Ian Simplification

Percentage Accurate: 6.9% → 8.3%

Time: 24.7s

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: 6.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 8.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\\ \frac{{\pi}^{3} \cdot 0.125 - {\left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)\right)}^{3} \cdot 8}{\mathsf{fma}\left(2 \cdot t_0, \mathsf{fma}\left(2, t_0, \pi \cdot 0.5\right), {\pi}^{2} \cdot 0.25\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (asin (sqrt (- 0.5 (* 0.5 x))))))
   (/
    (-
     (* (pow PI 3.0) 0.125)
     (* (pow (- (* PI 0.5) (acos (sqrt (+ 0.5 (* x -0.5))))) 3.0) 8.0))
    (fma (* 2.0 t_0) (fma 2.0 t_0 (* PI 0.5)) (* (pow PI 2.0) 0.25)))))

C

double code(double x) {
	double t_0 = asin(sqrt((0.5 - (0.5 * x))));
	return ((pow(((double) M_PI), 3.0) * 0.125) - (pow(((((double) M_PI) * 0.5) - acos(sqrt((0.5 + (x * -0.5))))), 3.0) * 8.0)) / fma((2.0 * t_0), fma(2.0, t_0, (((double) M_PI) * 0.5)), (pow(((double) M_PI), 2.0) * 0.25));
}

Julia

function code(x)
	t_0 = asin(sqrt(Float64(0.5 - Float64(0.5 * x))))
	return Float64(Float64(Float64((pi ^ 3.0) * 0.125) - Float64((Float64(Float64(pi * 0.5) - acos(sqrt(Float64(0.5 + Float64(x * -0.5))))) ^ 3.0) * 8.0)) / fma(Float64(2.0 * t_0), fma(2.0, t_0, Float64(pi * 0.5)), Float64((pi ^ 2.0) * 0.25)))
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[(N[Power[Pi, 3.0], $MachinePrecision] * 0.125), $MachinePrecision] - N[(N[Power[N[(N[(Pi * 0.5), $MachinePrecision] - N[ArcCos[N[Sqrt[N[(0.5 + N[(x * -0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * 8.0), $MachinePrecision]), $MachinePrecision] / N[(N[(2.0 * t$95$0), $MachinePrecision] * N[(2.0 * t$95$0 + N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[Power[Pi, 2.0], $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 8.4%

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

   1. flip3-- 8.4%

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

3. Applied egg-rr 8.4%

   \[\leadsto \color{blue}{\frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}^{3} \cdot 8}{\mathsf{fma}\left(2 \cdot \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right), {\pi}^{2} \cdot 0.25\right)}} \]
4. Step-by-step derivation

   1. asin-acos 9.8%

      \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - {\color{blue}{\left(\frac{\pi}{2} - \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)\right)}}^{3} \cdot 8}{\mathsf{fma}\left(2 \cdot \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right), {\pi}^{2} \cdot 0.25\right)} \]

   2. div-inv 9.8%

      \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - {\left(\color{blue}{\pi \cdot \frac{1}{2}} - \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)\right)}^{3} \cdot 8}{\mathsf{fma}\left(2 \cdot \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right), {\pi}^{2} \cdot 0.25\right)} \]

   3. metadata-eval 9.8%

      \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - {\left(\pi \cdot \color{blue}{0.5} - \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)\right)}^{3} \cdot 8}{\mathsf{fma}\left(2 \cdot \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right), {\pi}^{2} \cdot 0.25\right)} \]

   4. sub-neg 9.8%

      \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - {\left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\color{blue}{0.5 + \left(-x \cdot 0.5\right)}}\right)\right)}^{3} \cdot 8}{\mathsf{fma}\left(2 \cdot \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right), {\pi}^{2} \cdot 0.25\right)} \]

   5. distribute-rgt-neg-in 9.8%

      \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - {\left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{0.5 + \color{blue}{x \cdot \left(-0.5\right)}}\right)\right)}^{3} \cdot 8}{\mathsf{fma}\left(2 \cdot \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right), {\pi}^{2} \cdot 0.25\right)} \]

   6. metadata-eval 9.8%

      \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - {\left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{0.5 + x \cdot \color{blue}{-0.5}}\right)\right)}^{3} \cdot 8}{\mathsf{fma}\left(2 \cdot \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right), {\pi}^{2} \cdot 0.25\right)} \]

5. Applied egg-rr 9.8%

   \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - {\color{blue}{\left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)\right)}}^{3} \cdot 8}{\mathsf{fma}\left(2 \cdot \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right), {\pi}^{2} \cdot 0.25\right)} \]

6. Final simplification 9.8%

   \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - {\left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)\right)}^{3} \cdot 8}{\mathsf{fma}\left(2 \cdot \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right), \mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right), \pi \cdot 0.5\right), {\pi}^{2} \cdot 0.25\right)} \]

