Ian Simplification

Percentage Accurate: 7.1% → 8.5%

Time: 27.1s

Alternatives: 6

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: 7.1% 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

\[\begin{array}{l} \\ {\left(\sqrt[3]{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left({\left({\left(0.5 - 0.5 \cdot x\right)}^{0.25}\right)}^{2}\right) \cdot -2\right)}\right)}^{2} \cdot \sqrt[3]{\mathsf{fma}\left(\pi, 0.5, -2 \cdot \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)\right)\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (pow
   (cbrt (fma PI 0.5 (* (asin (pow (pow (- 0.5 (* 0.5 x)) 0.25) 2.0)) -2.0)))
   2.0)
  (cbrt (fma PI 0.5 (* -2.0 (- (* PI 0.5) (acos (sqrt (fma x -0.5 0.5)))))))))

C

double code(double x) {
	return pow(cbrt(fma(((double) M_PI), 0.5, (asin(pow(pow((0.5 - (0.5 * x)), 0.25), 2.0)) * -2.0))), 2.0) * cbrt(fma(((double) M_PI), 0.5, (-2.0 * ((((double) M_PI) * 0.5) - acos(sqrt(fma(x, -0.5, 0.5)))))));
}

Julia

function code(x)
	return Float64((cbrt(fma(pi, 0.5, Float64(asin(((Float64(0.5 - Float64(0.5 * x)) ^ 0.25) ^ 2.0)) * -2.0))) ^ 2.0) * cbrt(fma(pi, 0.5, Float64(-2.0 * Float64(Float64(pi * 0.5) - acos(sqrt(fma(x, -0.5, 0.5))))))))
end

Wolfram

code[x_] := N[(N[Power[N[Power[N[(Pi * 0.5 + N[(N[ArcSin[N[Power[N[Power[N[(0.5 - N[(0.5 * x), $MachinePrecision]), $MachinePrecision], 0.25], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision] * N[Power[N[(Pi * 0.5 + N[(-2.0 * N[(N[(Pi * 0.5), $MachinePrecision] - N[ArcCos[N[Sqrt[N[(x * -0.5 + 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 5.9%

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

   1. add-cube-cbrt 5.9%

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

4. Applied egg-rr 5.9%

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

   1. asin-acos 7.3%

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

   2. div-inv 7.3%

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

   3. metadata-eval 7.3%

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

   4. *-commutative 7.3%

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

   5. cancel-sign-sub-inv 7.3%

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

   6. metadata-eval 7.3%

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

   7. *-commutative 7.3%

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

   8. +-commutative 7.3%

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

   9. fma-define 7.3%

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

6. Applied egg-rr 7.3%

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

   1. add-sqr-sqrt 7.3%

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

   2. pow2 7.3%

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

   3. pow1/2 7.3%

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

   4. sqrt-pow1 7.3%

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

   5. metadata-eval 7.3%

      \[\leadsto {\left(\sqrt[3]{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left({\left({\left(0.5 - x \cdot 0.5\right)}^{\color{blue}{0.25}}\right)}^{2}\right) \cdot -2\right)}\right)}^{2} \cdot \sqrt[3]{\mathsf{fma}\left(\pi, 0.5, \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)\right) \cdot -2\right)} \]

8. Applied egg-rr 7.3%

   \[\leadsto {\left(\sqrt[3]{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \color{blue}{\left({\left({\left(0.5 - x \cdot 0.5\right)}^{0.25}\right)}^{2}\right)} \cdot -2\right)}\right)}^{2} \cdot \sqrt[3]{\mathsf{fma}\left(\pi, 0.5, \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)\right) \cdot -2\right)} \]

9. Final simplification 7.3%

   \[\leadsto {\left(\sqrt[3]{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left({\left({\left(0.5 - 0.5 \cdot x\right)}^{0.25}\right)}^{2}\right) \cdot -2\right)}\right)}^{2} \cdot \sqrt[3]{\mathsf{fma}\left(\pi, 0.5, -2 \cdot \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)\right)\right)} \]
10. Add Preprocessing

