Result for 1 - sqrt( (3.5041384 * 10e15 * x * x * x * x * x * x * x * x) / ((20.598997 * 20.598997 + x * x) * (20.598997 * 20.598997 + x * x) * (107.65265 * 107.65265 + x*x) * (737.86223 * 737.86223 + x*x) * (12194.217 * 12194.217 + x*x) * (12194.217 * 12194.217 + x*x)))

Specification

\[0 \leq x \land x \leq 44100\]

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 12194.217 \cdot 12194.217 + x \cdot x\\ t_1 := 20.598997 \cdot 20.598997 + x \cdot x\\ 1 - \sqrt{\frac{\left(\left(\left(\left(\left(\left(\left(\left(3.5041384 \cdot 10^{+16}\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x}{\left(\left(\left(\left(t\_1 \cdot t\_1\right) \cdot \left(107.65265 \cdot 107.65265 + x \cdot x\right)\right) \cdot \left(737.86223 \cdot 737.86223 + x \cdot x\right)\right) \cdot t\_0\right) \cdot t\_0}} \end{array} \end{array} \]

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (* 12194.217 12194.217) (* x x)))
        (t_1 (+ (* 20.598997 20.598997) (* x x))))
   (-
    1.0
    (sqrt
     (/
      (* (* (* (* (* (* (* (* (* 3.5041384 1e+16) x) x) x) x) x) x) x) x)
      (*
       (*
        (*
         (* (* t_1 t_1) (+ (* 107.65265 107.65265) (* x x)))
         (+ (* 737.86223 737.86223) (* x x)))
        t_0)
       t_0))))))

double code(double x) {
	double t_0 = (12194.217 * 12194.217) + (x * x);
	double t_1 = (20.598997 * 20.598997) + (x * x);
	return 1.0 - sqrt(((((((((((3.5041384 * 1e+16) * x) * x) * x) * x) * x) * x) * x) * x) / (((((t_1 * t_1) * ((107.65265 * 107.65265) + (x * x))) * ((737.86223 * 737.86223) + (x * x))) * t_0) * t_0)));
}

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    t_0 = (12194.217d0 * 12194.217d0) + (x * x)
    t_1 = (20.598997d0 * 20.598997d0) + (x * x)
    code = 1.0d0 - sqrt(((((((((((3.5041384d0 * 1d+16) * x) * x) * x) * x) * x) * x) * x) * x) / (((((t_1 * t_1) * ((107.65265d0 * 107.65265d0) + (x * x))) * ((737.86223d0 * 737.86223d0) + (x * x))) * t_0) * t_0)))
end function

public static double code(double x) {
	double t_0 = (12194.217 * 12194.217) + (x * x);
	double t_1 = (20.598997 * 20.598997) + (x * x);
	return 1.0 - Math.sqrt(((((((((((3.5041384 * 1e+16) * x) * x) * x) * x) * x) * x) * x) * x) / (((((t_1 * t_1) * ((107.65265 * 107.65265) + (x * x))) * ((737.86223 * 737.86223) + (x * x))) * t_0) * t_0)));
}

def code(x):
	t_0 = (12194.217 * 12194.217) + (x * x)
	t_1 = (20.598997 * 20.598997) + (x * x)
	return 1.0 - math.sqrt(((((((((((3.5041384 * 1e+16) * x) * x) * x) * x) * x) * x) * x) * x) / (((((t_1 * t_1) * ((107.65265 * 107.65265) + (x * x))) * ((737.86223 * 737.86223) + (x * x))) * t_0) * t_0)))

function code(x)
	t_0 = Float64(Float64(12194.217 * 12194.217) + Float64(x * x))
	t_1 = Float64(Float64(20.598997 * 20.598997) + Float64(x * x))
	return Float64(1.0 - sqrt(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(3.5041384 * 1e+16) * x) * x) * x) * x) * x) * x) * x) * x) / Float64(Float64(Float64(Float64(Float64(t_1 * t_1) * Float64(Float64(107.65265 * 107.65265) + Float64(x * x))) * Float64(Float64(737.86223 * 737.86223) + Float64(x * x))) * t_0) * t_0))))
end

function tmp = code(x)
	t_0 = (12194.217 * 12194.217) + (x * x);
	t_1 = (20.598997 * 20.598997) + (x * x);
	tmp = 1.0 - sqrt(((((((((((3.5041384 * 1e+16) * x) * x) * x) * x) * x) * x) * x) * x) / (((((t_1 * t_1) * ((107.65265 * 107.65265) + (x * x))) * ((737.86223 * 737.86223) + (x * x))) * t_0) * t_0)));
end

code[x_] := Block[{t$95$0 = N[(N[(12194.217 * 12194.217), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(20.598997 * 20.598997), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]}, N[(1.0 - N[Sqrt[N[(N[(N[(N[(N[(N[(N[(N[(N[(N[(3.5041384 * 1e+16), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] / N[(N[(N[(N[(N[(t$95$1 * t$95$1), $MachinePrecision] * N[(N[(107.65265 * 107.65265), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(737.86223 * 737.86223), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

\begin{array}{l}

\\
\begin{array}{l}
t_0 := 12194.217 \cdot 12194.217 + x \cdot x\\
t_1 := 20.598997 \cdot 20.598997 + x \cdot x\\
1 - \sqrt{\frac{\left(\left(\left(\left(\left(\left(\left(\left(3.5041384 \cdot 10^{+16}\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x}{\left(\left(\left(\left(t\_1 \cdot t\_1\right) \cdot \left(107.65265 \cdot 107.65265 + x \cdot x\right)\right) \cdot \left(737.86223 \cdot 737.86223 + x \cdot x\right)\right) \cdot t\_0\right) \cdot t\_0}}
\end{array}
\end{array}

Sampling outcomes in binary64 precision:

Initial Program: 100.0% accurate, 1.0× speedup?

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (* 12194.217 12194.217) (* x x)))
        (t_1 (+ (* 20.598997 20.598997) (* x x))))
   (-
    1.0
    (sqrt
     (/
      (* (* (* (* (* (* (* (* (* 3.5041384 1e+16) x) x) x) x) x) x) x) x)
      (*
       (*
        (*
         (* (* t_1 t_1) (+ (* 107.65265 107.65265) (* x x)))
         (+ (* 737.86223 737.86223) (* x x)))
        t_0)
       t_0))))))

double code(double x) {
	double t_0 = (12194.217 * 12194.217) + (x * x);
	double t_1 = (20.598997 * 20.598997) + (x * x);
	return 1.0 - sqrt(((((((((((3.5041384 * 1e+16) * x) * x) * x) * x) * x) * x) * x) * x) / (((((t_1 * t_1) * ((107.65265 * 107.65265) + (x * x))) * ((737.86223 * 737.86223) + (x * x))) * t_0) * t_0)));
}

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    t_0 = (12194.217d0 * 12194.217d0) + (x * x)
    t_1 = (20.598997d0 * 20.598997d0) + (x * x)
    code = 1.0d0 - sqrt(((((((((((3.5041384d0 * 1d+16) * x) * x) * x) * x) * x) * x) * x) * x) / (((((t_1 * t_1) * ((107.65265d0 * 107.65265d0) + (x * x))) * ((737.86223d0 * 737.86223d0) + (x * x))) * t_0) * t_0)))
end function

public static double code(double x) {
	double t_0 = (12194.217 * 12194.217) + (x * x);
	double t_1 = (20.598997 * 20.598997) + (x * x);
	return 1.0 - Math.sqrt(((((((((((3.5041384 * 1e+16) * x) * x) * x) * x) * x) * x) * x) * x) / (((((t_1 * t_1) * ((107.65265 * 107.65265) + (x * x))) * ((737.86223 * 737.86223) + (x * x))) * t_0) * t_0)));
}

def code(x):
	t_0 = (12194.217 * 12194.217) + (x * x)
	t_1 = (20.598997 * 20.598997) + (x * x)
	return 1.0 - math.sqrt(((((((((((3.5041384 * 1e+16) * x) * x) * x) * x) * x) * x) * x) * x) / (((((t_1 * t_1) * ((107.65265 * 107.65265) + (x * x))) * ((737.86223 * 737.86223) + (x * x))) * t_0) * t_0)))

function code(x)
	t_0 = Float64(Float64(12194.217 * 12194.217) + Float64(x * x))
	t_1 = Float64(Float64(20.598997 * 20.598997) + Float64(x * x))
	return Float64(1.0 - sqrt(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(3.5041384 * 1e+16) * x) * x) * x) * x) * x) * x) * x) * x) / Float64(Float64(Float64(Float64(Float64(t_1 * t_1) * Float64(Float64(107.65265 * 107.65265) + Float64(x * x))) * Float64(Float64(737.86223 * 737.86223) + Float64(x * x))) * t_0) * t_0))))
end

function tmp = code(x)
	t_0 = (12194.217 * 12194.217) + (x * x);
	t_1 = (20.598997 * 20.598997) + (x * x);
	tmp = 1.0 - sqrt(((((((((((3.5041384 * 1e+16) * x) * x) * x) * x) * x) * x) * x) * x) / (((((t_1 * t_1) * ((107.65265 * 107.65265) + (x * x))) * ((737.86223 * 737.86223) + (x * x))) * t_0) * t_0)));
end

code[x_] := Block[{t$95$0 = N[(N[(12194.217 * 12194.217), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(20.598997 * 20.598997), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]}, N[(1.0 - N[Sqrt[N[(N[(N[(N[(N[(N[(N[(N[(N[(N[(3.5041384 * 1e+16), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] / N[(N[(N[(N[(N[(t$95$1 * t$95$1), $MachinePrecision] * N[(N[(107.65265 * 107.65265), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(737.86223 * 737.86223), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

\begin{array}{l}

\\
\begin{array}{l}
t_0 := 12194.217 \cdot 12194.217 + x \cdot x\\
t_1 := 20.598997 \cdot 20.598997 + x \cdot x\\
1 - \sqrt{\frac{\left(\left(\left(\left(\left(\left(\left(\left(3.5041384 \cdot 10^{+16}\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x}{\left(\left(\left(\left(t\_1 \cdot t\_1\right) \cdot \left(107.65265 \cdot 107.65265 + x \cdot x\right)\right) \cdot \left(737.86223 \cdot 737.86223 + x \cdot x\right)\right) \cdot t\_0\right) \cdot t\_0}}
\end{array}
\end{array}

Alternative 1: 100.0% accurate, 0.8× speedup?

\[\begin{array}{l} \\ 1 - \left(\sqrt{3.5041384 \cdot 10^{+16} \cdot x} \cdot \left(x \cdot x\right)\right) \cdot \sqrt{\left(x \cdot x\right) \cdot \frac{x}{\left(\mathsf{fma}\left(x, x, 11589.0930520225\right) \cdot \mathsf{fma}\left(x, x, 544440.6704605728\right)\right) \cdot {\left(\mathsf{fma}\left(x, x, 424.31867740600904\right) \cdot \mathsf{fma}\left(x, x, 148698928.24308902\right)\right)}^{2}}} \end{array} \]

