Ian Simplification

Percentage Accurate: 6.7% → 8.1%

Time: 35.9s

Alternatives: 7

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: 6.7% 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: 8.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)\\ t_1 := \sqrt[3]{\mathsf{fma}\left(2, t\_0, 0.5 \cdot \pi\right)}\\ \frac{1}{{t\_1}^{2} \cdot \frac{t\_1}{\mathsf{fma}\left(0.25, {\left(\sqrt[3]{{\pi}^{3}}\right)}^{2}, -4 \cdot {t\_0}^{2}\right)}} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (asin (sqrt (+ 0.5 (* x -0.5)))))
        (t_1 (cbrt (fma 2.0 t_0 (* 0.5 PI)))))
   (/
    1.0
    (*
     (pow t_1 2.0)
     (/
      t_1
      (fma 0.25 (pow (cbrt (pow PI 3.0)) 2.0) (* -4.0 (pow t_0 2.0))))))))

C

double code(double x) {
	double t_0 = asin(sqrt((0.5 + (x * -0.5))));
	double t_1 = cbrt(fma(2.0, t_0, (0.5 * ((double) M_PI))));
	return 1.0 / (pow(t_1, 2.0) * (t_1 / fma(0.25, pow(cbrt(pow(((double) M_PI), 3.0)), 2.0), (-4.0 * pow(t_0, 2.0)))));
}

Julia

function code(x)
	t_0 = asin(sqrt(Float64(0.5 + Float64(x * -0.5))))
	t_1 = cbrt(fma(2.0, t_0, Float64(0.5 * pi)))
	return Float64(1.0 / Float64((t_1 ^ 2.0) * Float64(t_1 / fma(0.25, (cbrt((pi ^ 3.0)) ^ 2.0), Float64(-4.0 * (t_0 ^ 2.0))))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcSin[N[Sqrt[N[(0.5 + N[(x * -0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(2.0 * t$95$0 + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, N[(1.0 / N[(N[Power[t$95$1, 2.0], $MachinePrecision] * N[(t$95$1 / N[(0.25 * N[Power[N[Power[N[Power[Pi, 3.0], $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision] + N[(-4.0 * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 8.3%

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

   1. flip-- 8.3%

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

   2. clear-num 8.3%

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

4. Applied egg-rr 8.3%

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

   1. add-cube-cbrt 8.3%

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

   2. *-un-lft-identity 8.3%

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

   3. times-frac 8.3%

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

6. Applied egg-rr 8.3%

   \[\leadsto \frac{1}{\color{blue}{\frac{{\left(\sqrt[3]{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right), \pi \cdot 0.5\right)}\right)}^{2}}{1} \cdot \frac{\sqrt[3]{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right), \pi \cdot 0.5\right)}}{\mathsf{fma}\left(0.25, {\pi}^{2}, -4 \cdot {\sin^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)}^{2}\right)}}} \]
7. Step-by-step derivation

   1. add-cbrt-cube 9.4%

      \[\leadsto \frac{1}{\frac{{\left(\sqrt[3]{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right), \pi \cdot 0.5\right)}\right)}^{2}}{1} \cdot \frac{\sqrt[3]{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right), \pi \cdot 0.5\right)}}{\mathsf{fma}\left(0.25, {\color{blue}{\left(\sqrt[3]{\left(\pi \cdot \pi\right) \cdot \pi}\right)}}^{2}, -4 \cdot {\sin^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)}^{2}\right)}} \]

   2. pow3 9.4%

      \[\leadsto \frac{1}{\frac{{\left(\sqrt[3]{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right), \pi \cdot 0.5\right)}\right)}^{2}}{1} \cdot \frac{\sqrt[3]{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right), \pi \cdot 0.5\right)}}{\mathsf{fma}\left(0.25, {\left(\sqrt[3]{\color{blue}{{\pi}^{3}}}\right)}^{2}, -4 \cdot {\sin^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)}^{2}\right)}} \]

8. Applied egg-rr 9.4%

   \[\leadsto \frac{1}{\frac{{\left(\sqrt[3]{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right), \pi \cdot 0.5\right)}\right)}^{2}}{1} \cdot \frac{\sqrt[3]{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right), \pi \cdot 0.5\right)}}{\mathsf{fma}\left(0.25, {\color{blue}{\left(\sqrt[3]{{\pi}^{3}}\right)}}^{2}, -4 \cdot {\sin^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)}^{2}\right)}} \]

9. Final simplification 9.4%

   \[\leadsto \frac{1}{{\left(\sqrt[3]{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right), 0.5 \cdot \pi\right)}\right)}^{2} \cdot \frac{\sqrt[3]{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right), 0.5 \cdot \pi\right)}}{\mathsf{fma}\left(0.25, {\left(\sqrt[3]{{\pi}^{3}}\right)}^{2}, -4 \cdot {\sin^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)}^{2}\right)}} \]
10. Add Preprocessing

