Ian Simplification

Percentage Accurate: 6.8% → 8.2%

Time: 24.2s

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: 8.2% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.8%

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

   1. asin-acos 9.2%

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

   2. div-inv 9.2%

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

   3. metadata-eval 9.2%

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

   4. div-sub 9.2%

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

   5. metadata-eval 9.2%

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

   6. div-inv 9.2%

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

   7. metadata-eval 9.2%

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

3. Applied egg-rr 9.2%

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

   1. div-inv 9.2%

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

   2. metadata-eval 9.2%

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

   3. log1p-expm1-u 9.2%

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

   4. log1p-udef 9.2%

      \[\leadsto \color{blue}{\log \left(1 + \mathsf{expm1}\left(\pi \cdot 0.5\right)\right)} - 2 \cdot \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)\right) \]

   5. metadata-eval 9.2%

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

   6. div-inv 9.2%

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

   7. asin-acos 7.8%

      \[\leadsto \log \left(1 + \mathsf{expm1}\left(\pi \cdot 0.5\right)\right) - 2 \cdot \color{blue}{\sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)} \]

   8. log1p-expm1-u 7.8%

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

   9. log1p-udef 7.8%

      \[\leadsto \log \left(1 + \mathsf{expm1}\left(\pi \cdot 0.5\right)\right) - \color{blue}{\log \left(1 + \mathsf{expm1}\left(2 \cdot \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)\right)\right)} \]

   10. log-div 7.8%

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

   11. clear-num 7.8%

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

   12. log-rec 7.8%

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

5. Applied egg-rr 7.8%

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

   1. pow1/2 7.8%

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

   2. metadata-eval 7.8%

      \[\leadsto -\log \left(e^{2 \cdot \sin^{-1} \left({\left(0.5 - 0.5 \cdot x\right)}^{\color{blue}{\left(0.25 \cdot 2\right)}}\right) - \pi \cdot 0.5}\right) \]

   3. pow-pow 7.8%

      \[\leadsto -\log \left(e^{2 \cdot \sin^{-1} \color{blue}{\left({\left({\left(0.5 - 0.5 \cdot x\right)}^{0.25}\right)}^{2}\right)} - \pi \cdot 0.5}\right) \]

   4. asin-acos 7.8%

      \[\leadsto -\log \left(e^{2 \cdot \color{blue}{\left(\frac{\pi}{2} - \cos^{-1} \left({\left({\left(0.5 - 0.5 \cdot x\right)}^{0.25}\right)}^{2}\right)\right)} - \pi \cdot 0.5}\right) \]

   5. div-inv 7.8%

      \[\leadsto -\log \left(e^{2 \cdot \left(\color{blue}{\pi \cdot \frac{1}{2}} - \cos^{-1} \left({\left({\left(0.5 - 0.5 \cdot x\right)}^{0.25}\right)}^{2}\right)\right) - \pi \cdot 0.5}\right) \]

   6. metadata-eval 7.8%

      \[\leadsto -\log \left(e^{2 \cdot \left(\pi \cdot \color{blue}{0.5} - \cos^{-1} \left({\left({\left(0.5 - 0.5 \cdot x\right)}^{0.25}\right)}^{2}\right)\right) - \pi \cdot 0.5}\right) \]

   7. pow-pow 9.2%

      \[\leadsto -\log \left(e^{2 \cdot \left(\pi \cdot 0.5 - \cos^{-1} \color{blue}{\left({\left(0.5 - 0.5 \cdot x\right)}^{\left(0.25 \cdot 2\right)}\right)}\right) - \pi \cdot 0.5}\right) \]

   8. *-commutative 9.2%

      \[\leadsto -\log \left(e^{2 \cdot \left(\pi \cdot 0.5 - \cos^{-1} \left({\left(0.5 - \color{blue}{x \cdot 0.5}\right)}^{\left(0.25 \cdot 2\right)}\right)\right) - \pi \cdot 0.5}\right) \]