Alternative 2: 8.3% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (log
  (exp (fma PI 0.5 (* (- (* PI 0.5) (acos (sqrt (+ 0.5 (* x -0.5))))) -2.0)))))

C

double code(double x) {
	return log(exp(fma(((double) M_PI), 0.5, (((((double) M_PI) * 0.5) - acos(sqrt((0.5 + (x * -0.5))))) * -2.0))));
}

Julia

function code(x)
	return log(exp(fma(pi, 0.5, Float64(Float64(Float64(pi * 0.5) - acos(sqrt(Float64(0.5 + Float64(x * -0.5))))) * -2.0))))
end

Wolfram

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

Derivation

1. Initial program 8.4%

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

   1. add-log-exp 8.4%

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

   2. div-inv 8.4%

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

   3. metadata-eval 8.4%

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

   4. fma-neg 8.4%

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

   5. *-commutative 8.4%

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

   6. distribute-rgt-neg-in 8.4%

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

   7. div-sub 8.4%

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

   8. metadata-eval 8.4%

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

   9. div-inv 8.4%

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

   10. metadata-eval 8.4%

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

   11. metadata-eval 8.4%

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

3. Applied egg-rr 8.4%

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

   1. asin-acos 9.8%

      \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - {\color{blue}{\left(\frac{\pi}{2} - \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)\right)}}^{3} \cdot 8}{\mathsf{fma}\left(2 \cdot \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right), {\pi}^{2} \cdot 0.25\right)} \]

   2. div-inv 9.8%

      \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - {\left(\color{blue}{\pi \cdot \frac{1}{2}} - \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)\right)}^{3} \cdot 8}{\mathsf{fma}\left(2 \cdot \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right), {\pi}^{2} \cdot 0.25\right)} \]

   3. metadata-eval 9.8%

      \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - {\left(\pi \cdot \color{blue}{0.5} - \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)\right)}^{3} \cdot 8}{\mathsf{fma}\left(2 \cdot \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right), {\pi}^{2} \cdot 0.25\right)} \]

   4. sub-neg 9.8%

      \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - {\left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\color{blue}{0.5 + \left(-x \cdot 0.5\right)}}\right)\right)}^{3} \cdot 8}{\mathsf{fma}\left(2 \cdot \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right), {\pi}^{2} \cdot 0.25\right)} \]

   5. distribute-rgt-neg-in 9.8%

      \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - {\left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{0.5 + \color{blue}{x \cdot \left(-0.5\right)}}\right)\right)}^{3} \cdot 8}{\mathsf{fma}\left(2 \cdot \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right), {\pi}^{2} \cdot 0.25\right)} \]

   6. metadata-eval 9.8%

      \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - {\left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{0.5 + x \cdot \color{blue}{-0.5}}\right)\right)}^{3} \cdot 8}{\mathsf{fma}\left(2 \cdot \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right), {\pi}^{2} \cdot 0.25\right)} \]

5. Applied egg-rr 9.7%

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

6. Final simplification 9.7%

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

Alternative 3: 8.4% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 8.4%

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

   1. asin-acos 9.7%

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

   2. div-inv 9.7%

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

   3. metadata-eval 9.7%

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

   4. div-sub 9.7%

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

   5. metadata-eval 9.7%

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

   6. div-inv 9.7%

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

   7. metadata-eval 9.7%

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

3. Applied egg-rr 9.7%

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

4. Final simplification 9.7%

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

Alternative 4: 6.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 8.4%

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

2. Final simplification 8.4%

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

Alternative 5: 3.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 8.4%

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

   1. asin-acos 9.7%

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

   2. div-inv 9.7%

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

   3. metadata-eval 9.7%

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

   4. div-sub 9.7%

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

   5. metadata-eval 9.7%

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

   6. div-inv 9.7%

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

   7. metadata-eval 9.7%

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

3. Applied egg-rr 9.7%

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

   1. cancel-sign-sub-inv 9.7%

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

   2. div-inv 9.7%

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

   3. metadata-eval 9.7%

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

   4. metadata-eval 9.7%

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

   5. metadata-eval 9.7%

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

   6. div-inv 9.7%

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

   7. asin-acos 8.4%

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

   8. *-commutative 8.4%

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

   9. add-sqr-sqrt 0.0%

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

   10. sqrt-unprod 3.8%

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

   11. *-commutative 3.8%

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

   12. *-commutative 3.8%

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

   13. swap-sqr 3.8%

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

5. Applied egg-rr 3.8%

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

6. Taylor expanded in x around 0 3.8%

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

7. Final simplification 3.8%

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

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 8.4%

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

2. Taylor expanded in x around 0 4.3%

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

3. Final simplification 4.3%

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

Developer target: 100.0% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x)
	return asin(x)
end

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023334 
(FPCore (x)
  :name "Ian Simplification"
  :precision binary64

  :herbie-target
  (asin x)

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