(FPCore (x)
 :precision binary64
 (-
  1.0
  (*
   (* (sqrt (* 3.5041384e+16 x)) (* x x))
   (sqrt
    (*
     (* x x)
     (/
      x
      (*
       (* (fma x x 11589.0930520225) (fma x x 544440.6704605728))
       (pow
        (* (fma x x 424.31867740600904) (fma x x 148698928.24308902))
        2.0))))))))

double code(double x) {
	return 1.0 - ((sqrt((3.5041384e+16 * x)) * (x * x)) * sqrt(((x * x) * (x / ((fma(x, x, 11589.0930520225) * fma(x, x, 544440.6704605728)) * pow((fma(x, x, 424.31867740600904) * fma(x, x, 148698928.24308902)), 2.0))))));
}

function code(x)
	return Float64(1.0 - Float64(Float64(sqrt(Float64(3.5041384e+16 * x)) * Float64(x * x)) * sqrt(Float64(Float64(x * x) * Float64(x / Float64(Float64(fma(x, x, 11589.0930520225) * fma(x, x, 544440.6704605728)) * (Float64(fma(x, x, 424.31867740600904) * fma(x, x, 148698928.24308902)) ^ 2.0)))))))
end

code[x_] := N[(1.0 - N[(N[(N[Sqrt[N[(3.5041384e+16 * x), $MachinePrecision]], $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(x * x), $MachinePrecision] * N[(x / N[(N[(N[(x * x + 11589.0930520225), $MachinePrecision] * N[(x * x + 544440.6704605728), $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(x * x + 424.31867740600904), $MachinePrecision] * N[(x * x + 148698928.24308902), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\begin{array}{l}

\\
1 - \left(\sqrt{3.5041384 \cdot 10^{+16} \cdot x} \cdot \left(x \cdot x\right)\right) \cdot \sqrt{\left(x \cdot x\right) \cdot \frac{x}{\left(\mathsf{fma}\left(x, x, 11589.0930520225\right) \cdot \mathsf{fma}\left(x, x, 544440.6704605728\right)\right) \cdot {\left(\mathsf{fma}\left(x, x, 424.31867740600904\right) \cdot \mathsf{fma}\left(x, x, 148698928.24308902\right)\right)}^{2}}}
\end{array}

Derivation

Initial program 100.0%
\[1 - \sqrt{\frac{\left(\left(\left(\left(\left(\left(\left(\left(3.5041384 \cdot 10^{+16}\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x}{\left(\left(\left(\left(\left(20.598997 \cdot 20.598997 + x \cdot x\right) \cdot \left(20.598997 \cdot 20.598997 + x \cdot x\right)\right) \cdot \left(107.65265 \cdot 107.65265 + x \cdot x\right)\right) \cdot \left(737.86223 \cdot 737.86223 + x \cdot x\right)\right) \cdot \left(12194.217 \cdot 12194.217 + x \cdot x\right)\right) \cdot \left(12194.217 \cdot 12194.217 + x \cdot x\right)}} \]
Add Preprocessing
Applied rewrites100.0%
\[\leadsto 1 - \color{blue}{\sqrt{\frac{\left(x \cdot x\right) \cdot \left(3.5041384 \cdot 10^{+16} \cdot {x}^{4}\right)}{\mathsf{fma}\left(x, x, 148698928.24308902\right)}} \cdot \sqrt{\frac{x \cdot x}{\mathsf{fma}\left(x, x, 148698928.24308902\right) \cdot \left(\mathsf{fma}\left(x, x, 544440.6704605728\right) \cdot \left(\mathsf{fma}\left(x, x, 11589.0930520225\right) \cdot {\left(\mathsf{fma}\left(x, x, 424.31867740600904\right)\right)}^{2}\right)\right)}}} \]
Applied rewrites100.0%
\[\leadsto 1 - \color{blue}{\left(\sqrt{3.5041384 \cdot 10^{+16} \cdot x} \cdot \left(x \cdot x\right)\right) \cdot \sqrt{\left(x \cdot x\right) \cdot \frac{x}{\left(\mathsf{fma}\left(x, x, 11589.0930520225\right) \cdot \mathsf{fma}\left(x, x, 544440.6704605728\right)\right) \cdot {\left(\mathsf{fma}\left(x, x, 424.31867740600904\right) \cdot \mathsf{fma}\left(x, x, 148698928.24308902\right)\right)}^{2}}}} \]
Add Preprocessing

Alternative 2: 100.0% accurate, 1.1× speedup?

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 12194.217 \cdot 12194.217 + x \cdot x\\ 1 - \sqrt{\frac{\left(\left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot 3.5041384 \cdot 10^{+16}\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x}{\left(\left(\left(\left(\mathsf{fma}\left(x, x, 424.31867740600904\right) \cdot \mathsf{fma}\left(x, x, 424.31867740600904\right)\right) \cdot \left(107.65265 \cdot 107.65265 + x \cdot x\right)\right) \cdot \left(737.86223 \cdot 737.86223 + x \cdot x\right)\right) \cdot t\_0\right) \cdot t\_0}} \end{array} \end{array} \]

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (* 12194.217 12194.217) (* x x))))
   (-
    1.0
    (sqrt
     (/
      (* (* (* (* (* (* (* (* x x) 3.5041384e+16) x) x) x) x) x) x)
      (*
       (*
        (*
         (*
          (* (fma x x 424.31867740600904) (fma x x 424.31867740600904))
          (+ (* 107.65265 107.65265) (* x x)))
         (+ (* 737.86223 737.86223) (* x x)))
        t_0)
       t_0))))))

double code(double x) {
	double t_0 = (12194.217 * 12194.217) + (x * x);
	return 1.0 - sqrt((((((((((x * x) * 3.5041384e+16) * x) * x) * x) * x) * x) * x) / (((((fma(x, x, 424.31867740600904) * fma(x, x, 424.31867740600904)) * ((107.65265 * 107.65265) + (x * x))) * ((737.86223 * 737.86223) + (x * x))) * t_0) * t_0)));
}

function code(x)
	t_0 = Float64(Float64(12194.217 * 12194.217) + Float64(x * x))
	return Float64(1.0 - sqrt(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(x * x) * 3.5041384e+16) * x) * x) * x) * x) * x) * x) / Float64(Float64(Float64(Float64(Float64(fma(x, x, 424.31867740600904) * fma(x, x, 424.31867740600904)) * Float64(Float64(107.65265 * 107.65265) + Float64(x * x))) * Float64(Float64(737.86223 * 737.86223) + Float64(x * x))) * t_0) * t_0))))
end

code[x_] := Block[{t$95$0 = N[(N[(12194.217 * 12194.217), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]}, N[(1.0 - N[Sqrt[N[(N[(N[(N[(N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 3.5041384e+16), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] / N[(N[(N[(N[(N[(N[(x * x + 424.31867740600904), $MachinePrecision] * N[(x * x + 424.31867740600904), $MachinePrecision]), $MachinePrecision] * N[(N[(107.65265 * 107.65265), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(737.86223 * 737.86223), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

\begin{array}{l}

\\
\begin{array}{l}
t_0 := 12194.217 \cdot 12194.217 + x \cdot x\\
1 - \sqrt{\frac{\left(\left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot 3.5041384 \cdot 10^{+16}\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x}{\left(\left(\left(\left(\mathsf{fma}\left(x, x, 424.31867740600904\right) \cdot \mathsf{fma}\left(x, x, 424.31867740600904\right)\right) \cdot \left(107.65265 \cdot 107.65265 + x \cdot x\right)\right) \cdot \left(737.86223 \cdot 737.86223 + x \cdot x\right)\right) \cdot t\_0\right) \cdot t\_0}}
\end{array}
\end{array}

