Ian Simplification

Percentage Accurate: 6.8% → 8.3%

Time: 20.7s

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: 6.8% 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: 8.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \pi - \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)\\ \frac{1}{\frac{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right), 0.5 \cdot \pi\right)}{\frac{\log \left(e^{\mathsf{fma}\left({\pi}^{4}, 0.0625, {t_0}^{4} \cdot -16\right)}\right)}{\mathsf{fma}\left(4, {t_0}^{2}, 0.25 \cdot {\pi}^{2}\right)}}} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (* 0.5 PI) (acos (sqrt (fma x -0.5 0.5))))))
   (/
    1.0
    (/
     (fma 2.0 (asin (sqrt (- 0.5 (* 0.5 x)))) (* 0.5 PI))
     (/
      (log (exp (fma (pow PI 4.0) 0.0625 (* (pow t_0 4.0) -16.0))))
      (fma 4.0 (pow t_0 2.0) (* 0.25 (pow PI 2.0))))))))

C

double code(double x) {
	double t_0 = (0.5 * ((double) M_PI)) - acos(sqrt(fma(x, -0.5, 0.5)));
	return 1.0 / (fma(2.0, asin(sqrt((0.5 - (0.5 * x)))), (0.5 * ((double) M_PI))) / (log(exp(fma(pow(((double) M_PI), 4.0), 0.0625, (pow(t_0, 4.0) * -16.0)))) / fma(4.0, pow(t_0, 2.0), (0.25 * pow(((double) M_PI), 2.0)))));
}

Julia

function code(x)
	t_0 = Float64(Float64(0.5 * pi) - acos(sqrt(fma(x, -0.5, 0.5))))
	return Float64(1.0 / Float64(fma(2.0, asin(sqrt(Float64(0.5 - Float64(0.5 * x)))), Float64(0.5 * pi)) / Float64(log(exp(fma((pi ^ 4.0), 0.0625, Float64((t_0 ^ 4.0) * -16.0)))) / fma(4.0, (t_0 ^ 2.0), Float64(0.25 * (pi ^ 2.0))))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(0.5 * Pi), $MachinePrecision] - N[ArcCos[N[Sqrt[N[(x * -0.5 + 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(1.0 / N[(N[(2.0 * N[ArcSin[N[Sqrt[N[(0.5 - N[(0.5 * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision] / N[(N[Log[N[Exp[N[(N[Power[Pi, 4.0], $MachinePrecision] * 0.0625 + N[(N[Power[t$95$0, 4.0], $MachinePrecision] * -16.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / N[(4.0 * N[Power[t$95$0, 2.0], $MachinePrecision] + N[(0.25 * N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 6.1%

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

   1. flip-- 6.1%

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

   2. clear-num 6.1%

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

3. Applied egg-rr 6.1%

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

   1. asin-acos 7.4%

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

   2. div-inv 7.4%

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

   3. metadata-eval 7.4%

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

   4. *-commutative 7.4%

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

5. Applied egg-rr 7.4%

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

   1. flip-- 7.4%

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

7. Applied egg-rr 7.4%

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

9. Simplified 7.4%

   \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)}{\color{blue}{\frac{\mathsf{fma}\left({\pi}^{4}, 0.0625, {\left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)\right)}^{4} \cdot -16\right)}{\mathsf{fma}\left(4, {\left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)\right)}^{2}, 0.25 \cdot {\pi}^{2}\right)}}}} \]
10. Step-by-step derivation

    1. add-log-exp 7.4%

       \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)}{\frac{\color{blue}{\log \left(e^{\mathsf{fma}\left({\pi}^{4}, 0.0625, {\left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)\right)}^{4} \cdot -16\right)}\right)}}{\mathsf{fma}\left(4, {\left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)\right)}^{2}, 0.25 \cdot {\pi}^{2}\right)}}} \]

