Ian Simplification

Percentage Accurate: 6.9% → 8.4%

Time: 33.9s

Alternatives: 5

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- (/ PI 2.0) (* 2.0 (asin (sqrt (/ (- 1.0 x) 2.0))))))

C

double code(double x) {
	return (((double) M_PI) / 2.0) - (2.0 * asin(sqrt(((1.0 - x) / 2.0))));
}

Java

public static double code(double x) {
	return (Math.PI / 2.0) - (2.0 * Math.asin(Math.sqrt(((1.0 - x) / 2.0))));
}

Python

def code(x):
	return (math.pi / 2.0) - (2.0 * math.asin(math.sqrt(((1.0 - x) / 2.0))))

Julia

function code(x)
	return Float64(Float64(pi / 2.0) - Float64(2.0 * asin(sqrt(Float64(Float64(1.0 - x) / 2.0)))))
end

MATLAB

function tmp = code(x)
	tmp = (pi / 2.0) - (2.0 * asin(sqrt(((1.0 - x) / 2.0))));
end

Wolfram

code[x_] := N[(N[(Pi / 2.0), $MachinePrecision] - N[(2.0 * N[ArcSin[N[Sqrt[N[(N[(1.0 - x), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 6.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- (/ PI 2.0) (* 2.0 (asin (sqrt (/ (- 1.0 x) 2.0))))))

C

double code(double x) {
	return (((double) M_PI) / 2.0) - (2.0 * asin(sqrt(((1.0 - x) / 2.0))));
}

Java

public static double code(double x) {
	return (Math.PI / 2.0) - (2.0 * Math.asin(Math.sqrt(((1.0 - x) / 2.0))));
}

Python

def code(x):
	return (math.pi / 2.0) - (2.0 * math.asin(math.sqrt(((1.0 - x) / 2.0))))

Julia

function code(x)
	return Float64(Float64(pi / 2.0) - Float64(2.0 * asin(sqrt(Float64(Float64(1.0 - x) / 2.0)))))
end

MATLAB

function tmp = code(x)
	tmp = (pi / 2.0) - (2.0 * asin(sqrt(((1.0 - x) / 2.0))));
end

Wolfram

code[x_] := N[(N[(Pi / 2.0), $MachinePrecision] - N[(2.0 * N[ArcSin[N[Sqrt[N[(N[(1.0 - x), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 8.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{0.5 - 0.5 \cdot x}\\ \frac{0.25 \cdot {\pi}^{2} - 4 \cdot {\left(\pi \cdot 0.5 - \cos^{-1} t_0\right)}^{2}}{2 \cdot \sin^{-1} t_0 + \pi \cdot 0.5} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (- 0.5 (* 0.5 x)))))
   (/
    (- (* 0.25 (pow PI 2.0)) (* 4.0 (pow (- (* PI 0.5) (acos t_0)) 2.0)))
    (+ (* 2.0 (asin t_0)) (* PI 0.5)))))

C

double code(double x) {
	double t_0 = sqrt((0.5 - (0.5 * x)));
	return ((0.25 * pow(((double) M_PI), 2.0)) - (4.0 * pow(((((double) M_PI) * 0.5) - acos(t_0)), 2.0))) / ((2.0 * asin(t_0)) + (((double) M_PI) * 0.5));
}

Java

public static double code(double x) {
	double t_0 = Math.sqrt((0.5 - (0.5 * x)));
	return ((0.25 * Math.pow(Math.PI, 2.0)) - (4.0 * Math.pow(((Math.PI * 0.5) - Math.acos(t_0)), 2.0))) / ((2.0 * Math.asin(t_0)) + (Math.PI * 0.5));
}

Python

def code(x):
	t_0 = math.sqrt((0.5 - (0.5 * x)))
	return ((0.25 * math.pow(math.pi, 2.0)) - (4.0 * math.pow(((math.pi * 0.5) - math.acos(t_0)), 2.0))) / ((2.0 * math.asin(t_0)) + (math.pi * 0.5))

Julia

function code(x)
	t_0 = sqrt(Float64(0.5 - Float64(0.5 * x)))
	return Float64(Float64(Float64(0.25 * (pi ^ 2.0)) - Float64(4.0 * (Float64(Float64(pi * 0.5) - acos(t_0)) ^ 2.0))) / Float64(Float64(2.0 * asin(t_0)) + Float64(pi * 0.5)))
end

