Ian Simplification

Percentage Accurate: 6.8% → 8.3%

Time: 35.6s

Alternatives: 8

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, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.6%

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

   1. flip-- 6.6%

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

   2. div-inv 6.6%

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

   3. *-commutative 6.6%

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

3. Applied egg-rr 6.6%

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

   1. add-cbrt-cube 8.2%

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

   2. pow3 8.2%

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

   3. +-commutative 8.2%

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

   4. fma-def 8.2%

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

5. Applied egg-rr 8.2%

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

   1. asin-acos 8.2%

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

   2. div-inv 8.2%

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

   3. metadata-eval 8.2%

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

   4. +-commutative 8.2%

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

   5. fma-def 8.2%

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

7. Applied egg-rr 8.2%

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

8. Final simplification 8.2%

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

Alternative 2: 8.3% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.6%

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

   1. flip-- 6.6%

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

   2. div-inv 6.6%

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

   3. *-commutative 6.6%

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

3. Applied egg-rr 6.6%

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

   1. add-cbrt-cube 8.2%

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

   2. pow3 8.2%

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

   3. +-commutative 8.2%

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

   4. fma-def 8.2%

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

5. Applied egg-rr 8.2%

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

6. Final simplification 8.2%

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

Alternative 3: 8.3% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Python

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

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 * 0.5))))
end

MATLAB

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

Derivation

1. Initial program 6.6%

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

   1. asin-acos 8.1%

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

   2. div-inv 8.1%

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

   3. metadata-eval 8.1%

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

   4. div-sub 8.1%

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

   5. sub-neg 8.1%

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

   6. metadata-eval 8.1%

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

   7. div-inv 8.1%

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

   8. distribute-rgt-neg-in 8.1%

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

   9. metadata-eval 8.1%

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

   10. metadata-eval 8.1%

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

3. Applied egg-rr 8.1%

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

4. Final simplification 8.1%

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

Alternative 4: 6.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.6%

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

   1. clear-num 6.6%

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

   2. sqrt-div 6.8%

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

   3. metadata-eval 6.8%

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

3. Applied egg-rr 6.8%

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

4. Final simplification 6.8%

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

Alternative 5: 6.8% 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 6.6%

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

   1. clear-num 6.6%

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

   2. sqrt-div 6.8%

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

   3. metadata-eval 6.8%

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

3. Applied egg-rr 6.8%

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

   1. inv-pow 6.8%

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

   2. metadata-eval 6.8%

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

   3. sqrt-pow2 6.6%

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

   4. metadata-eval 6.6%

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

   5. metadata-eval 6.6%

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

5. Applied egg-rr 6.6%

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

6. Final simplification 6.6%

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

Alternative 6: 5.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\frac{1}{\sqrt{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1e-310)
   (- (/ PI 2.0) (* 2.0 (asin (sqrt 0.5))))
   (- (/ PI 2.0) (* 2.0 (asin (/ 1.0 (sqrt 2.0)))))))

C

double code(double x) {
	double tmp;
	if (x <= -1e-310) {
		tmp = (((double) M_PI) / 2.0) - (2.0 * asin(sqrt(0.5)));
	} else {
		tmp = (((double) M_PI) / 2.0) - (2.0 * asin((1.0 / sqrt(2.0))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -1e-310) {
		tmp = (Math.PI / 2.0) - (2.0 * Math.asin(Math.sqrt(0.5)));
	} else {
		tmp = (Math.PI / 2.0) - (2.0 * Math.asin((1.0 / Math.sqrt(2.0))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1e-310:
		tmp = (math.pi / 2.0) - (2.0 * math.asin(math.sqrt(0.5)))
	else:
		tmp = (math.pi / 2.0) - (2.0 * math.asin((1.0 / math.sqrt(2.0))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1e-310)
		tmp = Float64(Float64(pi / 2.0) - Float64(2.0 * asin(sqrt(0.5))));
	else
		tmp = Float64(Float64(pi / 2.0) - Float64(2.0 * asin(Float64(1.0 / sqrt(2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1e-310)
		tmp = (pi / 2.0) - (2.0 * asin(sqrt(0.5)));
	else
		tmp = (pi / 2.0) - (2.0 * asin((1.0 / sqrt(2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1e-310], N[(N[(Pi / 2.0), $MachinePrecision] - N[(2.0 * N[ArcSin[N[Sqrt[0.5], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi / 2.0), $MachinePrecision] - N[(2.0 * N[ArcSin[N[(1.0 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -9.999999999999969e-311

   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 5.6%

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

   if -9.999999999999969e-311 < x

   1. Initial program 4.8%

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

      1. clear-num 4.8%

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

      2. sqrt-div 7.8%

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

      3. metadata-eval 7.8%

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

   3. Applied egg-rr 7.8%

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

   4. Taylor expanded in x around 0 5.8%

      \[\leadsto \frac{\pi}{2} - 2 \cdot \sin^{-1} \color{blue}{\left(\frac{1}{\sqrt{2}}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 5.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\frac{1}{\sqrt{2}}\right)\\ \end{array} \]

Alternative 7: 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 6.6%

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

2. Final simplification 6.6%

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

Alternative 8: 4.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.6%

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

2. Taylor expanded in x around 0 3.9%

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

3. Final simplification 3.9%

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

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 2023326 
(FPCore (x)
  :name "Ian Simplification"
  :precision binary64

  :herbie-target
  (asin x)

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