x\right)} \]

  5. Applied rewrites98.2%

    \[\leadsto \sqrt{1 - {x}^{2}} \cdot \color{blue}{\left(-18.229321614212964 \cdot x\right)} \]

  6. Taylor expanded in x around 0

    \[\leadsto \color{blue}{1} \cdot \left(\frac{-1282774469202913}{70368744177664} \cdot x\right) \]
  7. Step-by-step derivation

    1. Applied rewrites98.2%

      \[\leadsto \color{blue}{1} \cdot \left(-18.229321614212964 \cdot x\right) \]
    2. Add Preprocessing

    Reproduce

    ?
    herbie shell --seed 1 
(FPCore (x)
  :name "sqrt(1 - pow(x, 2))*(346759527252.22327*pow(x, 39) - 3252516578403.7652*pow(x, 37) + 14066078189720.179*pow(x, 35) - 37196962323926.696*pow(x, 33) + 67260260640524.985*pow(x, 31) - 88101468162941.178*pow(x, 29) + 86399024333705.6*pow(x, 27) - 64661103499212.506*pow(x, 25) + 37304482788007.215*pow(x, 23) - 16645562866606.394*pow(x, 21) + 5730439675389.0865*pow(x, 19) - 1509869313546.277*pow(x, 17) + 300207933687.56385*pow(x, 15) - 44086479772.299586*pow(x, 13) + 4634427792.7754282*pow(x, 11) - 333194155.03614189*pow(x, 9) + 15299731.608802434*pow(x, 7) - 402120.6054879238*pow(x, 5) + 4964.4519196039976*pow(x, 3) - 18.229321614212965*x);"
  :precision binary64
  :pre (and (<= -1.0 x) (<= x 1.0))
  (* (sqrt (- 1.0 (pow x 2.0))) (- (+ (- (+ (- (+ (- (+ (- (+ (- (+ (- (+ (- (+ (- (+ (- (* 346759527252.22327 (pow x 39.0)) (* 3252516578403.765 (pow x 37.0))) (* 14066078189720.18 (pow x 35.0))) (* 37196962323926.695 (pow x 33.0))) (* 67260260640524.984 (pow x 31.0))) (* 88101468162941.17 (pow x 29.0))) (* 86399024333705.6 (pow x 27.0))) (* 64661103499212.51 (pow x 25.0))) (* 37304482788007.22 (pow x 23.0))) (* 16645562866606.395 (pow x 21.0))) (* 5730439675389.087 (pow x 19.0))) (* 1509869313546.277 (pow x 17.0))) (* 300207933687.56384 (pow x 15.0))) (* 44086479772.29958 (pow x 13.0))) (* 4634427792.775428 (pow x 11.0))) (* 333194155.0361419 (pow x 9.0))) (* 15299731.608802434 (pow x 7.0))) (* 402120.6054879238 (pow x 5.0))) (* 4964.451919603997 (pow x 3.0))) (* 18.229321614212964 x))))