Derivation

Initial program 100.0%
\[1 - \sqrt{\frac{\left(\left(\left(\left(\left(\left(\left(\left(3.5041384 \cdot 10^{+16}\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x}{\left(\left(\left(\left(\left(20.598997 \cdot 20.598997 + x \cdot x\right) \cdot \left(20.598997 \cdot 20.598997 + x \cdot x\right)\right) \cdot \left(107.65265 \cdot 107.65265 + x \cdot x\right)\right) \cdot \left(737.86223 \cdot 737.86223 + x \cdot x\right)\right) \cdot \left(12194.217 \cdot 12194.217 + x \cdot x\right)\right) \cdot \left(12194.217 \cdot 12194.217 + x \cdot x\right)}} \]
Add Preprocessing
Step-by-step derivation
1. lift-+.f64N/A
  \[\leadsto 1 - \sqrt{\frac{\left(\left(\left(\left(\left(\left(\left(\left(\frac{7890618196247323}{2251799813685248} \cdot 10000000000000000\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x}{\left(\left(\left(\left(\color{blue}{\left(\frac{5798102200837873}{281474976710656} \cdot \frac{5798102200837873}{281474976710656} + x \cdot x\right)} \cdot \left(\frac{5798102200837873}{281474976710656} \cdot \frac{5798102200837873}{281474976710656} + x \cdot x\right)\right) \cdot \left(\frac{29591335108975}{274877906944} \cdot \frac{29591335108975}{274877906944} + x \cdot x\right)\right) \cdot \left(\frac{3245152406326917}{4398046511104} \cdot \frac{3245152406326917}{4398046511104} + x \cdot x\right)\right) \cdot \left(\frac{3351920845780943}{274877906944} \cdot \frac{3351920845780943}{274877906944} + x \cdot x\right)\right) \cdot \left(\frac{3351920845780943}{274877906944} \cdot \frac{3351920845780943}{274877906944} + x \cdot x\right)}} \]
2. +-commutativeN/A
  \[\leadsto 1 - \sqrt{\frac{\left(\left(\left(\left(\left(\left(\left(\left(\frac{7890618196247323}{2251799813685248} \cdot 10000000000000000\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x}{\left(\left(\left(\left(\color{blue}{\left(x \cdot x + \frac{5798102200837873}{281474976710656} \cdot \frac{5798102200837873}{281474976710656}\right)} \cdot \left(\frac{5798102200837873}{281474976710656} \cdot \frac{5798102200837873}{281474976710656} + x \cdot x\right)\right) \cdot \left(\frac{29591335108975}{274877906944} \cdot \frac{29591335108975}{274877906944} + x \cdot x\right)\right) \cdot \left(\frac{3245152406326917}{4398046511104} \cdot \frac{3245152406326917}{4398046511104} + x \cdot x\right)\right) \cdot \left(\frac{3351920845780943}{274877906944} \cdot \frac{3351920845780943}{274877906944} + x \cdot x\right)\right) \cdot \left(\frac{3351920845780943}{274877906944} \cdot \frac{3351920845780943}{274877906944} + x \cdot x\right)}} \]
3. lift-*.f64N/A
  \[\leadsto 1 - \sqrt{\frac{\left(\left(\left(\left(\left(\left(\left(\left(\frac{7890618196247323}{2251799813685248} \cdot 10000000000000000\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x}{\left(\left(\left(\left(\left(\color{blue}{x \cdot x} + \frac{5798102200837873}{281474976710656} \cdot \frac{5798102200837873}{281474976710656}\right) \cdot \left(\frac{5798102200837873}{281474976710656} \cdot \frac{5798102200837873}{281474976710656} + x \cdot x\right)\right) \cdot \left(\frac{29591335108975}{274877906944} \cdot \frac{29591335108975}{274877906944} + x \cdot x\right)\right) \cdot \left(\frac{3245152406326917}{4398046511104} \cdot \frac{3245152406326917}{4398046511104} + x \cdot x\right)\right) \cdot \left(\frac{3351920845780943}{274877906944} \cdot \frac{3351920845780943}{274877906944} + x \cdot x\right)\right) \cdot \left(\frac{3351920845780943}{274877906944} \cdot \frac{3351920845780943}{274877906944} + x \cdot x\right)}} \]
4. lower-fma.f64100.0
  \[\leadsto 1 - \sqrt{\frac{\left(\left(\left(\left(\left(\left(\left(\left(3.5041384 \cdot 10^{+16}\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x}{\left(\left(\left(\left(\color{blue}{\mathsf{fma}\left(x, x, 20.598997 \cdot 20.598997\right)} \cdot \left(20.598997 \cdot 20.598997 + x \cdot x\right)\right) \cdot \left(107.65265 \cdot 107.65265 + x \cdot x\right)\right) \cdot \left(737.86223 \cdot 737.86223 + x \cdot x\right)\right) \cdot \left(12194.217 \cdot 12194.217 + x \cdot x\right)\right) \cdot \left(12194.217 \cdot 12194.217 + x \cdot x\right)}} \]
5. lift-*.f64N/A
  \[\leadsto 1 - \sqrt{\frac{\left(\left(\left(\left(\left(\left(\left(\left(\frac{7890618196247323}{2251799813685248} \cdot 10000000000000000\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x}{\left(\left(\left(\left(\mathsf{fma}\left(x, x, \color{blue}{\frac{5798102200837873}{281474976710656} \cdot \frac{5798102200837873}{281474976710656}}\right) \cdot \left(\frac{5798102200837873}{281474976710656} \cdot \frac{5798102200837873}{281474976710656} + x \cdot x\right)\right) \cdot \left(\frac{29591335108975}{274877906944} \cdot \frac{29591335108975}{274877906944} + x \cdot x\right)\right) \cdot \left(\frac{3245152406326917}{4398046511104} \cdot \frac{3245152406326917}{4398046511104} + x \cdot x\right)\right) \cdot \left(\frac{3351920845780943}{274877906944} \cdot \frac{3351920845780943}{274877906944} + x \cdot x\right)\right) \cdot \left(\frac{3351920845780943}{274877906944} \cdot \frac{3351920845780943}{274877906944} + x \cdot x\right)}} \]
6. metadata-eval100.0
  \[\leadsto 1 - \sqrt{\frac{\left(\left(\left(\left(\left(\left(\left(\left(3.5041384 \cdot 10^{+16}\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x}{\left(\left(\left(\left(\mathsf{fma}\left(x, x, \color{blue}{424.31867740600904}\right) \cdot \left(20.598997 \cdot 20.598997 + x \cdot x\right)\right) \cdot \left(107.65265 \cdot 107.65265 + x \cdot x\right)\right) \cdot \left(737.86223 \cdot 737.86223 + x \cdot x\right)\right) \cdot \left(12194.217 \cdot 12194.217 + x \cdot x\right)\right) \cdot \left(12194.217 \cdot 12194.217 + x \cdot x\right)}} \]
7. lift-+.f64N/A
  \[\leadsto 1 - \sqrt{\frac{\left(\left(\left(\left(\left(\left(\left(\left(\frac{7890618196247323}{2251799813685248} \cdot 10000000000000000\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x}{\left(\left(\left(\left(\mathsf{fma}\left(x, x, \frac{33617989131360986569943231164129}{79228162514264337593543950336}\right) \cdot \color{blue}{\left(\frac{5798102200837873}{281474976710656} \cdot \frac{5798102200837873}{281474976710656} + x \cdot x\right)}\right) \cdot \left(\frac{29591335108975}{274877906944} \cdot \frac{29591335108975}{274877906944} + x \cdot x\right)\right) \cdot \left(\frac{3245152406326917}{4398046511104} \cdot \frac{3245152406326917}{4398046511104} + x \cdot x\right)\right) \cdot \left(\frac{3351920845780943}{274877906944} \cdot \frac{3351920845780943}{274877906944} + x \cdot x\right)\right) \cdot \left(\frac{3351920845780943}{274877906944} \cdot \frac{3351920845780943}{274877906944} + x \cdot x\right)}} \]
8. +-commutativeN/A
  \[\leadsto 1 - \sqrt{\frac{\left(\left(\left(\left(\left(\left(\left(\left(\frac{7890618196247323}{2251799813685248} \cdot 10000000000000000\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x}{\left(\left(\left(\left(\mathsf{fma}\left(x, x, \frac{33617989131360986569943231164129}{79228162514264337593543950336}\right) \cdot \color{blue}{\left(x \cdot x + \frac{5798102200837873}{281474976710656} \cdot \frac{5798102200837873}{281474976710656}\right)}\right) \cdot \left(\frac{29591335108975}{274877906944} \cdot \frac{29591335108975}{274877906944} + x \cdot x\right)\right) \cdot \left(\frac{3245152406326917}{4398046511104} \cdot \frac{3245152406326917}{4398046511104} + x \cdot x\right)\right) \cdot \left(\frac{3351920845780943}{274877906944} \cdot \frac{3351920845780943}{274877906944} + x \cdot x\right)\right) \cdot \left(\frac{3351920845780943}{274877906944} \cdot \frac{3351920845780943}{274877906944} + x \cdot x\right)}} \]
9. lift-*.f64N/A
  \[\leadsto 1 - \sqrt{\frac{\left(\left(\left(\left(\left(\left(\left(\left(\frac{7890618196247323}{2251799813685248} \cdot 10000000000000000\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x}{\left(\left(\left(\left(\mathsf{fma}\left(x, x, \frac{33617989131360986569943231164129}{79228162514264337593543950336}\right) \cdot \left(\color{blue}{x \cdot x} + \frac{5798102200837873}{281474976710656} \cdot \frac{5798102200837873}{281474976710656}\right)\right) \cdot \left(\frac{29591335108975}{274877906944} \cdot \frac{29591335108975}{274877906944} + x \cdot x\right)\right) \cdot \left(\frac{3245152406326917}{4398046511104} \cdot \frac{3245152406326917}{4398046511104} + x \cdot x\right)\right) \cdot \left(\frac{3351920845780943}{274877906944} \cdot \frac{3351920845780943}{274877906944} + x \cdot x\right)\right) \cdot \left(\frac{3351920845780943}{274877906944} \cdot \frac{3351920845780943}{274877906944} + x \cdot x\right)}} \]
10. lower-fma.f64100.0
  \[\leadsto 1 - \sqrt{\frac{\left(\left(\left(\left(\left(\left(\left(\left(3.5041384 \cdot 10^{+16}\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x}{\left(\left(\left(\left(\mathsf{fma}\left(x, x, 424.31867740600904\right) \cdot \color{blue}{\mathsf{fma}\left(x, x, 20.598997 \cdot 20.598997\right)}\right) \cdot \left(107.65265 \cdot 107.65265 + x \cdot x\right)\right) \cdot \left(737.86223 \cdot 737.86223 + x \cdot x\right)\right) \cdot \left(12194.217 \cdot 12194.217 + x \cdot x\right)\right) \cdot \left(12194.217 \cdot 12194.217 + x \cdot x\right)}} \]
11. lift-*.f64N/A
  \[\leadsto 1 - \sqrt{\frac{\left(\left(\left(\left(\left(\left(\left(\left(\frac{7890618196247323}{2251799813685248} \cdot 10000000000000000\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x}{\left(\left(\left(\left(\mathsf{fma}\left(x, x, \frac{33617989131360986569943231164129}{79228162514264337593543950336}\right) \cdot \mathsf{fma}\left(x, x, \color{blue}{\frac{5798102200837873}{281474976710656} \cdot \frac{5798102200837873}{281474976710656}}\right)\right) \cdot \left(\frac{29591335108975}{274877906944} \cdot \frac{29591335108975}{274877906944} + x \cdot x\right)\right) \cdot \left(\frac{3245152406326917}{4398046511104} \cdot \frac{3245152406326917}{4398046511104} + x \cdot x\right)\right) \cdot \left(\frac{3351920845780943}{274877906944} \cdot \frac{3351920845780943}{274877906944} + x \cdot x\right)\right) \cdot \left(\frac{3351920845780943}{274877906944} \cdot \frac{3351920845780943}{274877906944} + x \cdot x\right)}} \]
12. metadata-eval100.0
  \[\leadsto 1 - \sqrt{\frac{\left(\left(\left(\left(\left(\left(\left(\left(3.5041384 \cdot 10^{+16}\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x}{\left(\left(\left(\left(\mathsf{fma}\left(x, x, 424.31867740600904\right) \cdot \mathsf{fma}\left(x, x, \color{blue}{424.31867740600904}\right)\right) \cdot \left(107.65265 \cdot 107.65265 + x \cdot x\right)\right) \cdot \left(737.86223 \cdot 737.86223 + x \cdot x\right)\right) \cdot \left(12194.217 \cdot 12194.217 + x \cdot x\right)\right) \cdot \left(12194.217 \cdot 12194.217 + x \cdot x\right)}} \]
Applied rewrites100.0%
\[\leadsto 1 - \sqrt{\frac{\left(\left(\left(\left(\left(\left(\left(\left(3.5041384 \cdot 10^{+16}\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x}{\left(\left(\left(\color{blue}{\left(\mathsf{fma}\left(x, x, 424.31867740600904\right) \cdot \mathsf{fma}\left(x, x, 424.31867740600904\right)\right)} \cdot \left(107.65265 \cdot 107.65265 + x \cdot x\right)\right) \cdot \left(737.86223 \cdot 737.86223 + x \cdot x\right)\right) \cdot \left(12194.217 \cdot 12194.217 + x \cdot x\right)\right) \cdot \left(12194.217 \cdot 12194.217 + x \cdot x\right)}} \]
Step-by-step derivation
1. lift-*.f64N/A
  \[\leadsto 1 - \sqrt{\frac{\left(\left(\left(\left(\left(\color{blue}{\left(\left(\left(\frac{7890618196247323}{2251799813685248} \cdot 10000000000000000\right) \cdot x\right) \cdot x\right)} \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x}{\left(\left(\left(\left(\mathsf{fma}\left(x, x, \frac{33617989131360986569943231164129}{79228162514264337593543950336}\right) \cdot \mathsf{fma}\left(x, x, \frac{33617989131360986569943231164129}{79228162514264337593543950336}\right)\right) \cdot \left(\frac{29591335108975}{274877906944} \cdot \frac{29591335108975}{274877906944} + x \cdot x\right)\right) \cdot \left(\frac{3245152406326917}{4398046511104} \cdot \frac{3245152406326917}{4398046511104} + x \cdot x\right)\right) \cdot \left(\frac{3351920845780943}{274877906944} \cdot \frac{3351920845780943}{274877906944} + x \cdot x\right)\right) \cdot \left(\frac{3351920845780943}{274877906944} \cdot \frac{3351920845780943}{274877906944} + x \cdot x\right)}} \]
2. lift-*.f64N/A
  \[\leadsto 1 - \sqrt{\frac{\left(\left(\left(\left(\left(\left(\color{blue}{\left(\left(\frac{7890618196247323}{2251799813685248} \cdot 10000000000000000\right) \cdot x\right)} \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x}{\left(\left(\left(\left(\mathsf{fma}\left(x, x, \frac{33617989131360986569943231164129}{79228162514264337593543950336}\right) \cdot \mathsf{fma}\left(x, x, \frac{33617989131360986569943231164129}{79228162514264337593543950336}\right)\right) \cdot \left(\frac{29591335108975}{274877906944} \cdot \frac{29591335108975}{274877906944} + x \cdot x\right)\right) \cdot \left(\frac{3245152406326917}{4398046511104} \cdot \frac{3245152406326917}{4398046511104} + x \cdot x\right)\right) \cdot \left(\frac{3351920845780943}{274877906944} \cdot \frac{3351920845780943}{274877906944} + x \cdot x\right)\right) \cdot \left(\frac{3351920845780943}{274877906944} \cdot \frac{3351920845780943}{274877906944} + x \cdot x\right)}} \]
3. lift-*.f64N/A
  \[\leadsto 1 - \sqrt{\frac{\left(\left(\left(\left(\left(\left(\left(\color{blue}{\left(\frac{7890618196247323}{2251799813685248} \cdot 10000000000000000\right)} \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x}{\left(\left(\left(\left(\mathsf{fma}\left(x, x, \frac{33617989131360986569943231164129}{79228162514264337593543950336}\right) \cdot \mathsf{fma}\left(x, x, \frac{33617989131360986569943231164129}{79228162514264337593543950336}\right)\right) \cdot \left(\frac{29591335108975}{274877906944} \cdot \frac{29591335108975}{274877906944} + x \cdot x\right)\right) \cdot \left(\frac{3245152406326917}{4398046511104} \cdot \frac{3245152406326917}{4398046511104} + x \cdot x\right)\right) \cdot \left(\frac{3351920845780943}{274877906944} \cdot \frac{3351920845780943}{274877906944} + x \cdot x\right)\right) \cdot \left(\frac{3351920845780943}{274877906944} \cdot \frac{3351920845780943}{274877906944} + x \cdot x\right)}} \]
4. metadata-evalN/A
  \[\leadsto 1 - \sqrt{\frac{\left(\left(\left(\left(\left(\left(\left(\color{blue}{\frac{1204012786292621307373046875}{34359738368}} \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x}{\left(\left(\left(\left(\mathsf{fma}\left(x, x, \frac{33617989131360986569943231164129}{79228162514264337593543950336}\right) \cdot \mathsf{fma}\left(x, x, \frac{33617989131360986569943231164129}{79228162514264337593543950336}\right)\right) \cdot \left(\frac{29591335108975}{274877906944} \cdot \frac{29591335108975}{274877906944} + x \cdot x\right)\right) \cdot \left(\frac{3245152406326917}{4398046511104} \cdot \frac{3245152406326917}{4398046511104} + x \cdot x\right)\right) \cdot \left(\frac{3351920845780943}{274877906944} \cdot \frac{3351920845780943}{274877906944} + x \cdot x\right)\right) \cdot \left(\frac{3351920845780943}{274877906944} \cdot \frac{3351920845780943}{274877906944} + x \cdot x\right)}} \]
5. associate-*l*N/A
  \[\leadsto 1 - \sqrt{\frac{\left(\left(\left(\left(\left(\color{blue}{\left(\frac{1204012786292621307373046875}{34359738368} \cdot \left(x \cdot x\right)\right)} \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x}{\left(\left(\left(\left(\mathsf{fma}\left(x, x, \frac{33617989131360986569943231164129}{79228162514264337593543950336}\right) \cdot \mathsf{fma}\left(x, x, \frac{33617989131360986569943231164129}{79228162514264337593543950336}\right)\right) \cdot \left(\frac{29591335108975}{274877906944} \cdot \frac{29591335108975}{274877906944} + x \cdot x\right)\right) \cdot \left(\frac{3245152406326917}{4398046511104} \cdot \frac{3245152406326917}{4398046511104} + x \cdot x\right)\right) \cdot \left(\frac{3351920845780943}{274877906944} \cdot \frac{3351920845780943}{274877906944} + x \cdot x\right)\right) \cdot \left(\frac{3351920845780943}{274877906944} \cdot \frac{3351920845780943}{274877906944} + x \cdot x\right)}} \]
6. lift-*.f64N/A
  \[\leadsto 1 - \sqrt{\frac{\left(\left(\left(\left(\left(\left(\frac{1204012786292621307373046875}{34359738368} \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x}{\left(\left(\left(\left(\mathsf{fma}\left(x, x, \frac{33617989131360986569943231164129}{79228162514264337593543950336}\right) \cdot \mathsf{fma}\left(x, x, \frac{33617989131360986569943231164129}{79228162514264337593543950336}\right)\right) \cdot \left(\frac{29591335108975}{274877906944} \cdot \frac{29591335108975}{274877906944} + x \cdot x\right)\right) \cdot \left(\frac{3245152406326917}{4398046511104} \cdot \frac{3245152406326917}{4398046511104} + x \cdot x\right)\right) \cdot \left(\frac{3351920845780943}{274877906944} \cdot \frac{3351920845780943}{274877906944} + x \cdot x\right)\right) \cdot \left(\frac{3351920845780943}{274877906944} \cdot \frac{3351920845780943}{274877906944} + x \cdot x\right)}} \]
7. *-commutativeN/A
  \[\leadsto 1 - \sqrt{\frac{\left(\left(\left(\left(\left(\color{blue}{\left(\left(x \cdot x\right) \cdot \frac{1204012786292621307373046875}{34359738368}\right)} \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x}{\left(\left(\left(\left(\mathsf{fma}\left(x, x, \frac{33617989131360986569943231164129}{79228162514264337593543950336}\right) \cdot \mathsf{fma}\left(x, x, \frac{33617989131360986569943231164129}{79228162514264337593543950336}\right)\right) \cdot \left(\frac{29591335108975}{274877906944} \cdot \frac{29591335108975}{274877906944} + x \cdot x\right)\right) \cdot \left(\frac{3245152406326917}{4398046511104} \cdot \frac{3245152406326917}{4398046511104} + x \cdot x\right)\right) \cdot \left(\frac{3351920845780943}{274877906944} \cdot \frac{3351920845780943}{274877906944} + x \cdot x\right)\right) \cdot \left(\frac{3351920845780943}{274877906944} \cdot \frac{3351920845780943}{274877906944} + x \cdot x\right)}} \]
8. lower-*.f64100.0
  \[\leadsto 1 - \sqrt{\frac{\left(\left(\left(\left(\left(\color{blue}{\left(\left(x \cdot x\right) \cdot 3.5041384 \cdot 10^{+16}\right)} \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x}{\left(\left(\left(\left(\mathsf{fma}\left(x, x, 424.31867740600904\right) \cdot \mathsf{fma}\left(x, x, 424.31867740600904\right)\right) \cdot \left(107.65265 \cdot 107.65265 + x \cdot x\right)\right) \cdot \left(737.86223 \cdot 737.86223 + x \cdot x\right)\right) \cdot \left(12194.217 \cdot 12194.217 + x \cdot x\right)\right) \cdot \left(12194.217 \cdot 12194.217 + x \cdot x\right)}} \]
Applied rewrites100.0%
\[\leadsto 1 - \sqrt{\frac{\left(\left(\left(\left(\left(\color{blue}{\left(\left(x \cdot x\right) \cdot 3.5041384 \cdot 10^{+16}\right)} \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x}{\left(\left(\left(\left(\mathsf{fma}\left(x, x, 424.31867740600904\right) \cdot \mathsf{fma}\left(x, x, 424.31867740600904\right)\right) \cdot \left(107.65265 \cdot 107.65265 + x \cdot x\right)\right) \cdot \left(737.86223 \cdot 737.86223 + x \cdot x\right)\right) \cdot \left(12194.217 \cdot 12194.217 + x \cdot x\right)\right) \cdot \left(12194.217 \cdot 12194.217 + x \cdot x\right)}} \]
Add Preprocessing