MATLAB

function tmp = code(x)
	t_0 = sqrt((0.5 - (0.5 * x)));
	tmp = ((0.25 * (pi ^ 2.0)) - (4.0 * (((pi * 0.5) - acos(t_0)) ^ 2.0))) / ((2.0 * asin(t_0)) + (pi * 0.5));
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(0.5 - N[(0.5 * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(0.25 * N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision] - N[(4.0 * N[Power[N[(N[(Pi * 0.5), $MachinePrecision] - N[ArcCos[t$95$0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(2.0 * N[ArcSin[t$95$0], $MachinePrecision]), $MachinePrecision] + N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 7.4%

   \[\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
2. Step-by-step derivation

   1. asin-acos 8.5%

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

   2. div-inv 8.5%

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

   3. metadata-eval 8.5%

      \[\leadsto \frac{\pi}{2} - 2 \cdot \left(\pi \cdot \color{blue}{0.5} - \cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right) \]

   4. div-sub 8.5%

      \[\leadsto \frac{\pi}{2} - 2 \cdot \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\color{blue}{\frac{1}{2} - \frac{x}{2}}}\right)\right) \]

   5. metadata-eval 8.5%

      \[\leadsto \frac{\pi}{2} - 2 \cdot \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\color{blue}{0.5} - \frac{x}{2}}\right)\right) \]

   6. div-inv 8.5%

      \[\leadsto \frac{\pi}{2} - 2 \cdot \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{0.5 - \color{blue}{x \cdot \frac{1}{2}}}\right)\right) \]

   7. metadata-eval 8.5%

      \[\leadsto \frac{\pi}{2} - 2 \cdot \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{0.5 - x \cdot \color{blue}{0.5}}\right)\right) \]

3. Applied egg-rr 8.5%

   \[\leadsto \frac{\pi}{2} - 2 \cdot \color{blue}{\left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)\right)} \]
4. Step-by-step derivation

   1. flip-- 8.5%

      \[\leadsto \color{blue}{\frac{\frac{\pi}{2} \cdot \frac{\pi}{2} - \left(2 \cdot \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)\right)\right) \cdot \left(2 \cdot \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)\right)\right)}{\frac{\pi}{2} + 2 \cdot \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)\right)}} \]

5. Applied egg-rr 7.4%

   \[\leadsto \color{blue}{\frac{\frac{{\pi}^{2}}{4} - {\left(2 \cdot \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\right)}^{2}}{\mathsf{fma}\left(\pi, 0.5, 2 \cdot \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\right)}} \]
6. Step-by-step derivation

   1. asin-acos 8.5%

      \[\leadsto \frac{\frac{{\pi}^{2}}{4} - {\left(2 \cdot \color{blue}{\left(\frac{\pi}{2} - \cos^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\right)}\right)}^{2}}{\mathsf{fma}\left(\pi, 0.5, 2 \cdot \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\right)} \]

   2. add-sqr-sqrt 6.7%

      \[\leadsto \frac{\frac{{\pi}^{2}}{4} - {\left(2 \cdot \left(\frac{\color{blue}{\sqrt{\pi} \cdot \sqrt{\pi}}}{2} - \cos^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\right)\right)}^{2}}{\mathsf{fma}\left(\pi, 0.5, 2 \cdot \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\right)} \]

   3. sqrt-prod 8.5%

      \[\leadsto \frac{\frac{{\pi}^{2}}{4} - {\left(2 \cdot \left(\frac{\color{blue}{\sqrt{\pi \cdot \pi}}}{2} - \cos^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\right)\right)}^{2}}{\mathsf{fma}\left(\pi, 0.5, 2 \cdot \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\right)} \]

   4. unpow2 8.5%

      \[\leadsto \frac{\frac{{\pi}^{2}}{4} - {\left(2 \cdot \left(\frac{\sqrt{\color{blue}{{\pi}^{2}}}}{2} - \cos^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\right)\right)}^{2}}{\mathsf{fma}\left(\pi, 0.5, 2 \cdot \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\right)} \]