Alternative 2: 8.5% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	t_0 = sqrt(Float64(0.5 - Float64(0.5 * x)))
	return Float64((cbrt(fma(pi, 0.5, Float64(-2.0 * asin(t_0)))) ^ 2.0) * cbrt(Float64(Float64(pi * 0.5) - Float64(-2.0 * Float64(acos(t_0) - Float64(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[Power[N[Power[N[(Pi * 0.5 + N[(-2.0 * N[ArcSin[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision] * N[Power[N[(N[(Pi * 0.5), $MachinePrecision] - N[(-2.0 * N[(N[ArcCos[t$95$0], $MachinePrecision] - N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 5.9%

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

   1. add-cube-cbrt 5.9%

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

4. Applied egg-rr 5.9%

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

   1. asin-acos 7.3%

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

   2. div-inv 7.3%

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

   3. metadata-eval 7.3%

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

   4. *-commutative 7.3%

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

   5. cancel-sign-sub-inv 7.3%

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

   6. metadata-eval 7.3%

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

   7. *-commutative 7.3%

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

   8. +-commutative 7.3%

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

   9. fma-define 7.3%

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

6. Applied egg-rr 7.3%

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

7. Taylor expanded in x around inf 7.3%

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

8. Final simplification 7.3%

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

Alternative 3: 8.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 5.9%

   \[\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
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. add-cube-cbrt 5.5%

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

   3. associate-/l* 5.5%

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

   4. fma-neg 5.5%

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

   5. pow2 5.5%

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

   6. div-sub 5.5%

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

   7. metadata-eval 5.5%

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

   8. div-inv 5.5%

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

   9. metadata-eval 5.5%

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

4. Applied egg-rr 5.5%

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

   1. fma-neg 5.5%

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

   2. associate-*r/ 5.5%

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

   3. unpow2 5.5%

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

   4. rem-3cbrt-lft 7.3%

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

   5. sub-neg 7.3%

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

   6. distribute-rgt-neg-in 7.3%

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

   7. metadata-eval 7.3%

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

6. Simplified 7.3%

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

7. Final simplification 7.3%

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

Alternative 4: 7.1% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (- (/ PI 2.0) (* 2.0 (asin (pow (/ 2.0 (- 1.0 x)) -0.5)))))

C

double code(double x) {
	return (((double) M_PI) / 2.0) - (2.0 * asin(pow((2.0 / (1.0 - x)), -0.5)));
}

Java

public static double code(double x) {
	return (Math.PI / 2.0) - (2.0 * Math.asin(Math.pow((2.0 / (1.0 - x)), -0.5)));
}

Python

def code(x):
	return (math.pi / 2.0) - (2.0 * math.asin(math.pow((2.0 / (1.0 - x)), -0.5)))

Julia

function code(x)
	return Float64(Float64(pi / 2.0) - Float64(2.0 * asin((Float64(2.0 / Float64(1.0 - x)) ^ -0.5))))
end

MATLAB

function tmp = code(x)
	tmp = (pi / 2.0) - (2.0 * asin(((2.0 / (1.0 - x)) ^ -0.5)));
end

Wolfram

code[x_] := N[(N[(Pi / 2.0), $MachinePrecision] - N[(2.0 * N[ArcSin[N[Power[N[(2.0 / N[(1.0 - x), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 5.9%

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

   1. clear-num 5.9%

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

   2. sqrt-div 5.7%

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

   3. metadata-eval 5.7%

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

4. Applied egg-rr 5.7%

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

   1. *-un-lft-identity 5.7%

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

   2. inv-pow 5.7%

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

   3. sqrt-pow2 5.9%

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

   4. metadata-eval 5.9%

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

6. Applied egg-rr 5.9%

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

   1. *-lft-identity 5.9%

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

8. Simplified 5.9%

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

Alternative 5: 7.1% 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 5.9%

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

Alternative 6: 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 5.9%

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

3. Taylor expanded in x around 0 4.2%

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

Developer target: 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 2024100 
(FPCore (x)
  :name "Ian Simplification"
  :precision binary64

  :alt
  (asin x)

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