Ian Simplification

Percentage Accurate: 6.9% → 8.3%

Time: 46.2s

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: 6.9% 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, 0.7× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (+ (/ PI 2.0) (* 2.0 (- (acos (* (sqrt (- 1.0 x)) (sqrt 0.5))) (/ PI 2.0)))))

C

double code(double x) {
	return (((double) M_PI) / 2.0) + (2.0 * (acos((sqrt((1.0 - x)) * sqrt(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((1.0 - x)) * Math.sqrt(0.5))) - (Math.PI / 2.0)));
}

Python

def code(x):
	return (math.pi / 2.0) + (2.0 * (math.acos((math.sqrt((1.0 - x)) * math.sqrt(0.5))) - (math.pi / 2.0)))

Julia

function code(x)
	return Float64(Float64(pi / 2.0) + Float64(2.0 * Float64(acos(Float64(sqrt(Float64(1.0 - x)) * sqrt(0.5))) - Float64(pi / 2.0))))
end

MATLAB

function tmp = code(x)
	tmp = (pi / 2.0) + (2.0 * (acos((sqrt((1.0 - x)) * sqrt(0.5))) - (pi / 2.0)));
end

Wolfram

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

Derivation

1. Initial program 7.9%

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

   1. asin-acos 9.1%

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

   2. add-sqr-sqrt 7.4%

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

   3. associate-/l* 7.4%

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

   4. fma-neg 7.4%

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

   5. div-inv 7.4%

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

   6. metadata-eval 7.4%

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

4. Applied egg-rr 7.4%

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

   1. fma-neg 7.4%

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

   2. associate-*r/ 7.4%

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

   3. rem-square-sqrt 9.1%

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

   4. *-commutative 9.1%

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

6. Simplified 9.1%

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

   1. *-commutative 9.1%

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

   2. sqrt-prod 9.1%

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

8. Applied egg-rr 9.1%

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

9. Final simplification 9.1%

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

Alternative 2: 8.4% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (+ (/ PI 2.0) (* 2.0 (- (acos (sqrt (* (- 1.0 x) 0.5))) (/ PI 2.0)))))

C

double code(double x) {
	return (((double) M_PI) / 2.0) + (2.0 * (acos(sqrt(((1.0 - 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(((1.0 - x) * 0.5))) - (Math.PI / 2.0)));
}

Python

def code(x):
	return (math.pi / 2.0) + (2.0 * (math.acos(math.sqrt(((1.0 - x) * 0.5))) - (math.pi / 2.0)))

Julia

function code(x)
	return Float64(Float64(pi / 2.0) + Float64(2.0 * Float64(acos(sqrt(Float64(Float64(1.0 - x) * 0.5))) - Float64(pi / 2.0))))
end

MATLAB

function tmp = code(x)
	tmp = (pi / 2.0) + (2.0 * (acos(sqrt(((1.0 - 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[(N[(1.0 - x), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] - N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.9%

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

   1. asin-acos 9.1%

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

   2. add-sqr-sqrt 7.4%

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

   3. associate-/l* 7.4%

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

   4. fma-neg 7.4%

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

   5. div-inv 7.4%

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

   6. metadata-eval 7.4%

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

4. Applied egg-rr 7.4%

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

   1. fma-neg 7.4%

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

   2. associate-*r/ 7.4%

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

   3. rem-square-sqrt 9.1%

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

   4. *-commutative 9.1%

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

6. Simplified 9.1%

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

7. Final simplification 9.1%

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

Alternative 3: 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 7.9%

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

Alternative 4: 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 7.9%

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

3. Taylor expanded in x around 0 4.2%

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

Alternative 5: 3.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.9%

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

   1. asin-acos 9.1%

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

   2. add-sqr-sqrt 7.4%

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

   3. associate-/l* 7.4%

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

   4. fma-neg 7.4%

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

   5. div-inv 7.4%

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

   6. metadata-eval 7.4%

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

4. Applied egg-rr 7.4%

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

   1. fma-neg 7.4%

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

   2. associate-*r/ 7.4%

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

   3. rem-square-sqrt 9.1%

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

   4. *-commutative 9.1%

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

6. Simplified 9.1%

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

   1. cancel-sign-sub-inv 9.1%

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

   2. div-inv 9.1%

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

   3. metadata-eval 9.1%

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

   4. metadata-eval 9.1%

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

   5. *-commutative 9.1%

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

   6. asin-acos 7.9%

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

   7. *-commutative 7.9%

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

   8. add-sqr-sqrt 0.0%

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

   9. sqrt-unprod 3.8%

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

   10. swap-sqr 3.8%

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

   11. pow2 3.8%

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

   12. metadata-eval 3.8%

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

   13. metadata-eval 3.8%

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

8. Applied egg-rr 3.8%

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

9. Taylor expanded in x around 0 3.8%

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

10. Final simplification 3.8%

    \[\leadsto 2 \cdot \sin^{-1} \left(\sqrt{0.5}\right) + \pi \cdot 0.5 \]
11. 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 2024096 
(FPCore (x)
  :name "Ian Simplification"
  :precision binary64

  :alt
  (asin x)

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