   5. metadata-eval 8.5%

      \[\leadsto \frac{\frac{{\pi}^{2}}{4} - {\left(2 \cdot \left(\frac{\sqrt{{\pi}^{2}}}{\color{blue}{\sqrt{4}}} - \cos^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\right)\right)}^{2}}{\mathsf{fma}\left(\pi, 0.5, 2 \cdot \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\right)} \]

   6. sqrt-div 8.5%

      \[\leadsto \frac{\frac{{\pi}^{2}}{4} - {\left(2 \cdot \left(\color{blue}{\sqrt{\frac{{\pi}^{2}}{4}}} - \cos^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\right)\right)}^{2}}{\mathsf{fma}\left(\pi, 0.5, 2 \cdot \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\right)} \]

   7. div-inv 8.5%

      \[\leadsto \frac{\frac{{\pi}^{2}}{4} - {\left(2 \cdot \left(\sqrt{\color{blue}{{\pi}^{2} \cdot \frac{1}{4}}} - \cos^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\right)\right)}^{2}}{\mathsf{fma}\left(\pi, 0.5, 2 \cdot \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\right)} \]

   8. metadata-eval 8.5%

      \[\leadsto \frac{\frac{{\pi}^{2}}{4} - {\left(2 \cdot \left(\sqrt{{\pi}^{2} \cdot \color{blue}{0.25}} - \cos^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\right)\right)}^{2}}{\mathsf{fma}\left(\pi, 0.5, 2 \cdot \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\right)} \]

   9. metadata-eval 8.5%

      \[\leadsto \frac{\frac{{\pi}^{2}}{4} - {\left(2 \cdot \left(\sqrt{{\pi}^{2} \cdot \color{blue}{\left(0.5 \cdot 0.5\right)}} - \cos^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\right)\right)}^{2}}{\mathsf{fma}\left(\pi, 0.5, 2 \cdot \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\right)} \]

   10. sqrt-prod 8.5%

       \[\leadsto \frac{\frac{{\pi}^{2}}{4} - {\left(2 \cdot \left(\color{blue}{\sqrt{{\pi}^{2}} \cdot \sqrt{0.5 \cdot 0.5}} - \cos^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\right)\right)}^{2}}{\mathsf{fma}\left(\pi, 0.5, 2 \cdot \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\right)} \]

   11. unpow2 8.5%

       \[\leadsto \frac{\frac{{\pi}^{2}}{4} - {\left(2 \cdot \left(\sqrt{\color{blue}{\pi \cdot \pi}} \cdot \sqrt{0.5 \cdot 0.5} - \cos^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\right)\right)}^{2}}{\mathsf{fma}\left(\pi, 0.5, 2 \cdot \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\right)} \]

   12. sqrt-prod 6.7%

       \[\leadsto \frac{\frac{{\pi}^{2}}{4} - {\left(2 \cdot \left(\color{blue}{\left(\sqrt{\pi} \cdot \sqrt{\pi}\right)} \cdot \sqrt{0.5 \cdot 0.5} - \cos^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\right)\right)}^{2}}{\mathsf{fma}\left(\pi, 0.5, 2 \cdot \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\right)} \]

   13. add-sqr-sqrt 8.5%

       \[\leadsto \frac{\frac{{\pi}^{2}}{4} - {\left(2 \cdot \left(\color{blue}{\pi} \cdot \sqrt{0.5 \cdot 0.5} - \cos^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\right)\right)}^{2}}{\mathsf{fma}\left(\pi, 0.5, 2 \cdot \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\right)} \]

   14. metadata-eval 8.5%

       \[\leadsto \frac{\frac{{\pi}^{2}}{4} - {\left(2 \cdot \left(\pi \cdot \sqrt{\color{blue}{0.25}} - \cos^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\right)\right)}^{2}}{\mathsf{fma}\left(\pi, 0.5, 2 \cdot \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\right)} \]

   15. metadata-eval 8.5%

       \[\leadsto \frac{\frac{{\pi}^{2}}{4} - {\left(2 \cdot \left(\pi \cdot \color{blue}{0.5} - \cos^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\right)\right)}^{2}}{\mathsf{fma}\left(\pi, 0.5, 2 \cdot \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\right)} \]

   16. cancel-sign-sub-inv 8.5%

       \[\leadsto \frac{\frac{{\pi}^{2}}{4} - {\left(2 \cdot \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\color{blue}{0.5 + \left(-0.5\right) \cdot x}}\right)\right)\right)}^{2}}{\mathsf{fma}\left(\pi, 0.5, 2 \cdot \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\right)} \]