Alternative 2: 8.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \frac{{\left(0.5 \cdot \sqrt[3]{{\pi}^{3}}\right)}^{2} - {\left(2 \cdot \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\right)}^{2}}{\mathsf{fma}\left(2, e^{\mathsf{log1p}\left(\sin^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)\right)} + -1, 0.5 \cdot \pi\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (-
   (pow (* 0.5 (cbrt (pow PI 3.0))) 2.0)
   (pow (* 2.0 (asin (sqrt (- 0.5 (* 0.5 x))))) 2.0))
  (fma 2.0 (+ (exp (log1p (asin (sqrt (fma x -0.5 0.5))))) -1.0) (* 0.5 PI))))

C

double code(double x) {
	return (pow((0.5 * cbrt(pow(((double) M_PI), 3.0))), 2.0) - pow((2.0 * asin(sqrt((0.5 - (0.5 * x))))), 2.0)) / fma(2.0, (exp(log1p(asin(sqrt(fma(x, -0.5, 0.5))))) + -1.0), (0.5 * ((double) M_PI)));
}

Julia

function code(x)
	return Float64(Float64((Float64(0.5 * cbrt((pi ^ 3.0))) ^ 2.0) - (Float64(2.0 * asin(sqrt(Float64(0.5 - Float64(0.5 * x))))) ^ 2.0)) / fma(2.0, Float64(exp(log1p(asin(sqrt(fma(x, -0.5, 0.5))))) + -1.0), Float64(0.5 * pi)))
end

Wolfram

code[x_] := N[(N[(N[Power[N[(0.5 * N[Power[N[Power[Pi, 3.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[(2.0 * N[ArcSin[N[Sqrt[N[(0.5 - N[(0.5 * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(2.0 * N[(N[Exp[N[Log[1 + N[ArcSin[N[Sqrt[N[(x * -0.5 + 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision] + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 8.3%

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

   1. flip-- 8.3%

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

   2. pow2 8.3%

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

   3. div-inv 8.3%

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

   4. metadata-eval 8.3%

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

   5. pow2 8.3%

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

   6. div-sub 8.3%

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

   7. metadata-eval 8.3%

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

   8. div-inv 8.3%

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

   9. metadata-eval 8.3%

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

   10. +-commutative 8.3%

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

4. Applied egg-rr 8.3%

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

   1. add-cbrt-cube 9.4%

      \[\leadsto \frac{1}{\frac{{\left(\sqrt[3]{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right), \pi \cdot 0.5\right)}\right)}^{2}}{1} \cdot \frac{\sqrt[3]{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right), \pi \cdot 0.5\right)}}{\mathsf{fma}\left(0.25, {\color{blue}{\left(\sqrt[3]{\left(\pi \cdot \pi\right) \cdot \pi}\right)}}^{2}, -4 \cdot {\sin^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)}^{2}\right)}} \]

   2. pow3 9.4%

      \[\leadsto \frac{1}{\frac{{\left(\sqrt[3]{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right), \pi \cdot 0.5\right)}\right)}^{2}}{1} \cdot \frac{\sqrt[3]{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right), \pi \cdot 0.5\right)}}{\mathsf{fma}\left(0.25, {\left(\sqrt[3]{\color{blue}{{\pi}^{3}}}\right)}^{2}, -4 \cdot {\sin^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)}^{2}\right)}} \]

6. Applied egg-rr 9.4%

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

   1. asin-acos 9.4%

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

   2. *-commutative 9.4%

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

   3. cancel-sign-sub-inv 9.4%

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

   4. metadata-eval 9.4%

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

   5. *-commutative 9.4%

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

   6. expm1-log1p-u 9.4%

      \[\leadsto \frac{{\left(\sqrt[3]{{\pi}^{3}} \cdot 0.5\right)}^{2} - {\left(2 \cdot \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)\right)}^{2}}{\mathsf{fma}\left(2, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\pi}{2} - \cos^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)\right)\right)}, \pi \cdot 0.5\right)} \]

   7. expm1-undefine 9.4%

      \[\leadsto \frac{{\left(\sqrt[3]{{\pi}^{3}} \cdot 0.5\right)}^{2} - {\left(2 \cdot \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)\right)}^{2}}{\mathsf{fma}\left(2, \color{blue}{e^{\mathsf{log1p}\left(\frac{\pi}{2} - \cos^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)\right)} - 1}, \pi \cdot 0.5\right)} \]

8. Applied egg-rr 9.4%

   \[\leadsto \frac{{\left(\sqrt[3]{{\pi}^{3}} \cdot 0.5\right)}^{2} - {\left(2 \cdot \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)\right)}^{2}}{\mathsf{fma}\left(2, \color{blue}{e^{\mathsf{log1p}\left(\sin^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)\right)} - 1}, \pi \cdot 0.5\right)} \]

9. Final simplification 9.4%

   \[\leadsto \frac{{\left(0.5 \cdot \sqrt[3]{{\pi}^{3}}\right)}^{2} - {\left(2 \cdot \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\right)}^{2}}{\mathsf{fma}\left(2, e^{\mathsf{log1p}\left(\sin^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)\right)} + -1, 0.5 \cdot \pi\right)} \]
10. Add Preprocessing