Alternative 3: 98.9% accurate, 1.7× speedup?

\[\begin{array}{l} \\ 1 - \sqrt{\frac{\left(\left(\left(\left(\left(\left(\left(\left(3.5041384 \cdot 10^{+16}\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(1.2315361913695067 \cdot 10^{+22}, x \cdot x, 1.499525677491431 \cdot 10^{+26}\right), x \cdot x, 1.2061015046255844 \cdot 10^{+29}\right), x \cdot x, 2.5118863397964406 \cdot 10^{+31}\right)}} \end{array} \]

(FPCore (x)
 :precision binary64
 (-
  1.0
  (sqrt
   (/
    (* (* (* (* (* (* (* (* (* 3.5041384 1e+16) x) x) x) x) x) x) x) x)
    (fma
     (fma
      (fma 1.2315361913695067e+22 (* x x) 1.499525677491431e+26)
      (* x x)
      1.2061015046255844e+29)
     (* x x)
     2.5118863397964406e+31)))))

double code(double x) {
	return 1.0 - sqrt(((((((((((3.5041384 * 1e+16) * x) * x) * x) * x) * x) * x) * x) * x) / fma(fma(fma(1.2315361913695067e+22, (x * x), 1.499525677491431e+26), (x * x), 1.2061015046255844e+29), (x * x), 2.5118863397964406e+31)));
}

function code(x)
	return Float64(1.0 - sqrt(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(3.5041384 * 1e+16) * x) * x) * x) * x) * x) * x) * x) * x) / fma(fma(fma(1.2315361913695067e+22, Float64(x * x), 1.499525677491431e+26), Float64(x * x), 1.2061015046255844e+29), Float64(x * x), 2.5118863397964406e+31))))
end

code[x_] := N[(1.0 - N[Sqrt[N[(N[(N[(N[(N[(N[(N[(N[(N[(N[(3.5041384 * 1e+16), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] / N[(N[(N[(1.2315361913695067e+22 * N[(x * x), $MachinePrecision] + 1.499525677491431e+26), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.2061015046255844e+29), $MachinePrecision] * N[(x * x), $MachinePrecision] + 2.5118863397964406e+31), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

\begin{array}{l}

\\
1 - \sqrt{\frac{\left(\left(\left(\left(\left(\left(\left(\left(3.5041384 \cdot 10^{+16}\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(1.2315361913695067 \cdot 10^{+22}, x \cdot x, 1.499525677491431 \cdot 10^{+26}\right), x \cdot x, 1.2061015046255844 \cdot 10^{+29}\right), x \cdot x, 2.5118863397964406 \cdot 10^{+31}\right)}}
\end{array}