   17. metadata-eval 8.5%

       \[\leadsto \frac{\frac{{\pi}^{2}}{4} - {\left(2 \cdot \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{0.5 + \color{blue}{-0.5} \cdot x}\right)\right)\right)}^{2}}{\mathsf{fma}\left(\pi, 0.5, 2 \cdot \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\right)} \]

7. Applied egg-rr 8.5%

   \[\leadsto \frac{\frac{{\pi}^{2}}{4} - {\left(2 \cdot \color{blue}{\left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{0.5 + -0.5 \cdot x}\right)\right)}\right)}^{2}}{\mathsf{fma}\left(\pi, 0.5, 2 \cdot \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\right)} \]
8. Step-by-step derivation

   1. *-commutative 8.5%

      \[\leadsto \frac{\frac{{\pi}^{2}}{4} - {\left(2 \cdot \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{0.5 + \color{blue}{x \cdot -0.5}}\right)\right)\right)}^{2}}{\mathsf{fma}\left(\pi, 0.5, 2 \cdot \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\right)} \]

9. Simplified 8.5%

   \[\leadsto \frac{\frac{{\pi}^{2}}{4} - {\left(2 \cdot \color{blue}{\left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)\right)}\right)}^{2}}{\mathsf{fma}\left(\pi, 0.5, 2 \cdot \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\right)} \]

10. Taylor expanded in x around inf 8.5%

    \[\leadsto \color{blue}{\frac{0.25 \cdot {\pi}^{2} - 4 \cdot {\left(0.5 \cdot \pi - \cos^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\right)}^{2}}{2 \cdot \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) + 0.5 \cdot \pi}} \]

11. Final simplification 8.5%

    \[\leadsto \frac{0.25 \cdot {\pi}^{2} - 4 \cdot {\left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\right)}^{2}}{2 \cdot \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) + \pi \cdot 0.5} \]

Alternative 2: 8.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ {\left(\sqrt[3]{\mathsf{fma}\left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right), -2, \pi \cdot 0.5\right)}\right)}^{3} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (pow
  (cbrt (fma (- (* PI 0.5) (acos (sqrt (- 0.5 (* 0.5 x))))) -2.0 (* PI 0.5)))
  3.0))

C

double code(double x) {
	return pow(cbrt(fma(((((double) M_PI) * 0.5) - acos(sqrt((0.5 - (0.5 * x))))), -2.0, (((double) M_PI) * 0.5))), 3.0);
}

Julia

function code(x)
	return cbrt(fma(Float64(Float64(pi * 0.5) - acos(sqrt(Float64(0.5 - Float64(0.5 * x))))), -2.0, Float64(pi * 0.5))) ^ 3.0
end

Wolfram

code[x_] := N[Power[N[Power[N[(N[(N[(Pi * 0.5), $MachinePrecision] - N[ArcCos[N[Sqrt[N[(0.5 - N[(0.5 * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * -2.0 + N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]

Derivation

1. Initial program 7.4%

   \[\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
2. Step-by-step derivation

   1. add-cube-cbrt 7.4%

      \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)} \cdot \sqrt[3]{\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)}\right) \cdot \sqrt[3]{\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)}} \]

   2. pow3 7.4%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)}\right)}^{3}} \]

3. Applied egg-rr 7.4%

   \[\leadsto \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(\sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), -2, \pi \cdot 0.5\right)}\right)}^{3}} \]
4. Step-by-step derivation

   1. asin-acos 8.5%

      \[\leadsto {\left(\sqrt[3]{\mathsf{fma}\left(\color{blue}{\frac{\pi}{2} - \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}, -2, \pi \cdot 0.5\right)}\right)}^{3} \]

   2. div-inv 8.5%

      \[\leadsto {\left(\sqrt[3]{\mathsf{fma}\left(\color{blue}{\pi \cdot \frac{1}{2}} - \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), -2, \pi \cdot 0.5\right)}\right)}^{3} \]

   3. metadata-eval 8.5%

      \[\leadsto {\left(\sqrt[3]{\mathsf{fma}\left(\pi \cdot \color{blue}{0.5} - \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), -2, \pi \cdot 0.5\right)}\right)}^{3} \]