11. Applied egg-rr 7.4%

    \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)}{\frac{\color{blue}{\log \left(e^{\mathsf{fma}\left({\pi}^{4}, 0.0625, {\left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)\right)}^{4} \cdot -16\right)}\right)}}{\mathsf{fma}\left(4, {\left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)\right)}^{2}, 0.25 \cdot {\pi}^{2}\right)}}} \]

12. Final simplification 7.4%

    \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right), 0.5 \cdot \pi\right)}{\frac{\log \left(e^{\mathsf{fma}\left({\pi}^{4}, 0.0625, {\left(0.5 \cdot \pi - \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)\right)}^{4} \cdot -16\right)}\right)}{\mathsf{fma}\left(4, {\left(0.5 \cdot \pi - \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)\right)}^{2}, 0.25 \cdot {\pi}^{2}\right)}}} \]

Alternative 2: 8.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \pi - \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)\\ \frac{1}{\frac{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right), 0.5 \cdot \pi\right)}{\frac{{t_0}^{4} \cdot -16 + {\pi}^{4} \cdot 0.0625}{\mathsf{fma}\left(4, {t_0}^{2}, 0.25 \cdot {\pi}^{2}\right)}}} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (* 0.5 PI) (acos (sqrt (fma x -0.5 0.5))))))
   (/
    1.0
    (/
     (fma 2.0 (asin (sqrt (- 0.5 (* 0.5 x)))) (* 0.5 PI))
     (/
      (+ (* (pow t_0 4.0) -16.0) (* (pow PI 4.0) 0.0625))
      (fma 4.0 (pow t_0 2.0) (* 0.25 (pow PI 2.0))))))))

C

double code(double x) {
	double t_0 = (0.5 * ((double) M_PI)) - acos(sqrt(fma(x, -0.5, 0.5)));
	return 1.0 / (fma(2.0, asin(sqrt((0.5 - (0.5 * x)))), (0.5 * ((double) M_PI))) / (((pow(t_0, 4.0) * -16.0) + (pow(((double) M_PI), 4.0) * 0.0625)) / fma(4.0, pow(t_0, 2.0), (0.25 * pow(((double) M_PI), 2.0)))));
}

Julia

function code(x)
	t_0 = Float64(Float64(0.5 * pi) - acos(sqrt(fma(x, -0.5, 0.5))))
	return Float64(1.0 / Float64(fma(2.0, asin(sqrt(Float64(0.5 - Float64(0.5 * x)))), Float64(0.5 * pi)) / Float64(Float64(Float64((t_0 ^ 4.0) * -16.0) + Float64((pi ^ 4.0) * 0.0625)) / fma(4.0, (t_0 ^ 2.0), Float64(0.25 * (pi ^ 2.0))))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(0.5 * Pi), $MachinePrecision] - N[ArcCos[N[Sqrt[N[(x * -0.5 + 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(1.0 / N[(N[(2.0 * N[ArcSin[N[Sqrt[N[(0.5 - N[(0.5 * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[Power[t$95$0, 4.0], $MachinePrecision] * -16.0), $MachinePrecision] + N[(N[Power[Pi, 4.0], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision] / N[(4.0 * N[Power[t$95$0, 2.0], $MachinePrecision] + N[(0.25 * N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 6.1%

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

   1. flip-- 6.1%

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

   2. clear-num 6.1%

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

3. Applied egg-rr 6.1%

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

   1. asin-acos 7.4%

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

   2. div-inv 7.4%

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

   3. metadata-eval 7.4%

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

   4. *-commutative 7.4%

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

5. Applied egg-rr 7.4%

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

   1. flip-- 7.4%

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

7. Applied egg-rr 7.4%

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

9. Simplified 7.4%

   \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)}{\color{blue}{\frac{\mathsf{fma}\left({\pi}^{4}, 0.0625, {\left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)\right)}^{4} \cdot -16\right)}{\mathsf{fma}\left(4, {\left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)\right)}^{2}, 0.25 \cdot {\pi}^{2}\right)}}}} \]
10. Step-by-step derivation

