Ian Simplification

Percentage Accurate: 7.0% → 8.5%

Time: 12.9s

Alternatives: 4

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: 7.0% 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: 8.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{0.5 - 0.5 \cdot x}\\ {\left({\left(\mathsf{fma}\left(\pi, 0.5, \sin^{-1} t\_0 \cdot -2\right)\right)}^{2}\right)}^{0.3333333333333333} \cdot \sqrt[3]{\mathsf{fma}\left(\pi, 0.5, -2 \cdot \left(\pi \cdot 0.5 - \cos^{-1} t\_0\right)\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (- 0.5 (* 0.5 x)))))
   (*
    (pow (pow (fma PI 0.5 (* (asin t_0) -2.0)) 2.0) 0.3333333333333333)
    (cbrt (fma PI 0.5 (* -2.0 (- (* PI 0.5) (acos t_0))))))))

C

double code(double x) {
	double t_0 = sqrt((0.5 - (0.5 * x)));
	return pow(pow(fma(((double) M_PI), 0.5, (asin(t_0) * -2.0)), 2.0), 0.3333333333333333) * cbrt(fma(((double) M_PI), 0.5, (-2.0 * ((((double) M_PI) * 0.5) - acos(t_0)))));
}

Julia

function code(x)
	t_0 = sqrt(Float64(0.5 - Float64(0.5 * x)))
	return Float64(((fma(pi, 0.5, Float64(asin(t_0) * -2.0)) ^ 2.0) ^ 0.3333333333333333) * cbrt(fma(pi, 0.5, Float64(-2.0 * Float64(Float64(pi * 0.5) - acos(t_0))))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(0.5 - N[(0.5 * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(N[Power[N[Power[N[(Pi * 0.5 + N[(N[ArcSin[t$95$0], $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], 0.3333333333333333], $MachinePrecision] * N[Power[N[(Pi * 0.5 + N[(-2.0 * N[(N[(Pi * 0.5), $MachinePrecision] - N[ArcCos[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 6.3%

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

   1. expm1-log1p-u 6.3%

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

   2. div-inv 6.3%

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

   3. metadata-eval 6.3%

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

   4. fmm-def 6.3%

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

   5. *-commutative 6.3%

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

   6. distribute-rgt-neg-in 6.3%

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

   7. div-sub 6.3%

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

   8. metadata-eval 6.3%

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

   9. div-inv 6.3%

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

   10. metadata-eval 6.3%

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

   11. metadata-eval 6.3%

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

4. Applied egg-rr 6.3%

   \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right) \cdot -2\right)\right)\right)} \]
5. Step-by-step derivation

   1. expm1-log1p-u 6.3%

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

   2. rem-cbrt-cube 6.3%

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

   3. unpow1/3 0.6%

      \[\leadsto \color{blue}{{\left({\left(\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right) \cdot -2\right)\right)}^{3}\right)}^{0.3333333333333333}} \]

   4. unpow3 0.6%

      \[\leadsto {\color{blue}{\left(\left(\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right) \cdot -2\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right) \cdot -2\right)\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right) \cdot -2\right)\right)}}^{0.3333333333333333} \]

   5. unpow-prod-down 0.6%

      \[\leadsto \color{blue}{{\left(\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right) \cdot -2\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right) \cdot -2\right)\right)}^{0.3333333333333333} \cdot {\left(\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right) \cdot -2\right)\right)}^{0.3333333333333333}} \]

6. Applied egg-rr 6.3%

   \[\leadsto \color{blue}{{\left({\left(\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot -2\right)\right)}^{2}\right)}^{0.3333333333333333} \cdot \sqrt[3]{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot -2\right)}} \]
7. Step-by-step derivation

   1. asin-acos 7.3%

      \[\leadsto {\left({\left(\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot -2\right)\right)}^{2}\right)}^{0.3333333333333333} \cdot \sqrt[3]{\mathsf{fma}\left(\pi, 0.5, \color{blue}{\left(\frac{\pi}{2} - \cos^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\right)} \cdot -2\right)} \]