   4. *-commutative 8.5%

      \[\leadsto {\left(\sqrt[3]{\mathsf{fma}\left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{0.5 - \color{blue}{0.5 \cdot x}}\right), -2, \pi \cdot 0.5\right)}\right)}^{3} \]

5. Applied egg-rr 8.5%

   \[\leadsto {\left(\sqrt[3]{\mathsf{fma}\left(\color{blue}{\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)}, -2, \pi \cdot 0.5\right)}\right)}^{3} \]

6. Final simplification 8.5%

   \[\leadsto {\left(\sqrt[3]{\mathsf{fma}\left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right), -2, \pi \cdot 0.5\right)}\right)}^{3} \]

Alternative 3: 8.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{\pi}{2} + 2 \cdot \left(\cos^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) - \pi \cdot 0.5\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (+ (/ PI 2.0) (* 2.0 (- (acos (sqrt (- 0.5 (* 0.5 x)))) (* PI 0.5)))))

C

double code(double x) {
	return (((double) M_PI) / 2.0) + (2.0 * (acos(sqrt((0.5 - (0.5 * x)))) - (((double) M_PI) * 0.5)));
}

Java

public static double code(double x) {
	return (Math.PI / 2.0) + (2.0 * (Math.acos(Math.sqrt((0.5 - (0.5 * x)))) - (Math.PI * 0.5)));
}

Python

def code(x):
	return (math.pi / 2.0) + (2.0 * (math.acos(math.sqrt((0.5 - (0.5 * x)))) - (math.pi * 0.5)))

Julia

function code(x)
	return Float64(Float64(pi / 2.0) + Float64(2.0 * Float64(acos(sqrt(Float64(0.5 - Float64(0.5 * x)))) - Float64(pi * 0.5))))
end

MATLAB

function tmp = code(x)
	tmp = (pi / 2.0) + (2.0 * (acos(sqrt((0.5 - (0.5 * x)))) - (pi * 0.5)));
end

Wolfram

code[x_] := N[(N[(Pi / 2.0), $MachinePrecision] + N[(2.0 * N[(N[ArcCos[N[Sqrt[N[(0.5 - N[(0.5 * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] - N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.4%

   \[\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
2. Step-by-step derivation

   1. asin-acos 8.5%

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

   2. div-inv 8.5%

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

   3. metadata-eval 8.5%

      \[\leadsto \frac{\pi}{2} - 2 \cdot \left(\pi \cdot \color{blue}{0.5} - \cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right) \]

   4. div-sub 8.5%

      \[\leadsto \frac{\pi}{2} - 2 \cdot \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\color{blue}{\frac{1}{2} - \frac{x}{2}}}\right)\right) \]

   5. metadata-eval 8.5%

      \[\leadsto \frac{\pi}{2} - 2 \cdot \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\color{blue}{0.5} - \frac{x}{2}}\right)\right) \]

   6. div-inv 8.5%

      \[\leadsto \frac{\pi}{2} - 2 \cdot \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{0.5 - \color{blue}{x \cdot \frac{1}{2}}}\right)\right) \]

   7. metadata-eval 8.5%

      \[\leadsto \frac{\pi}{2} - 2 \cdot \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{0.5 - x \cdot \color{blue}{0.5}}\right)\right) \]

3. Applied egg-rr 8.5%

   \[\leadsto \frac{\pi}{2} - 2 \cdot \color{blue}{\left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)\right)} \]

4. Final simplification 8.5%

   \[\leadsto \frac{\pi}{2} + 2 \cdot \left(\cos^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) - \pi \cdot 0.5\right) \]

Alternative 4: 6.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- (/ PI 2.0) (* 2.0 (asin (sqrt (/ (- 1.0 x) 2.0))))))

C

double code(double x) {
	return (((double) M_PI) / 2.0) - (2.0 * asin(sqrt(((1.0 - x) / 2.0))));
}

Java

public static double code(double x) {
	return (Math.PI / 2.0) - (2.0 * Math.asin(Math.sqrt(((1.0 - x) / 2.0))));
}

Python

def code(x):
	return (math.pi / 2.0) - (2.0 * math.asin(math.sqrt(((1.0 - x) / 2.0))))