   9. metadata-eval 9.2%

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

   10. pow1/2 9.2%

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

   11. sub-neg 9.2%

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

   12. distribute-rgt-neg-in 9.2%

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

   13. metadata-eval 9.2%

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

7. Applied egg-rr 9.2%

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

8. Final simplification 9.2%

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

Alternative 2: 8.2% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[Log[N[Exp[N[(N[(Pi * 0.5), $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]], $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 7.8%

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

   1. add-log-exp 7.8%

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

   2. sub-neg 7.8%

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

   3. +-commutative 7.8%

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

   4. *-commutative 7.8%

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

   5. distribute-rgt-neg-in 7.8%

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

   6. fma-def 7.8%

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

   7. div-sub 7.8%

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

   8. metadata-eval 7.8%

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

   9. div-inv 7.8%

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

   10. metadata-eval 7.8%

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

   11. metadata-eval 7.8%

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

   12. div-inv 7.8%

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

   13. metadata-eval 7.8%

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

3. Applied egg-rr 7.8%

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

   1. asin-acos 9.2%

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

   2. div-inv 9.2%

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

   3. metadata-eval 9.2%

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

   4. *-commutative 9.2%

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

5. Applied egg-rr 9.2%

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

6. Taylor expanded in x around 0 9.2%

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

7. Final simplification 9.2%

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

Alternative 3: 8.2% 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 7.8%

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

   1. asin-acos 9.2%

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

   2. div-inv 9.2%

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

   3. metadata-eval 9.2%

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

   4. div-sub 9.2%

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

   5. metadata-eval 9.2%

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

   6. div-inv 9.2%

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

   7. metadata-eval 9.2%

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

3. Applied egg-rr 9.2%

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

4. Final simplification 9.2%

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

Alternative 4: 5.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.32 \cdot 10^{-300}:\\ \;\;\;\;\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot 0.5 + 2 \cdot \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.32e-300)
   (- (/ PI 2.0) (* 2.0 (asin (sqrt 0.5))))
   (+ (* PI 0.5) (* 2.0 (asin (sqrt (- 0.5 (* 0.5 x))))))))

C

double code(double x) {
	double tmp;
	if (x <= 1.32e-300) {
		tmp = (((double) M_PI) / 2.0) - (2.0 * asin(sqrt(0.5)));
	} else {
		tmp = (((double) M_PI) * 0.5) + (2.0 * asin(sqrt((0.5 - (0.5 * x)))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.32e-300) {
		tmp = (Math.PI / 2.0) - (2.0 * Math.asin(Math.sqrt(0.5)));
	} else {
		tmp = (Math.PI * 0.5) + (2.0 * Math.asin(Math.sqrt((0.5 - (0.5 * x)))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.32e-300:
		tmp = (math.pi / 2.0) - (2.0 * math.asin(math.sqrt(0.5)))
	else:
		tmp = (math.pi * 0.5) + (2.0 * math.asin(math.sqrt((0.5 - (0.5 * x)))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.32e-300)
		tmp = Float64(Float64(pi / 2.0) - Float64(2.0 * asin(sqrt(0.5))));
	else
		tmp = Float64(Float64(pi * 0.5) + Float64(2.0 * asin(sqrt(Float64(0.5 - Float64(0.5 * x))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.32e-300)
		tmp = (pi / 2.0) - (2.0 * asin(sqrt(0.5)));
	else
		tmp = (pi * 0.5) + (2.0 * asin(sqrt((0.5 - (0.5 * x)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.32e-300], N[(N[(Pi / 2.0), $MachinePrecision] - N[(2.0 * N[ArcSin[N[Sqrt[0.5], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * 0.5), $MachinePrecision] + N[(2.0 * N[ArcSin[N[Sqrt[N[(0.5 - N[(0.5 * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.32e-300

   1. Initial program 9.0%

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

   2. Taylor expanded in x around 0 5.9%

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

   if 1.32e-300 < x

   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. sub-neg 6.6%

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

      2. +-commutative 6.6%

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

      3. add-sqr-sqrt 6.6%

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

      4. distribute-rgt-neg-in 6.6%

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

      5. fma-def 6.6%

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

   3. Applied egg-rr 6.6%

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

      1. fma-udef 6.6%

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

   5. Applied egg-rr 5.6%

      \[\leadsto \color{blue}{2 \cdot \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) + \pi \cdot 0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 5.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.32 \cdot 10^{-300}:\\ \;\;\;\;\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot 0.5 + 2 \cdot \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\\ \end{array} \]