Derivation

Initial program 100.0%
\[1 - \sqrt{\frac{\left(\left(\left(\left(\left(\left(\left(\left(3.5041384 \cdot 10^{+16}\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x}{\left(\left(\left(\left(\left(20.598997 \cdot 20.598997 + x \cdot x\right) \cdot \left(20.598997 \cdot 20.598997 + x \cdot x\right)\right) \cdot \left(107.65265 \cdot 107.65265 + x \cdot x\right)\right) \cdot \left(737.86223 \cdot 737.86223 + x \cdot x\right)\right) \cdot \left(12194.217 \cdot 12194.217 + x \cdot x\right)\right) \cdot \left(12194.217 \cdot 12194.217 + x \cdot x\right)}} \]
Add Preprocessing
Taylor expanded in x around 0
\[\leadsto 1 - \sqrt{\frac{\left(\left(\left(\left(\left(\left(\left(\left(\frac{7890618196247323}{2251799813685248} \cdot 10000000000000000\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x}{\color{blue}{\frac{1315581624446765670896680980461586616360623897704795715865349413371368441985234365242636457474613230454287489183683904180293545031734252213064606094900258918769550433855941923780605625}{52374249726338269920211035149241586435466272736689036631732661889538140742474792878132321477214466514414186946040961136147476104734166288853256441430016} + {x}^{2} \cdot \left(\frac{83603027247721846947400001807748686585593809928679218857034661437178107499566497870512394123262412067236255967698478030577310303374589078484741568855549153099}{693167423530203714894603546035770925859109268843954143792619895153655326951406405759993601526034894524347802740350892957243539456} + {x}^{2} \cdot \left(\frac{1375664026387268463856426913120470294500076676243929134801266166161504374607081765151624544909447479868312852629968476540989780957035}{9173994463960286046443283581208347763186259956673124494950355357547691504353939232280074212440502746218496} + \frac{1495291905389808157732411293717000243787935966429477398989981141751109634705859831631687929428611871965145}{121416805764108066932466369176469931665150427440758720078238275608681517825325531136} \cdot {x}^{2}\right)\right)}}} \]
Step-by-step derivation
1. +-commutativeN/A
  \[\leadsto 1 - \sqrt{\frac{\left(\left(\left(\left(\left(\left(\left(\left(\frac{7890618196247323}{2251799813685248} \cdot 10000000000000000\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x}{\color{blue}{{x}^{2} \cdot \left(\frac{83603027247721846947400001807748686585593809928679218857034661437178107499566497870512394123262412067236255967698478030577310303374589078484741568855549153099}{693167423530203714894603546035770925859109268843954143792619895153655326951406405759993601526034894524347802740350892957243539456} + {x}^{2} \cdot \left(\frac{1375664026387268463856426913120470294500076676243929134801266166161504374607081765151624544909447479868312852629968476540989780957035}{9173994463960286046443283581208347763186259956673124494950355357547691504353939232280074212440502746218496} + \frac{1495291905389808157732411293717000243787935966429477398989981141751109634705859831631687929428611871965145}{121416805764108066932466369176469931665150427440758720078238275608681517825325531136} \cdot {x}^{2}\right)\right) + \frac{1315581624446765670896680980461586616360623897704795715865349413371368441985234365242636457474613230454287489183683904180293545031734252213064606094900258918769550433855941923780605625}{52374249726338269920211035149241586435466272736689036631732661889538140742474792878132321477214466514414186946040961136147476104734166288853256441430016}}}} \]
2. *-commutativeN/A
  \[\leadsto 1 - \sqrt{\frac{\left(\left(\left(\left(\left(\left(\left(\left(\frac{7890618196247323}{2251799813685248} \cdot 10000000000000000\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x}{\color{blue}{\left(\frac{83603027247721846947400001807748686585593809928679218857034661437178107499566497870512394123262412067236255967698478030577310303374589078484741568855549153099}{693167423530203714894603546035770925859109268843954143792619895153655326951406405759993601526034894524347802740350892957243539456} + {x}^{2} \cdot \left(\frac{1375664026387268463856426913120470294500076676243929134801266166161504374607081765151624544909447479868312852629968476540989780957035}{9173994463960286046443283581208347763186259956673124494950355357547691504353939232280074212440502746218496} + \frac{1495291905389808157732411293717000243787935966429477398989981141751109634705859831631687929428611871965145}{121416805764108066932466369176469931665150427440758720078238275608681517825325531136} \cdot {x}^{2}\right)\right) \cdot {x}^{2}} + \frac{1315581624446765670896680980461586616360623897704795715865349413371368441985234365242636457474613230454287489183683904180293545031734252213064606094900258918769550433855941923780605625}{52374249726338269920211035149241586435466272736689036631732661889538140742474792878132321477214466514414186946040961136147476104734166288853256441430016}}} \]
3. lower-fma.f64N/A
  \[\leadsto 1 - \sqrt{\frac{\left(\left(\left(\left(\left(\left(\left(\left(\frac{7890618196247323}{2251799813685248} \cdot 10000000000000000\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x}{\color{blue}{\mathsf{fma}\left(\frac{83603027247721846947400001807748686585593809928679218857034661437178107499566497870512394123262412067236255967698478030577310303374589078484741568855549153099}{693167423530203714894603546035770925859109268843954143792619895153655326951406405759993601526034894524347802740350892957243539456} + {x}^{2} \cdot \left(\frac{1375664026387268463856426913120470294500076676243929134801266166161504374607081765151624544909447479868312852629968476540989780957035}{9173994463960286046443283581208347763186259956673124494950355357547691504353939232280074212440502746218496} + \frac{1495291905389808157732411293717000243787935966429477398989981141751109634705859831631687929428611871965145}{121416805764108066932466369176469931665150427440758720078238275608681517825325531136} \cdot {x}^{2}\right), {x}^{2}, \frac{1315581624446765670896680980461586616360623897704795715865349413371368441985234365242636457474613230454287489183683904180293545031734252213064606094900258918769550433855941923780605625}{52374249726338269920211035149241586435466272736689036631732661889538140742474792878132321477214466514414186946040961136147476104734166288853256441430016}\right)}}} \]
4. +-commutativeN/A
  \[\leadsto 1 - \sqrt{\frac{\left(\left(\left(\left(\left(\left(\left(\left(\frac{7890618196247323}{2251799813685248} \cdot 10000000000000000\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x}{\mathsf{fma}\left(\color{blue}{{x}^{2} \cdot \left(\frac{1375664026387268463856426913120470294500076676243929134801266166161504374607081765151624544909447479868312852629968476540989780957035}{9173994463960286046443283581208347763186259956673124494950355357547691504353939232280074212440502746218496} + \frac{1495291905389808157732411293717000243787935966429477398989981141751109634705859831631687929428611871965145}{121416805764108066932466369176469931665150427440758720078238275608681517825325531136} \cdot {x}^{2}\right) + \frac{83603027247721846947400001807748686585593809928679218857034661437178107499566497870512394123262412067236255967698478030577310303374589078484741568855549153099}{693167423530203714894603546035770925859109268843954143792619895153655326951406405759993601526034894524347802740350892957243539456}}, {x}^{2}, \frac{1315581624446765670896680980461586616360623897704795715865349413371368441985234365242636457474613230454287489183683904180293545031734252213064606094900258918769550433855941923780605625}{52374249726338269920211035149241586435466272736689036631732661889538140742474792878132321477214466514414186946040961136147476104734166288853256441430016}\right)}} \]
5. *-commutativeN/A
  \[\leadsto 1 - \sqrt{\frac{\left(\left(\left(\left(\left(\left(\left(\left(\frac{7890618196247323}{2251799813685248} \cdot 10000000000000000\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x}{\mathsf{fma}\left(\color{blue}{\left(\frac{1375664026387268463856426913120470294500076676243929134801266166161504374607081765151624544909447479868312852629968476540989780957035}{9173994463960286046443283581208347763186259956673124494950355357547691504353939232280074212440502746218496} + \frac{1495291905389808157732411293717000243787935966429477398989981141751109634705859831631687929428611871965145}{121416805764108066932466369176469931665150427440758720078238275608681517825325531136} \cdot {x}^{2}\right) \cdot {x}^{2}} + \frac{83603027247721846947400001807748686585593809928679218857034661437178107499566497870512394123262412067236255967698478030577310303374589078484741568855549153099}{693167423530203714894603546035770925859109268843954143792619895153655326951406405759993601526034894524347802740350892957243539456}, {x}^{2}, \frac{1315581624446765670896680980461586616360623897704795715865349413371368441985234365242636457474613230454287489183683904180293545031734252213064606094900258918769550433855941923780605625}{52374249726338269920211035149241586435466272736689036631732661889538140742474792878132321477214466514414186946040961136147476104734166288853256441430016}\right)}} \]
6. lower-fma.f64N/A
  \[\leadsto 1 - \sqrt{\frac{\left(\left(\left(\left(\left(\left(\left(\left(\frac{7890618196247323}{2251799813685248} \cdot 10000000000000000\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1375664026387268463856426913120470294500076676243929134801266166161504374607081765151624544909447479868312852629968476540989780957035}{9173994463960286046443283581208347763186259956673124494950355357547691504353939232280074212440502746218496} + \frac{1495291905389808157732411293717000243787935966429477398989981141751109634705859831631687929428611871965145}{121416805764108066932466369176469931665150427440758720078238275608681517825325531136} \cdot {x}^{2}, {x}^{2}, \frac{83603027247721846947400001807748686585593809928679218857034661437178107499566497870512394123262412067236255967698478030577310303374589078484741568855549153099}{693167423530203714894603546035770925859109268843954143792619895153655326951406405759993601526034894524347802740350892957243539456}\right)}, {x}^{2}, \frac{1315581624446765670896680980461586616360623897704795715865349413371368441985234365242636457474613230454287489183683904180293545031734252213064606094900258918769550433855941923780605625}{52374249726338269920211035149241586435466272736689036631732661889538140742474792878132321477214466514414186946040961136147476104734166288853256441430016}\right)}} \]
7. +-commutativeN/A
  \[\leadsto 1 - \sqrt{\frac{\left(\left(\left(\left(\left(\left(\left(\left(\frac{7890618196247323}{2251799813685248} \cdot 10000000000000000\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1495291905389808157732411293717000243787935966429477398989981141751109634705859831631687929428611871965145}{121416805764108066932466369176469931665150427440758720078238275608681517825325531136} \cdot {x}^{2} + \frac{1375664026387268463856426913120470294500076676243929134801266166161504374607081765151624544909447479868312852629968476540989780957035}{9173994463960286046443283581208347763186259956673124494950355357547691504353939232280074212440502746218496}}, {x}^{2}, \frac{83603027247721846947400001807748686585593809928679218857034661437178107499566497870512394123262412067236255967698478030577310303374589078484741568855549153099}{693167423530203714894603546035770925859109268843954143792619895153655326951406405759993601526034894524347802740350892957243539456}\right), {x}^{2}, \frac{1315581624446765670896680980461586616360623897704795715865349413371368441985234365242636457474613230454287489183683904180293545031734252213064606094900258918769550433855941923780605625}{52374249726338269920211035149241586435466272736689036631732661889538140742474792878132321477214466514414186946040961136147476104734166288853256441430016}\right)}} \]
8. lower-fma.f64N/A
  \[\leadsto 1 - \sqrt{\frac{\left(\left(\left(\left(\left(\left(\left(\left(\frac{7890618196247323}{2251799813685248} \cdot 10000000000000000\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1495291905389808157732411293717000243787935966429477398989981141751109634705859831631687929428611871965145}{121416805764108066932466369176469931665150427440758720078238275608681517825325531136}, {x}^{2}, \frac{1375664026387268463856426913120470294500076676243929134801266166161504374607081765151624544909447479868312852629968476540989780957035}{9173994463960286046443283581208347763186259956673124494950355357547691504353939232280074212440502746218496}\right)}, {x}^{2}, \frac{83603027247721846947400001807748686585593809928679218857034661437178107499566497870512394123262412067236255967698478030577310303374589078484741568855549153099}{693167423530203714894603546035770925859109268843954143792619895153655326951406405759993601526034894524347802740350892957243539456}\right), {x}^{2}, \frac{1315581624446765670896680980461586616360623897704795715865349413371368441985234365242636457474613230454287489183683904180293545031734252213064606094900258918769550433855941923780605625}{52374249726338269920211035149241586435466272736689036631732661889538140742474792878132321477214466514414186946040961136147476104734166288853256441430016}\right)}} \]
9. unpow2N/A
  \[\leadsto 1 - \sqrt{\frac{\left(\left(\left(\left(\left(\left(\left(\left(\frac{7890618196247323}{2251799813685248} \cdot 10000000000000000\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1495291905389808157732411293717000243787935966429477398989981141751109634705859831631687929428611871965145}{121416805764108066932466369176469931665150427440758720078238275608681517825325531136}, \color{blue}{x \cdot x}, \frac{1375664026387268463856426913120470294500076676243929134801266166161504374607081765151624544909447479868312852629968476540989780957035}{9173994463960286046443283581208347763186259956673124494950355357547691504353939232280074212440502746218496}\right), {x}^{2}, \frac{83603027247721846947400001807748686585593809928679218857034661437178107499566497870512394123262412067236255967698478030577310303374589078484741568855549153099}{693167423530203714894603546035770925859109268843954143792619895153655326951406405759993601526034894524347802740350892957243539456}\right), {x}^{2}, \frac{1315581624446765670896680980461586616360623897704795715865349413371368441985234365242636457474613230454287489183683904180293545031734252213064606094900258918769550433855941923780605625}{52374249726338269920211035149241586435466272736689036631732661889538140742474792878132321477214466514414186946040961136147476104734166288853256441430016}\right)}} \]
10. lower-*.f64N/A
  \[\leadsto 1 - \sqrt{\frac{\left(\left(\left(\left(\left(\left(\left(\left(\frac{7890618196247323}{2251799813685248} \cdot 10000000000000000\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1495291905389808157732411293717000243787935966429477398989981141751109634705859831631687929428611871965145}{121416805764108066932466369176469931665150427440758720078238275608681517825325531136}, \color{blue}{x \cdot x}, \frac{1375664026387268463856426913120470294500076676243929134801266166161504374607081765151624544909447479868312852629968476540989780957035}{9173994463960286046443283581208347763186259956673124494950355357547691504353939232280074212440502746218496}\right), {x}^{2}, \frac{83603027247721846947400001807748686585593809928679218857034661437178107499566497870512394123262412067236255967698478030577310303374589078484741568855549153099}{693167423530203714894603546035770925859109268843954143792619895153655326951406405759993601526034894524347802740350892957243539456}\right), {x}^{2}, \frac{1315581624446765670896680980461586616360623897704795715865349413371368441985234365242636457474613230454287489183683904180293545031734252213064606094900258918769550433855941923780605625}{52374249726338269920211035149241586435466272736689036631732661889538140742474792878132321477214466514414186946040961136147476104734166288853256441430016}\right)}} \]
11. unpow2N/A
  \[\leadsto 1 - \sqrt{\frac{\left(\left(\left(\left(\left(\left(\left(\left(\frac{7890618196247323}{2251799813685248} \cdot 10000000000000000\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1495291905389808157732411293717000243787935966429477398989981141751109634705859831631687929428611871965145}{121416805764108066932466369176469931665150427440758720078238275608681517825325531136}, x \cdot x, \frac{1375664026387268463856426913120470294500076676243929134801266166161504374607081765151624544909447479868312852629968476540989780957035}{9173994463960286046443283581208347763186259956673124494950355357547691504353939232280074212440502746218496}\right), \color{blue}{x \cdot x}, \frac{83603027247721846947400001807748686585593809928679218857034661437178107499566497870512394123262412067236255967698478030577310303374589078484741568855549153099}{693167423530203714894603546035770925859109268843954143792619895153655326951406405759993601526034894524347802740350892957243539456}\right), {x}^{2}, \frac{1315581624446765670896680980461586616360623897704795715865349413371368441985234365242636457474613230454287489183683904180293545031734252213064606094900258918769550433855941923780605625}{52374249726338269920211035149241586435466272736689036631732661889538140742474792878132321477214466514414186946040961136147476104734166288853256441430016}\right)}} \]
12. lower-*.f64N/A
  \[\leadsto 1 - \sqrt{\frac{\left(\left(\left(\left(\left(\left(\left(\left(\frac{7890618196247323}{2251799813685248} \cdot 10000000000000000\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1495291905389808157732411293717000243787935966429477398989981141751109634705859831631687929428611871965145}{121416805764108066932466369176469931665150427440758720078238275608681517825325531136}, x \cdot x, \frac{1375664026387268463856426913120470294500076676243929134801266166161504374607081765151624544909447479868312852629968476540989780957035}{9173994463960286046443283581208347763186259956673124494950355357547691504353939232280074212440502746218496}\right), \color{blue}{x \cdot x}, \frac{83603027247721846947400001807748686585593809928679218857034661437178107499566497870512394123262412067236255967698478030577310303374589078484741568855549153099}{693167423530203714894603546035770925859109268843954143792619895153655326951406405759993601526034894524347802740350892957243539456}\right), {x}^{2}, \frac{1315581624446765670896680980461586616360623897704795715865349413371368441985234365242636457474613230454287489183683904180293545031734252213064606094900258918769550433855941923780605625}{52374249726338269920211035149241586435466272736689036631732661889538140742474792878132321477214466514414186946040961136147476104734166288853256441430016}\right)}} \]
13. unpow2N/A
  \[\leadsto 1 - \sqrt{\frac{\left(\left(\left(\left(\left(\left(\left(\left(\frac{7890618196247323}{2251799813685248} \cdot 10000000000000000\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1495291905389808157732411293717000243787935966429477398989981141751109634705859831631687929428611871965145}{121416805764108066932466369176469931665150427440758720078238275608681517825325531136}, x \cdot x, \frac{1375664026387268463856426913120470294500076676243929134801266166161504374607081765151624544909447479868312852629968476540989780957035}{9173994463960286046443283581208347763186259956673124494950355357547691504353939232280074212440502746218496}\right), x \cdot x, \frac{83603027247721846947400001807748686585593809928679218857034661437178107499566497870512394123262412067236255967698478030577310303374589078484741568855549153099}{693167423530203714894603546035770925859109268843954143792619895153655326951406405759993601526034894524347802740350892957243539456}\right), \color{blue}{x \cdot x}, \frac{1315581624446765670896680980461586616360623897704795715865349413371368441985234365242636457474613230454287489183683904180293545031734252213064606094900258918769550433855941923780605625}{52374249726338269920211035149241586435466272736689036631732661889538140742474792878132321477214466514414186946040961136147476104734166288853256441430016}\right)}} \]
14. lower-*.f6498.5
  \[\leadsto 1 - \sqrt{\frac{\left(\left(\left(\left(\left(\left(\left(\left(3.5041384 \cdot 10^{+16}\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(1.2315361913695067 \cdot 10^{+22}, x \cdot x, 1.499525677491431 \cdot 10^{+26}\right), x \cdot x, 1.2061015046255844 \cdot 10^{+29}\right), \color{blue}{x \cdot x}, 2.5118863397964406 \cdot 10^{+31}\right)}} \]
Applied rewrites98.5%
\[\leadsto 1 - \sqrt{\frac{\left(\left(\left(\left(\left(\left(\left(\left(3.5041384 \cdot 10^{+16}\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(1.2315361913695067 \cdot 10^{+22}, x \cdot x, 1.499525677491431 \cdot 10^{+26}\right), x \cdot x, 1.2061015046255844 \cdot 10^{+29}\right), x \cdot x, 2.5118863397964406 \cdot 10^{+31}\right)}}} \]
Add Preprocessing