Julia

function code(x)
	return Float64(Float64(pi / 2.0) - Float64(2.0 * asin(sqrt(Float64(Float64(1.0 - x) / 2.0)))))
end

MATLAB

function tmp = code(x)
	tmp = (pi / 2.0) - (2.0 * asin(sqrt(((1.0 - x) / 2.0))));
end

Wolfram

code[x_] := N[(N[(Pi / 2.0), $MachinePrecision] - N[(2.0 * N[ArcSin[N[Sqrt[N[(N[(1.0 - x), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.4%

   \[\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]

2. Final simplification 7.4%

   \[\leadsto \frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]

Alternative 5: 4.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{0.5}\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (/ PI 2.0) (* 2.0 (asin (sqrt 0.5)))))

C

double code(double x) {
	return (((double) M_PI) / 2.0) - (2.0 * asin(sqrt(0.5)));
}

Java

public static double code(double x) {
	return (Math.PI / 2.0) - (2.0 * Math.asin(Math.sqrt(0.5)));
}

Python

def code(x):
	return (math.pi / 2.0) - (2.0 * math.asin(math.sqrt(0.5)))

Julia

function code(x)
	return Float64(Float64(pi / 2.0) - Float64(2.0 * asin(sqrt(0.5))))
end

MATLAB

function tmp = code(x)
	tmp = (pi / 2.0) - (2.0 * asin(sqrt(0.5)));
end

Wolfram

code[x_] := N[(N[(Pi / 2.0), $MachinePrecision] - N[(2.0 * N[ArcSin[N[Sqrt[0.5], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.4%

   \[\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]

2. Taylor expanded in x around 0 7.4%

   \[\leadsto \frac{\pi}{2} - 2 \cdot \color{blue}{\sin^{-1} \left(\sqrt{0.5} \cdot \sqrt{1 - x}\right)} \]

3. Taylor expanded in x around 0 4.4%

   \[\leadsto \frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{0.5} \cdot \color{blue}{1}\right) \]
4. Step-by-step derivation

   1. expm1-log1p-u 4.4%

      \[\leadsto \frac{\pi}{2} - 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(\sqrt{0.5} \cdot 1\right)\right)\right)} \]

   2. expm1-udef 4.4%

      \[\leadsto \frac{\pi}{2} - 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sin^{-1} \left(\sqrt{0.5} \cdot 1\right)\right)} - 1\right)} \]

   3. *-rgt-identity 4.4%

      \[\leadsto \frac{\pi}{2} - 2 \cdot \left(e^{\mathsf{log1p}\left(\sin^{-1} \color{blue}{\left(\sqrt{0.5}\right)}\right)} - 1\right) \]

5. Applied egg-rr 4.4%

   \[\leadsto \frac{\pi}{2} - 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sin^{-1} \left(\sqrt{0.5}\right)\right)} - 1\right)} \]
6. Step-by-step derivation

   1. expm1-def 4.4%

      \[\leadsto \frac{\pi}{2} - 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(\sqrt{0.5}\right)\right)\right)} \]

   2. expm1-log1p 4.4%

      \[\leadsto \frac{\pi}{2} - 2 \cdot \color{blue}{\sin^{-1} \left(\sqrt{0.5}\right)} \]

7. Simplified 4.4%

   \[\leadsto \frac{\pi}{2} - 2 \cdot \color{blue}{\sin^{-1} \left(\sqrt{0.5}\right)} \]

8. Final simplification 4.4%

   \[\leadsto \frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{0.5}\right) \]

Developer target: 100.0% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \sin^{-1} x \end{array} \]

FPCore

(FPCore (x) :precision binary64 (asin x))

C

double code(double x) {
	return asin(x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = asin(x)
end function

Java

public static double code(double x) {
	return Math.asin(x);
}

Python

def code(x):
	return math.asin(x)

Julia

function code(x)
	return asin(x)
end

MATLAB

function tmp = code(x)
	tmp = asin(x);
end

Wolfram

code[x_] := N[ArcSin[x], $MachinePrecision]

Reproduce

herbie shell --seed 2023279 
(FPCore (x)
  :name "Ian Simplification"
  :precision binary64

  :herbie-target
  (asin x)

  (- (/ PI 2.0) (* 2.0 (asin (sqrt (/ (- 1.0 x) 2.0))))))