Alternative 5: 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 7.8%

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

   1. clear-num 7.8%

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

   2. sqrt-div 7.9%

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

   3. metadata-eval 7.9%

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

3. Applied egg-rr 7.9%

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

4. Final simplification 7.9%

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

Alternative 6: 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 7.8%

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

2. Final simplification 7.8%

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

Alternative 7: 5.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \sin^{-1} \left(\sqrt{0.5}\right)\\ \mathbf{if}\;x \leq 1.32 \cdot 10^{-300}:\\ \;\;\;\;\frac{\pi}{2} - t_0\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot 0.5 + t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* 2.0 (asin (sqrt 0.5)))))
   (if (<= x 1.32e-300) (- (/ PI 2.0) t_0) (+ (* PI 0.5) t_0))))

C

double code(double x) {
	double t_0 = 2.0 * asin(sqrt(0.5));
	double tmp;
	if (x <= 1.32e-300) {
		tmp = (((double) M_PI) / 2.0) - t_0;
	} else {
		tmp = (((double) M_PI) * 0.5) + t_0;
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = 2.0 * Math.asin(Math.sqrt(0.5));
	double tmp;
	if (x <= 1.32e-300) {
		tmp = (Math.PI / 2.0) - t_0;
	} else {
		tmp = (Math.PI * 0.5) + t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = 2.0 * math.asin(math.sqrt(0.5))
	tmp = 0
	if x <= 1.32e-300:
		tmp = (math.pi / 2.0) - t_0
	else:
		tmp = (math.pi * 0.5) + t_0
	return tmp

Julia

function code(x)
	t_0 = Float64(2.0 * asin(sqrt(0.5)))
	tmp = 0.0
	if (x <= 1.32e-300)
		tmp = Float64(Float64(pi / 2.0) - t_0);
	else
		tmp = Float64(Float64(pi * 0.5) + t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 2.0 * asin(sqrt(0.5));
	tmp = 0.0;
	if (x <= 1.32e-300)
		tmp = (pi / 2.0) - t_0;
	else
		tmp = (pi * 0.5) + t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(2.0 * N[ArcSin[N[Sqrt[0.5], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1.32e-300], N[(N[(Pi / 2.0), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(Pi * 0.5), $MachinePrecision] + t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.32e-300

   1. Initial program 9.0%

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

   2. Taylor expanded in x around 0 5.9%

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

   if 1.32e-300 < x

   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. sub-neg 6.6%

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

      2. +-commutative 6.6%

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

      3. add-sqr-sqrt 6.6%

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

      4. distribute-rgt-neg-in 6.6%

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

      5. fma-def 6.6%

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

   3. Applied egg-rr 6.6%

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

      1. fma-udef 6.6%

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

   5. Applied egg-rr 5.6%

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

   6. Taylor expanded in x around 0 5.6%

      \[\leadsto 2 \cdot \sin^{-1} \color{blue}{\left(\sqrt{0.5}\right)} + \pi \cdot 0.5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 5.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.32 \cdot 10^{-300}:\\ \;\;\;\;\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot 0.5 + 2 \cdot \sin^{-1} \left(\sqrt{0.5}\right)\\ \end{array} \]

Alternative 8: 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 7.8%

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

   1. sub-neg 7.8%

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

   2. +-commutative 7.8%

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

   3. add-sqr-sqrt 7.7%

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

   4. distribute-rgt-neg-in 7.7%

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

   5. fma-def 7.7%

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

3. Applied egg-rr 7.7%

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

   1. fma-udef 7.7%

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

5. Applied egg-rr 3.9%

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

6. Taylor expanded in x around 0 3.9%

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

7. Final simplification 3.9%

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

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 2023187 
(FPCore (x)
  :name "Ian Simplification"
  :precision binary64

  :herbie-target
  (asin x)

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