Alternative 4: 98.7% accurate, 2.0× speedup?

\[\begin{array}{l} \\ 1 - \left(\sqrt{3.5041384 \cdot 10^{+16} \cdot x} \cdot \left(x \cdot x\right)\right) \cdot \sqrt{\left(x \cdot x\right) \cdot \frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(1.499525677491431 \cdot 10^{+26}, x \cdot x, 1.2061015046255844 \cdot 10^{+29}\right), x \cdot x, 2.5118863397964406 \cdot 10^{+31}\right)}} \end{array} \]

(FPCore (x)
 :precision binary64
 (-
  1.0
  (*
   (* (sqrt (* 3.5041384e+16 x)) (* x x))
   (sqrt
    (*
     (* x x)
     (/
      x
      (fma
       (fma 1.499525677491431e+26 (* x x) 1.2061015046255844e+29)
       (* x x)
       2.5118863397964406e+31)))))))

double code(double x) {
	return 1.0 - ((sqrt((3.5041384e+16 * x)) * (x * x)) * sqrt(((x * x) * (x / fma(fma(1.499525677491431e+26, (x * x), 1.2061015046255844e+29), (x * x), 2.5118863397964406e+31)))));
}

function code(x)
	return Float64(1.0 - Float64(Float64(sqrt(Float64(3.5041384e+16 * x)) * Float64(x * x)) * sqrt(Float64(Float64(x * x) * Float64(x / fma(fma(1.499525677491431e+26, Float64(x * x), 1.2061015046255844e+29), Float64(x * x), 2.5118863397964406e+31))))))
end

code[x_] := N[(1.0 - N[(N[(N[Sqrt[N[(3.5041384e+16 * x), $MachinePrecision]], $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(x * x), $MachinePrecision] * N[(x / N[(N[(1.499525677491431e+26 * N[(x * x), $MachinePrecision] + 1.2061015046255844e+29), $MachinePrecision] * N[(x * x), $MachinePrecision] + 2.5118863397964406e+31), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\begin{array}{l}