   2. div-inv 7.3%

      \[\leadsto {\left({\left(\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot -2\right)\right)}^{2}\right)}^{0.3333333333333333} \cdot \sqrt[3]{\mathsf{fma}\left(\pi, 0.5, \left(\color{blue}{\pi \cdot \frac{1}{2}} - \cos^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\right) \cdot -2\right)} \]

   3. metadata-eval 7.3%

      \[\leadsto {\left({\left(\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot -2\right)\right)}^{2}\right)}^{0.3333333333333333} \cdot \sqrt[3]{\mathsf{fma}\left(\pi, 0.5, \left(\pi \cdot \color{blue}{0.5} - \cos^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\right) \cdot -2\right)} \]

   4. *-commutative 7.3%

      \[\leadsto {\left({\left(\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot -2\right)\right)}^{2}\right)}^{0.3333333333333333} \cdot \sqrt[3]{\mathsf{fma}\left(\pi, 0.5, \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{0.5 - \color{blue}{x \cdot 0.5}}\right)\right) \cdot -2\right)} \]

   5. sub-neg 7.3%

      \[\leadsto {\left({\left(\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot -2\right)\right)}^{2}\right)}^{0.3333333333333333} \cdot \sqrt[3]{\mathsf{fma}\left(\pi, 0.5, \color{blue}{\left(\pi \cdot 0.5 + \left(-\cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)\right)\right)} \cdot -2\right)} \]

   6. *-commutative 7.3%

      \[\leadsto {\left({\left(\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot -2\right)\right)}^{2}\right)}^{0.3333333333333333} \cdot \sqrt[3]{\mathsf{fma}\left(\pi, 0.5, \left(\color{blue}{0.5 \cdot \pi} + \left(-\cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)\right)\right) \cdot -2\right)} \]

   7. *-commutative 7.3%

      \[\leadsto {\left({\left(\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot -2\right)\right)}^{2}\right)}^{0.3333333333333333} \cdot \sqrt[3]{\mathsf{fma}\left(\pi, 0.5, \left(0.5 \cdot \pi + \left(-\cos^{-1} \left(\sqrt{0.5 - \color{blue}{0.5 \cdot x}}\right)\right)\right) \cdot -2\right)} \]

   8. sub-neg 7.3%

      \[\leadsto {\left({\left(\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot -2\right)\right)}^{2}\right)}^{0.3333333333333333} \cdot \sqrt[3]{\mathsf{fma}\left(\pi, 0.5, \left(0.5 \cdot \pi + \left(-\cos^{-1} \left(\sqrt{\color{blue}{0.5 + \left(-0.5 \cdot x\right)}}\right)\right)\right) \cdot -2\right)} \]

   9. *-commutative 7.3%

      \[\leadsto {\left({\left(\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot -2\right)\right)}^{2}\right)}^{0.3333333333333333} \cdot \sqrt[3]{\mathsf{fma}\left(\pi, 0.5, \left(0.5 \cdot \pi + \left(-\cos^{-1} \left(\sqrt{0.5 + \left(-\color{blue}{x \cdot 0.5}\right)}\right)\right)\right) \cdot -2\right)} \]

   10. distribute-rgt-neg-in 7.3%

       \[\leadsto {\left({\left(\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot -2\right)\right)}^{2}\right)}^{0.3333333333333333} \cdot \sqrt[3]{\mathsf{fma}\left(\pi, 0.5, \left(0.5 \cdot \pi + \left(-\cos^{-1} \left(\sqrt{0.5 + \color{blue}{x \cdot \left(-0.5\right)}}\right)\right)\right) \cdot -2\right)} \]