    1. fma-udef 7.4%

       \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)}{\frac{\color{blue}{{\pi}^{4} \cdot 0.0625 + {\left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)\right)}^{4} \cdot -16}}{\mathsf{fma}\left(4, {\left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)\right)}^{2}, 0.25 \cdot {\pi}^{2}\right)}}} \]

11. Applied egg-rr 7.4%

    \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)}{\frac{\color{blue}{{\pi}^{4} \cdot 0.0625 + {\left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)\right)}^{4} \cdot -16}}{\mathsf{fma}\left(4, {\left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)\right)}^{2}, 0.25 \cdot {\pi}^{2}\right)}}} \]

12. Final simplification 7.4%

    \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right), 0.5 \cdot \pi\right)}{\frac{{\left(0.5 \cdot \pi - \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)\right)}^{4} \cdot -16 + {\pi}^{4} \cdot 0.0625}{\mathsf{fma}\left(4, {\left(0.5 \cdot \pi - \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)\right)}^{2}, 0.25 \cdot {\pi}^{2}\right)}}} \]

Alternative 3: 8.3% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double x) {
	double t_0 = Math.sqrt((0.5 - (0.5 * x)));
	return 1.0 / (((0.5 * Math.PI) + (2.0 * Math.asin(t_0))) / ((0.25 * Math.pow(Math.PI, 2.0)) - (4.0 * Math.pow(((Math.PI / 2.0) - Math.acos(t_0)), 2.0))));
}

Python

def code(x):
	t_0 = math.sqrt((0.5 - (0.5 * x)))
	return 1.0 / (((0.5 * math.pi) + (2.0 * math.asin(t_0))) / ((0.25 * math.pow(math.pi, 2.0)) - (4.0 * math.pow(((math.pi / 2.0) - math.acos(t_0)), 2.0))))

Julia

function code(x)
	t_0 = sqrt(Float64(0.5 - Float64(0.5 * x)))
	return Float64(1.0 / Float64(Float64(Float64(0.5 * pi) + Float64(2.0 * asin(t_0))) / Float64(Float64(0.25 * (pi ^ 2.0)) - Float64(4.0 * (Float64(Float64(pi / 2.0) - acos(t_0)) ^ 2.0)))))
end

MATLAB

function tmp = code(x)
	t_0 = sqrt((0.5 - (0.5 * x)));
	tmp = 1.0 / (((0.5 * pi) + (2.0 * asin(t_0))) / ((0.25 * (pi ^ 2.0)) - (4.0 * (((pi / 2.0) - acos(t_0)) ^ 2.0))));
end

Wolfram

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

Derivation

1. Initial program 6.1%

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

   1. flip-- 6.1%

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

   2. clear-num 6.1%

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

3. Applied egg-rr 6.1%

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

4. Taylor expanded in x around 0 6.1%

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

   1. asin-acos 7.4%

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

   2. *-commutative 7.4%

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

   3. *-commutative 7.4%

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

   4. cancel-sign-sub-inv 7.4%

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

   5. metadata-eval 7.4%

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

6. Applied egg-rr 7.4%

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

   1. metadata-eval 7.4%

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

   2. cancel-sign-sub-inv 7.4%

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

8. Simplified 7.4%

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

9. Final simplification 7.4%

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

Alternative 4: 8.3% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(x)
	tmp = (pi / 2.0) + (2.0 * (acos(sqrt((0.5 - (0.5 * x)))) - (0.5 * pi)));
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[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.1%

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

   1. asin-acos 7.4%

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

   2. div-inv 7.4%

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

   3. metadata-eval 7.4%

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

   4. div-sub 7.4%

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

   5. metadata-eval 7.4%

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

   6. div-inv 7.4%

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

   7. metadata-eval 7.4%

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

3. Applied egg-rr 7.4%

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

4. Final simplification 7.4%

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

Alternative 5: 6.8% 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 6.1%

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

2. Final simplification 6.1%

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

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 2023196 
(FPCore (x)
  :name "Ian Simplification"
  :precision binary64

  :herbie-target
  (asin x)

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