\\
1 - \left(\sqrt{3.5041384 \cdot 10^{+16} \cdot x} \cdot \left(x \cdot x\right)\right) \cdot \sqrt{\left(x \cdot x\right) \cdot \frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(1.499525677491431 \cdot 10^{+26}, x \cdot x, 1.2061015046255844 \cdot 10^{+29}\right), x \cdot x, 2.5118863397964406 \cdot 10^{+31}\right)}}
\end{array}

Derivation

Initial program 100.0%
\[1 - \sqrt{\frac{\left(\left(\left(\left(\left(\left(\left(\left(3.5041384 \cdot 10^{+16}\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x}{\left(\left(\left(\left(\left(20.598997 \cdot 20.598997 + x \cdot x\right) \cdot \left(20.598997 \cdot 20.598997 + x \cdot x\right)\right) \cdot \left(107.65265 \cdot 107.65265 + x \cdot x\right)\right) \cdot \left(737.86223 \cdot 737.86223 + x \cdot x\right)\right) \cdot \left(12194.217 \cdot 12194.217 + x \cdot x\right)\right) \cdot \left(12194.217 \cdot 12194.217 + x \cdot x\right)}} \]
Add Preprocessing
Taylor expanded in x around 0
\[\leadsto 1 - \sqrt{\frac{\left(\left(\left(\left(\left(\left(\left(\left(\frac{7890618196247323}{2251799813685248} \cdot 10000000000000000\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x}{\color{blue}{\frac{1315581624446765670896680980461586616360623897704795715865349413371368441985234365242636457474613230454287489183683904180293545031734252213064606094900258918769550433855941923780605625}{52374249726338269920211035149241586435466272736689036631732661889538140742474792878132321477214466514414186946040961136147476104734166288853256441430016} + {x}^{2} \cdot \left(\frac{83603027247721846947400001807748686585593809928679218857034661437178107499566497870512394123262412067236255967698478030577310303374589078484741568855549153099}{693167423530203714894603546035770925859109268843954143792619895153655326951406405759993601526034894524347802740350892957243539456} + \frac{1375664026387268463856426913120470294500076676243929134801266166161504374607081765151624544909447479868312852629968476540989780957035}{9173994463960286046443283581208347763186259956673124494950355357547691504353939232280074212440502746218496} \cdot {x}^{2}\right)}}} \]
Step-by-step derivation
1. +-commutativeN/A
  \[\leadsto 1 - \sqrt{\frac{\left(\left(\left(\left(\left(\left(\left(\left(\frac{7890618196247323}{2251799813685248} \cdot 10000000000000000\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x}{\color{blue}{{x}^{2} \cdot \left(\frac{83603027247721846947400001807748686585593809928679218857034661437178107499566497870512394123262412067236255967698478030577310303374589078484741568855549153099}{693167423530203714894603546035770925859109268843954143792619895153655326951406405759993601526034894524347802740350892957243539456} + \frac{1375664026387268463856426913120470294500076676243929134801266166161504374607081765151624544909447479868312852629968476540989780957035}{9173994463960286046443283581208347763186259956673124494950355357547691504353939232280074212440502746218496} \cdot {x}^{2}\right) + \frac{1315581624446765670896680980461586616360623897704795715865349413371368441985234365242636457474613230454287489183683904180293545031734252213064606094900258918769550433855941923780605625}{52374249726338269920211035149241586435466272736689036631732661889538140742474792878132321477214466514414186946040961136147476104734166288853256441430016}}}} \]
2. *-commutativeN/A
  \[\leadsto 1 - \sqrt{\frac{\left(\left(\left(\left(\left(\left(\left(\left(\frac{7890618196247323}{2251799813685248} \cdot 10000000000000000\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x}{\color{blue}{\left(\frac{83603027247721846947400001807748686585593809928679218857034661437178107499566497870512394123262412067236255967698478030577310303374589078484741568855549153099}{693167423530203714894603546035770925859109268843954143792619895153655326951406405759993601526034894524347802740350892957243539456} + \frac{1375664026387268463856426913120470294500076676243929134801266166161504374607081765151624544909447479868312852629968476540989780957035}{9173994463960286046443283581208347763186259956673124494950355357547691504353939232280074212440502746218496} \cdot {x}^{2}\right) \cdot {x}^{2}} + \frac{1315581624446765670896680980461586616360623897704795715865349413371368441985234365242636457474613230454287489183683904180293545031734252213064606094900258918769550433855941923780605625}{52374249726338269920211035149241586435466272736689036631732661889538140742474792878132321477214466514414186946040961136147476104734166288853256441430016}}} \]
3. lower-fma.f64N/A
  \[\leadsto 1 - \sqrt{\frac{\left(\left(\left(\left(\left(\left(\left(\left(\frac{7890618196247323}{2251799813685248} \cdot 10000000000000000\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x}{\color{blue}{\mathsf{fma}\left(\frac{83603027247721846947400001807748686585593809928679218857034661437178107499566497870512394123262412067236255967698478030577310303374589078484741568855549153099}{693167423530203714894603546035770925859109268843954143792619895153655326951406405759993601526034894524347802740350892957243539456} + \frac{1375664026387268463856426913120470294500076676243929134801266166161504374607081765151624544909447479868312852629968476540989780957035}{9173994463960286046443283581208347763186259956673124494950355357547691504353939232280074212440502746218496} \cdot {x}^{2}, {x}^{2}, \frac{1315581624446765670896680980461586616360623897704795715865349413371368441985234365242636457474613230454287489183683904180293545031734252213064606094900258918769550433855941923780605625}{52374249726338269920211035149241586435466272736689036631732661889538140742474792878132321477214466514414186946040961136147476104734166288853256441430016}\right)}}} \]
4. +-commutativeN/A
  \[\leadsto 1 - \sqrt{\frac{\left(\left(\left(\left(\left(\left(\left(\left(\frac{7890618196247323}{2251799813685248} \cdot 10000000000000000\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x}{\mathsf{fma}\left(\color{blue}{\frac{1375664026387268463856426913120470294500076676243929134801266166161504374607081765151624544909447479868312852629968476540989780957035}{9173994463960286046443283581208347763186259956673124494950355357547691504353939232280074212440502746218496} \cdot {x}^{2} + \frac{83603027247721846947400001807748686585593809928679218857034661437178107499566497870512394123262412067236255967698478030577310303374589078484741568855549153099}{693167423530203714894603546035770925859109268843954143792619895153655326951406405759993601526034894524347802740350892957243539456}}, {x}^{2}, \frac{1315581624446765670896680980461586616360623897704795715865349413371368441985234365242636457474613230454287489183683904180293545031734252213064606094900258918769550433855941923780605625}{52374249726338269920211035149241586435466272736689036631732661889538140742474792878132321477214466514414186946040961136147476104734166288853256441430016}\right)}} \]
5. lower-fma.f64N/A
  \[\leadsto 1 - \sqrt{\frac{\left(\left(\left(\left(\left(\left(\left(\left(\frac{7890618196247323}{2251799813685248} \cdot 10000000000000000\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1375664026387268463856426913120470294500076676243929134801266166161504374607081765151624544909447479868312852629968476540989780957035}{9173994463960286046443283581208347763186259956673124494950355357547691504353939232280074212440502746218496}, {x}^{2}, \frac{83603027247721846947400001807748686585593809928679218857034661437178107499566497870512394123262412067236255967698478030577310303374589078484741568855549153099}{693167423530203714894603546035770925859109268843954143792619895153655326951406405759993601526034894524347802740350892957243539456}\right)}, {x}^{2}, \frac{1315581624446765670896680980461586616360623897704795715865349413371368441985234365242636457474613230454287489183683904180293545031734252213064606094900258918769550433855941923780605625}{52374249726338269920211035149241586435466272736689036631732661889538140742474792878132321477214466514414186946040961136147476104734166288853256441430016}\right)}} \]
6. unpow2N/A
  \[\leadsto 1 - \sqrt{\frac{\left(\left(\left(\left(\left(\left(\left(\left(\frac{7890618196247323}{2251799813685248} \cdot 10000000000000000\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1375664026387268463856426913120470294500076676243929134801266166161504374607081765151624544909447479868312852629968476540989780957035}{9173994463960286046443283581208347763186259956673124494950355357547691504353939232280074212440502746218496}, \color{blue}{x \cdot x}, \frac{83603027247721846947400001807748686585593809928679218857034661437178107499566497870512394123262412067236255967698478030577310303374589078484741568855549153099}{693167423530203714894603546035770925859109268843954143792619895153655326951406405759993601526034894524347802740350892957243539456}\right), {x}^{2}, \frac{1315581624446765670896680980461586616360623897704795715865349413371368441985234365242636457474613230454287489183683904180293545031734252213064606094900258918769550433855941923780605625}{52374249726338269920211035149241586435466272736689036631732661889538140742474792878132321477214466514414186946040961136147476104734166288853256441430016}\right)}} \]
7. lower-*.f64N/A
  \[\leadsto 1 - \sqrt{\frac{\left(\left(\left(\left(\left(\left(\left(\left(\frac{7890618196247323}{2251799813685248} \cdot 10000000000000000\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1375664026387268463856426913120470294500076676243929134801266166161504374607081765151624544909447479868312852629968476540989780957035}{9173994463960286046443283581208347763186259956673124494950355357547691504353939232280074212440502746218496}, \color{blue}{x \cdot x}, \frac{83603027247721846947400001807748686585593809928679218857034661437178107499566497870512394123262412067236255967698478030577310303374589078484741568855549153099}{693167423530203714894603546035770925859109268843954143792619895153655326951406405759993601526034894524347802740350892957243539456}\right), {x}^{2}, \frac{1315581624446765670896680980461586616360623897704795715865349413371368441985234365242636457474613230454287489183683904180293545031734252213064606094900258918769550433855941923780605625}{52374249726338269920211035149241586435466272736689036631732661889538140742474792878132321477214466514414186946040961136147476104734166288853256441430016}\right)}} \]
8. unpow2N/A
  \[\leadsto 1 - \sqrt{\frac{\left(\left(\left(\left(\left(\left(\left(\left(\frac{7890618196247323}{2251799813685248} \cdot 10000000000000000\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1375664026387268463856426913120470294500076676243929134801266166161504374607081765151624544909447479868312852629968476540989780957035}{9173994463960286046443283581208347763186259956673124494950355357547691504353939232280074212440502746218496}, x \cdot x, \frac{83603027247721846947400001807748686585593809928679218857034661437178107499566497870512394123262412067236255967698478030577310303374589078484741568855549153099}{693167423530203714894603546035770925859109268843954143792619895153655326951406405759993601526034894524347802740350892957243539456}\right), \color{blue}{x \cdot x}, \frac{1315581624446765670896680980461586616360623897704795715865349413371368441985234365242636457474613230454287489183683904180293545031734252213064606094900258918769550433855941923780605625}{52374249726338269920211035149241586435466272736689036631732661889538140742474792878132321477214466514414186946040961136147476104734166288853256441430016}\right)}} \]
9. lower-*.f6498.4
  \[\leadsto 1 - \sqrt{\frac{\left(\left(\left(\left(\left(\left(\left(\left(3.5041384 \cdot 10^{+16}\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x}{\mathsf{fma}\left(\mathsf{fma}\left(1.499525677491431 \cdot 10^{+26}, x \cdot x, 1.2061015046255844 \cdot 10^{+29}\right), \color{blue}{x \cdot x}, 2.5118863397964406 \cdot 10^{+31}\right)}} \]
Applied rewrites98.4%
\[\leadsto 1 - \sqrt{\frac{\left(\left(\left(\left(\left(\left(\left(\left(3.5041384 \cdot 10^{+16}\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(1.499525677491431 \cdot 10^{+26}, x \cdot x, 1.2061015046255844 \cdot 10^{+29}\right), x \cdot x, 2.5118863397964406 \cdot 10^{+31}\right)}}} \]
Applied rewrites98.4%
\[\leadsto 1 - \color{blue}{\left(\sqrt{3.5041384 \cdot 10^{+16} \cdot x} \cdot \left(x \cdot x\right)\right) \cdot \sqrt{\left(x \cdot x\right) \cdot \frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(1.499525677491431 \cdot 10^{+26}, x \cdot x, 1.2061015046255844 \cdot 10^{+29}\right), x \cdot x, 2.5118863397964406 \cdot 10^{+31}\right)}}} \]
Add Preprocessing