   11. metadata-eval 7.3%

       \[\leadsto {\left({\left(\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot -2\right)\right)}^{2}\right)}^{0.3333333333333333} \cdot \sqrt[3]{\mathsf{fma}\left(\pi, 0.5, \left(0.5 \cdot \pi + \left(-\cos^{-1} \left(\sqrt{0.5 + x \cdot \color{blue}{-0.5}}\right)\right)\right) \cdot -2\right)} \]

8. Applied egg-rr 7.3%

   \[\leadsto {\left({\left(\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot -2\right)\right)}^{2}\right)}^{0.3333333333333333} \cdot \sqrt[3]{\mathsf{fma}\left(\pi, 0.5, \color{blue}{\left(0.5 \cdot \pi + \left(-\cos^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)\right)\right)} \cdot -2\right)} \]
9. Step-by-step derivation

   1. sub-neg 7.3%

      \[\leadsto {\left({\left(\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot -2\right)\right)}^{2}\right)}^{0.3333333333333333} \cdot \sqrt[3]{\mathsf{fma}\left(\pi, 0.5, \color{blue}{\left(0.5 \cdot \pi - \cos^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)\right)} \cdot -2\right)} \]

   2. *-commutative 7.3%

      \[\leadsto {\left({\left(\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot -2\right)\right)}^{2}\right)}^{0.3333333333333333} \cdot \sqrt[3]{\mathsf{fma}\left(\pi, 0.5, \left(0.5 \cdot \pi - \cos^{-1} \left(\sqrt{0.5 + \color{blue}{-0.5 \cdot x}}\right)\right) \cdot -2\right)} \]

   3. metadata-eval 7.3%

      \[\leadsto {\left({\left(\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot -2\right)\right)}^{2}\right)}^{0.3333333333333333} \cdot \sqrt[3]{\mathsf{fma}\left(\pi, 0.5, \left(0.5 \cdot \pi - \cos^{-1} \left(\sqrt{0.5 + \color{blue}{\left(-0.5\right)} \cdot x}\right)\right) \cdot -2\right)} \]

   4. cancel-sign-sub-inv 7.3%

      \[\leadsto {\left({\left(\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot -2\right)\right)}^{2}\right)}^{0.3333333333333333} \cdot \sqrt[3]{\mathsf{fma}\left(\pi, 0.5, \left(0.5 \cdot \pi - \cos^{-1} \left(\sqrt{\color{blue}{0.5 - 0.5 \cdot x}}\right)\right) \cdot -2\right)} \]

10. Simplified 7.3%

    \[\leadsto {\left({\left(\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot -2\right)\right)}^{2}\right)}^{0.3333333333333333} \cdot \sqrt[3]{\mathsf{fma}\left(\pi, 0.5, \color{blue}{\left(0.5 \cdot \pi - \cos^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\right)} \cdot -2\right)} \]

11. Final simplification 7.3%

    \[\leadsto {\left({\left(\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot -2\right)\right)}^{2}\right)}^{0.3333333333333333} \cdot \sqrt[3]{\mathsf{fma}\left(\pi, 0.5, -2 \cdot \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\right)\right)} \]
12. Add Preprocessing

Alternative 2: 8.5% accurate, 1.0× 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 6.3%

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

   1. asin-acos 7.3%

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

   2. div-inv 7.3%

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

   3. metadata-eval 7.3%

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

   4. div-sub 7.3%

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

   5. metadata-eval 7.3%

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

   6. div-inv 7.3%

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

   7. metadata-eval 7.3%

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

4. Applied egg-rr 7.3%

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

5. Final simplification 7.3%

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

Alternative 3: 7.0% 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 6.3%

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

Alternative 4: 4.1% 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 6.3%

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

3. Taylor expanded in x around 0 4.5%

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

Developer Target 1: 100.0% accurate, 2.1× 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 2024188 
(FPCore (x)
  :name "Ian Simplification"
  :precision binary64

  :alt
  (! :herbie-platform default (asin x))

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