Alternative 3: 8.1% accurate, 0.2× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (asin (sqrt (- 0.5 (* 0.5 x))))))
   (/
    (- (pow (* 0.5 (cbrt (pow PI 3.0))) 2.0) (pow (* 2.0 t_0) 2.0))
    (fma 2.0 t_0 (* 0.5 PI)))))

C

double code(double x) {
	double t_0 = asin(sqrt((0.5 - (0.5 * x))));
	return (pow((0.5 * cbrt(pow(((double) M_PI), 3.0))), 2.0) - pow((2.0 * t_0), 2.0)) / fma(2.0, t_0, (0.5 * ((double) M_PI)));
}

Julia

function code(x)
	t_0 = asin(sqrt(Float64(0.5 - Float64(0.5 * x))))
	return Float64(Float64((Float64(0.5 * cbrt((pi ^ 3.0))) ^ 2.0) - (Float64(2.0 * t_0) ^ 2.0)) / fma(2.0, t_0, Float64(0.5 * pi)))
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcSin[N[Sqrt[N[(0.5 - N[(0.5 * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, N[(N[(N[Power[N[(0.5 * N[Power[N[Power[Pi, 3.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[(2.0 * t$95$0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(2.0 * t$95$0 + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 8.3%

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

   1. flip-- 8.3%

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

   2. pow2 8.3%

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

   3. div-inv 8.3%

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

   4. metadata-eval 8.3%

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

   5. pow2 8.3%

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

   6. div-sub 8.3%

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

   7. metadata-eval 8.3%

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

   8. div-inv 8.3%

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

   9. metadata-eval 8.3%

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

   10. +-commutative 8.3%

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

4. Applied egg-rr 8.3%

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

   1. add-cbrt-cube 9.4%

      \[\leadsto \frac{1}{\frac{{\left(\sqrt[3]{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right), \pi \cdot 0.5\right)}\right)}^{2}}{1} \cdot \frac{\sqrt[3]{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right), \pi \cdot 0.5\right)}}{\mathsf{fma}\left(0.25, {\color{blue}{\left(\sqrt[3]{\left(\pi \cdot \pi\right) \cdot \pi}\right)}}^{2}, -4 \cdot {\sin^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)}^{2}\right)}} \]

   2. pow3 9.4%

      \[\leadsto \frac{1}{\frac{{\left(\sqrt[3]{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right), \pi \cdot 0.5\right)}\right)}^{2}}{1} \cdot \frac{\sqrt[3]{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right), \pi \cdot 0.5\right)}}{\mathsf{fma}\left(0.25, {\left(\sqrt[3]{\color{blue}{{\pi}^{3}}}\right)}^{2}, -4 \cdot {\sin^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)}^{2}\right)}} \]

6. Applied egg-rr 9.4%

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

7. Final simplification 9.4%

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

Alternative 4: 8.1% 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 8.3%

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

   1. asin-acos 9.4%

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

   2. add-cube-cbrt 7.4%

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

   3. associate-/l* 7.4%

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

   4. fma-neg 7.4%

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

   5. pow2 7.4%

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

   6. div-sub 7.4%

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

   7. metadata-eval 7.4%

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

   8. div-inv 7.4%

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

   9. metadata-eval 7.4%

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

4. Applied egg-rr 7.4%

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

   1. fma-neg 7.4%

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

   2. associate-*r/ 7.4%

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

   3. unpow2 7.4%

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

   4. rem-3cbrt-lft 9.4%

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

   5. sub-neg 9.4%

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

   6. distribute-rgt-neg-in 9.4%

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

   7. metadata-eval 9.4%

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

6. Simplified 9.4%

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

7. Final simplification 9.4%

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

Alternative 5: 6.7% 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 8.3%

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

3. Final simplification 8.3%

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

Alternative 6: 3.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 8.3%

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

   1. add-cbrt-cube 8.3%

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

   2. pow3 8.3%

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

4. Applied egg-rr 8.3%

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

   1. rem-cbrt-cube 8.3%

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

   2. fma-undefine 8.3%

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

   3. add-sqr-sqrt 0.0%

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

   4. sqrt-unprod 3.8%

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

   5. *-commutative 3.8%

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

   6. *-commutative 3.8%

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

   7. swap-sqr 3.8%

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

   8. metadata-eval 3.8%

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

   9. metadata-eval 3.8%

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

   10. swap-sqr 3.8%

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

   11. sqrt-prod 3.8%

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

   12. add-sqr-sqrt 3.8%

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

   13. sub-neg 3.8%

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

   14. distribute-rgt-neg-in 3.8%

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

   15. metadata-eval 3.8%

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

6. Applied egg-rr 3.8%

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

7. Taylor expanded in x around 0 3.8%

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

8. Final simplification 3.8%

   \[\leadsto 0.5 \cdot \pi + 2 \cdot \sin^{-1} \left(\sqrt{0.5}\right) \]
9. Add Preprocessing

Alternative 7: 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 8.3%

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

3. Taylor expanded in x around 0 4.4%

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

4. Final simplification 4.4%

   \[\leadsto \frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{0.5}\right) \]
5. 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 2024071 
(FPCore (x)
  :name "Ian Simplification"
  :precision binary64

  :alt
  (asin x)

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