Alternative 5: 98.6% accurate, 2.1× speedup?

\[\begin{array}{l} \\ 1 - \sqrt{\frac{\left(\left(\left(\left(\left(\left(\left(\left(3.5041384 \cdot 10^{+16}\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x}{\mathsf{fma}\left(1.2061015046255844 \cdot 10^{+29}, x \cdot x, 2.5118863397964406 \cdot 10^{+31}\right)}} \end{array} \]

(FPCore (x)
 :precision binary64
 (-
  1.0
  (sqrt
   (/
    (* (* (* (* (* (* (* (* (* 3.5041384 1e+16) x) x) x) x) x) x) x) x)
    (fma 1.2061015046255844e+29 (* x x) 2.5118863397964406e+31)))))

double code(double x) {
	return 1.0 - sqrt(((((((((((3.5041384 * 1e+16) * x) * x) * x) * x) * x) * x) * x) * x) / fma(1.2061015046255844e+29, (x * x), 2.5118863397964406e+31)));
}

function code(x)
	return Float64(1.0 - sqrt(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(3.5041384 * 1e+16) * x) * x) * x) * x) * x) * x) * x) * x) / fma(1.2061015046255844e+29, Float64(x * x), 2.5118863397964406e+31))))
end

code[x_] := N[(1.0 - N[Sqrt[N[(N[(N[(N[(N[(N[(N[(N[(N[(N[(3.5041384 * 1e+16), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] / N[(1.2061015046255844e+29 * N[(x * x), $MachinePrecision] + 2.5118863397964406e+31), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

\begin{array}{l}

\\
1 - \sqrt{\frac{\left(\left(\left(\left(\left(\left(\left(\left(3.5041384 \cdot 10^{+16}\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x}{\mathsf{fma}\left(1.2061015046255844 \cdot 10^{+29}, x \cdot x, 2.5118863397964406 \cdot 10^{+31}\right)}}
\end{array}

Derivation

Initial program 100.0%
\[1 - \sqrt{\frac{\left(\left(\left(\left(\left(\left(\left(\left(3.5041384 \cdot 10^{+16}\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x}{\left(\left(\left(\left(\left(20.598997 \cdot 20.598997 + x \cdot x\right) \cdot \left(20.598997 \cdot 20.598997 + x \cdot x\right)\right) \cdot \left(107.65265 \cdot 107.65265 + x \cdot x\right)\right) \cdot \left(737.86223 \cdot 737.86223 + x \cdot x\right)\right) \cdot \left(12194.217 \cdot 12194.217 + x \cdot x\right)\right) \cdot \left(12194.217 \cdot 12194.217 + x \cdot x\right)}} \]
Add Preprocessing
Taylor expanded in x around 0
\[\leadsto 1 - \sqrt{\frac{\left(\left(\left(\left(\left(\left(\left(\left(\frac{7890618196247323}{2251799813685248} \cdot 10000000000000000\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x}{\color{blue}{\frac{1315581624446765670896680980461586616360623897704795715865349413371368441985234365242636457474613230454287489183683904180293545031734252213064606094900258918769550433855941923780605625}{52374249726338269920211035149241586435466272736689036631732661889538140742474792878132321477214466514414186946040961136147476104734166288853256441430016} + \frac{83603027247721846947400001807748686585593809928679218857034661437178107499566497870512394123262412067236255967698478030577310303374589078484741568855549153099}{693167423530203714894603546035770925859109268843954143792619895153655326951406405759993601526034894524347802740350892957243539456} \cdot {x}^{2}}}} \]
Step-by-step derivation
1. +-commutativeN/A
  \[\leadsto 1 - \sqrt{\frac{\left(\left(\left(\left(\left(\left(\left(\left(\frac{7890618196247323}{2251799813685248} \cdot 10000000000000000\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x}{\color{blue}{\frac{83603027247721846947400001807748686585593809928679218857034661437178107499566497870512394123262412067236255967698478030577310303374589078484741568855549153099}{693167423530203714894603546035770925859109268843954143792619895153655326951406405759993601526034894524347802740350892957243539456} \cdot {x}^{2} + \frac{1315581624446765670896680980461586616360623897704795715865349413371368441985234365242636457474613230454287489183683904180293545031734252213064606094900258918769550433855941923780605625}{52374249726338269920211035149241586435466272736689036631732661889538140742474792878132321477214466514414186946040961136147476104734166288853256441430016}}}} \]
2. lower-fma.f64N/A
  \[\leadsto 1 - \sqrt{\frac{\left(\left(\left(\left(\left(\left(\left(\left(\frac{7890618196247323}{2251799813685248} \cdot 10000000000000000\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x}{\color{blue}{\mathsf{fma}\left(\frac{83603027247721846947400001807748686585593809928679218857034661437178107499566497870512394123262412067236255967698478030577310303374589078484741568855549153099}{693167423530203714894603546035770925859109268843954143792619895153655326951406405759993601526034894524347802740350892957243539456}, {x}^{2}, \frac{1315581624446765670896680980461586616360623897704795715865349413371368441985234365242636457474613230454287489183683904180293545031734252213064606094900258918769550433855941923780605625}{52374249726338269920211035149241586435466272736689036631732661889538140742474792878132321477214466514414186946040961136147476104734166288853256441430016}\right)}}} \]
3. unpow2N/A
  \[\leadsto 1 - \sqrt{\frac{\left(\left(\left(\left(\left(\left(\left(\left(\frac{7890618196247323}{2251799813685248} \cdot 10000000000000000\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x}{\mathsf{fma}\left(\frac{83603027247721846947400001807748686585593809928679218857034661437178107499566497870512394123262412067236255967698478030577310303374589078484741568855549153099}{693167423530203714894603546035770925859109268843954143792619895153655326951406405759993601526034894524347802740350892957243539456}, \color{blue}{x \cdot x}, \frac{1315581624446765670896680980461586616360623897704795715865349413371368441985234365242636457474613230454287489183683904180293545031734252213064606094900258918769550433855941923780605625}{52374249726338269920211035149241586435466272736689036631732661889538140742474792878132321477214466514414186946040961136147476104734166288853256441430016}\right)}} \]
4. lower-*.f6498.3
  \[\leadsto 1 - \sqrt{\frac{\left(\left(\left(\left(\left(\left(\left(\left(3.5041384 \cdot 10^{+16}\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x}{\mathsf{fma}\left(1.2061015046255844 \cdot 10^{+29}, \color{blue}{x \cdot x}, 2.5118863397964406 \cdot 10^{+31}\right)}} \]
Applied rewrites98.3%
\[\leadsto 1 - \sqrt{\frac{\left(\left(\left(\left(\left(\left(\left(\left(3.5041384 \cdot 10^{+16}\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x}{\color{blue}{\mathsf{fma}\left(1.2061015046255844 \cdot 10^{+29}, x \cdot x, 2.5118863397964406 \cdot 10^{+31}\right)}}} \]
Add Preprocessing

1 - sqrt( (3.5041384 * 10e15 * x * x * x * x * x * x * x * x) / ((20.598997 * 20.598997 + x * x) * (20.598997 * 20.598997 + x * x) * (107.65265 * 107.65265 + xx) (737.86223 * 737.86223 + xx) (12194.217 * 12194.217 + xx) (12194.217 * 12194.217 + x*x)))

Specification

Local Percentage Accuracy vs ?

Accuracy vs Speed?

Initial Program: 100.0% accurate, 1.0× speedup?

Alternative 1: 100.0% accurate, 0.8× speedup?

Alternative 2: 100.0% accurate, 1.1× speedup?

Alternative 3: 98.9% accurate, 1.7× speedup?

Alternative 4: 98.7% accurate, 2.0× speedup?

Alternative 5: 98.6% accurate, 2.1× speedup?

Alternative 6: 98.5% accurate, 173.0× speedup?

Reproduce

Specification

Local Percentage Accuracy vs ?

Accuracy vs Speed?

Initial Program: 100.0% accurate, 1.0× speedupMathFPCoreCFortranJavaPythonJuliaMATLABWolframTeX?

Alternative 1: 100.0% accurate, 0.8× speedupMathFPCoreCJuliaWolframTeX?

Alternative 2: 100.0% accurate, 1.1× speedupMathFPCoreCJuliaWolframTeX?

Alternative 3: 98.9% accurate, 1.7× speedupMathFPCoreCJuliaWolframTeX?

Alternative 4: 98.7% accurate, 2.0× speedupMathFPCoreCJuliaWolframTeX?

Alternative 5: 98.6% accurate, 2.1× speedupMathFPCoreCJuliaWolframTeX?

Alternative 6: 98.5% accurate, 173.0× speedupMathFPCoreCFortranJavaPythonJuliaMATLABWolframTeX?

Reproduce

Initial Program: 100.0% accurate, 1.0× speedup?

Alternative 1: 100.0% accurate, 0.8× speedup?

Alternative 2: 100.0% accurate, 1.1× speedup?

Alternative 3: 98.9% accurate, 1.7× speedup?

Alternative 4: 98.7% accurate, 2.0× speedup?

Alternative 5: 98.6% accurate, 2.1× speedup?

Alternative 6: 98.5% accurate, 173.0× speedup?