?

Average Error: 29.3 → 0.2
Time: 6.3s
Precision: binary64
Cost: 6980

?

\[0 \leq x \land x \leq 1.79 \cdot 10^{+308}\]
\[\sqrt{x + 1} - 1 \]
↓
\[\begin{array}{l} \mathbf{if}\;x + 1 \leq 1.02:\\ \;\;\;\;x \cdot \left(x \cdot -0.125\right) + x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + -1\\ \end{array} \]
(FPCore (x) :precision binary64 (- (sqrt (+ x 1.0)) 1.0))
↓
(FPCore (x)
 :precision binary64
 (if (<= (+ x 1.0) 1.02)
   (+ (* x (* x -0.125)) (* x 0.5))
   (+ (sqrt (+ x 1.0)) -1.0)))
double code(double x) {
	return sqrt((x + 1.0)) - 1.0;
}
↓
double code(double x) {
	double tmp;
	if ((x + 1.0) <= 1.02) {
		tmp = (x * (x * -0.125)) + (x * 0.5);
	} else {
		tmp = sqrt((x + 1.0)) + -1.0;
	}
	return tmp;
}

real(8) function code(x)
    real(8), intent (in) :: x
    code = sqrt((x + 1.0d0)) - 1.0d0
end function
↓
real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x + 1.0d0) <= 1.02d0) then
        tmp = (x * (x * (-0.125d0))) + (x * 0.5d0)
    else
        tmp = sqrt((x + 1.0d0)) + (-1.0d0)
    end if
    code = tmp
end function

public static double code(double x) {
	return Math.sqrt((x + 1.0)) - 1.0;
}
↓
public static double code(double x) {
	double tmp;
	if ((x + 1.0) <= 1.02) {
		tmp = (x * (x * -0.125)) + (x * 0.5);
	} else {
		tmp = Math.sqrt((x + 1.0)) + -1.0;
	}
	return tmp;
}

def code(x):
	return math.sqrt((x + 1.0)) - 1.0
↓
def code(x):
	tmp = 0
	if (x + 1.0) <= 1.02:
		tmp = (x * (x * -0.125)) + (x * 0.5)
	else:
		tmp = math.sqrt((x + 1.0)) + -1.0
	return tmp

function code(x)
	return Float64(sqrt(Float64(x + 1.0)) - 1.0)
end
↓
function code(x)
	tmp = 0.0
	if (Float64(x + 1.0) <= 1.02)
		tmp = Float64(Float64(x * Float64(x * -0.125)) + Float64(x * 0.5));
	else
		tmp = Float64(sqrt(Float64(x + 1.0)) + -1.0);
	end
	return tmp
end

function tmp = code(x)
	tmp = sqrt((x + 1.0)) - 1.0;
end
↓
function tmp_2 = code(x)
	tmp = 0.0;
	if ((x + 1.0) <= 1.02)
		tmp = (x * (x * -0.125)) + (x * 0.5);
	else
		tmp = sqrt((x + 1.0)) + -1.0;
	end
	tmp_2 = tmp;
end

code[x_] := N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - 1.0), $MachinePrecision]
↓
code[x_] := If[LessEqual[N[(x + 1.0), $MachinePrecision], 1.02], N[(N[(x * N[(x * -0.125), $MachinePrecision]), $MachinePrecision] + N[(x * 0.5), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]]

\sqrt{x + 1} - 1
↓
\begin{array}{l}
\mathbf{if}\;x + 1 \leq 1.02:\\
\;\;\;\;x \cdot \left(x \cdot -0.125\right) + x \cdot 0.5\\

\mathbf{else}:\\
\;\;\;\;\sqrt{x + 1} + -1\\


\end{array}

Error?

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation?

  1. Split input into 2 regimes

  2. if (+.f64 x 1) < 1.02

    1. Initial program 58.9

      \[\sqrt{x + 1} - 1 \]

    2. Taylor expanded in x around 0 0.4

      \[\leadsto \color{blue}{-0.125 \cdot {x}^{2} + 0.5 \cdot x} \]

    3. Simplified0.4

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, x, x \cdot \left(x \cdot -0.125\right)\right)} \]
      Proof

      [Start]0.4

      \[ -0.125 \cdot {x}^{2} + 0.5 \cdot x \]

      +-commutative [=>]0.4

      \[ \color{blue}{0.5 \cdot x + -0.125 \cdot {x}^{2}} \]

      fma-def [=>]0.4

      \[ \color{blue}{\mathsf{fma}\left(0.5, x, -0.125 \cdot {x}^{2}\right)} \]

      *-commutative [=>]0.4

      \[ \mathsf{fma}\left(0.5, x, \color{blue}{{x}^{2} \cdot -0.125}\right) \]

      unpow2 [=>]0.4

      \[ \mathsf{fma}\left(0.5, x, \color{blue}{\left(x \cdot x\right)} \cdot -0.125\right) \]

      associate-*l* [=>]0.4

      \[ \mathsf{fma}\left(0.5, x, \color{blue}{x \cdot \left(x \cdot -0.125\right)}\right) \]

    4. Applied egg-rr0.4

      \[\leadsto \color{blue}{x \cdot \left(x \cdot -0.125\right) + 0.5 \cdot x} \]

    if 1.02 < (+.f64 x 1)

    1. Initial program 0.0

      \[\sqrt{x + 1} - 1 \]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x + 1 \leq 1.02:\\ \;\;\;\;x \cdot \left(x \cdot -0.125\right) + x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + -1\\ \end{array} \]

Alternatives

Alternative 1
Error0.3
Cost6848
\[\frac{x}{1 + \sqrt{x + 1}} \]
Alternative 2
Error30.1
Cost448
\[\frac{x}{x \cdot 0.5 + 2} \]
Alternative 3
Error30.5
Cost192
\[x \cdot 0.5 \]

Error

Reproduce?

herbie shell --seed 1 
(FPCore (x)
  :name "sqrt(x + 1) - 1"
  :precision binary64
  :pre (and (<= 0.0 x) (<= x 1.79e+308))
  (- (sqrt (+ x 1.0)) 1.0))