Ian Simplification

Percentage Accurate: 6.9% → 8.4%

Time: 40.2s

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: 6.9% 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: 8.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \log \left(\frac{e^{\pi \cdot 0.5}}{1 + \mathsf{expm1}\left(2 \cdot \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\right)\right)}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (log
  (/
   (exp (* PI 0.5))
   (+ 1.0 (expm1 (* 2.0 (- (* PI 0.5) (acos (sqrt (- 0.5 (* 0.5 x)))))))))))

C

double code(double x) {
	return log((exp((((double) M_PI) * 0.5)) / (1.0 + expm1((2.0 * ((((double) M_PI) * 0.5) - acos(sqrt((0.5 - (0.5 * x))))))))));
}

Java

public static double code(double x) {
	return Math.log((Math.exp((Math.PI * 0.5)) / (1.0 + Math.expm1((2.0 * ((Math.PI * 0.5) - Math.acos(Math.sqrt((0.5 - (0.5 * x))))))))));
}

Python

def code(x):
	return math.log((math.exp((math.pi * 0.5)) / (1.0 + math.expm1((2.0 * ((math.pi * 0.5) - math.acos(math.sqrt((0.5 - (0.5 * x))))))))))

Julia

function code(x)
	return log(Float64(exp(Float64(pi * 0.5)) / Float64(1.0 + expm1(Float64(2.0 * Float64(Float64(pi * 0.5) - acos(sqrt(Float64(0.5 - Float64(0.5 * x))))))))))
end

Wolfram

code[x_] := N[Log[N[(N[Exp[N[(Pi * 0.5), $MachinePrecision]], $MachinePrecision] / N[(1.0 + N[(Exp[N[(2.0 * N[(N[(Pi * 0.5), $MachinePrecision] - N[ArcCos[N[Sqrt[N[(0.5 - N[(0.5 * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 7.5%

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

   1. log1p-expm1-u 7.5%

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

   2. log1p-undefine 7.5%

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

   3. log1p-expm1-u 7.5%

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

   4. log1p-undefine 7.4%

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

   5. diff-log 7.4%

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

   6. div-inv 7.4%

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

   7. metadata-eval 7.4%

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

   8. div-sub 7.4%

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

   9. metadata-eval 7.4%

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

   10. div-inv 7.4%

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

   11. metadata-eval 7.4%

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

4. Applied egg-rr 7.4%

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

   1. asin-acos 8.9%

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

   2. div-inv 8.9%

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

   3. metadata-eval 8.9%

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

   4. *-commutative 8.9%

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

6. Applied egg-rr 8.9%

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

   1. add-exp-log 8.9%

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

   2. log1p-define 8.9%

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

   3. log1p-expm1-u 8.9%

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

8. Applied egg-rr 8.9%

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

9. Final simplification 8.9%

   \[\leadsto \log \left(\frac{e^{\pi \cdot 0.5}}{1 + \mathsf{expm1}\left(2 \cdot \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\right)\right)}\right) \]
10. Add Preprocessing

Alternative 2: 5.6% accurate, 1.0× speedup

Math

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

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 (* x -0.5))))))))

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 + (x * -0.5)))));
	}
	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 + (x * -0.5)))));
	}
	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 + (x * -0.5)))))
	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(x * -0.5))))));
	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 + (x * -0.5)))));
	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[(x * -0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.32e-300

   1. Initial program 9.9%

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

   3. Taylor expanded in x around 0 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 5.1%

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

      1. log1p-expm1-u 5.1%

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

      2. log1p-undefine 5.1%

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

      3. log1p-expm1-u 5.1%

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

      4. log1p-undefine 5.0%

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

      5. diff-log 5.1%

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

      6. div-inv 5.1%

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

      7. metadata-eval 5.1%

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

      8. div-sub 5.1%

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

      9. metadata-eval 5.1%

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

      10. div-inv 5.1%

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

      11. metadata-eval 5.1%

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

   4. Applied egg-rr 5.1%

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

      1. asin-acos 7.7%

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

      2. div-inv 7.7%

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

      3. metadata-eval 7.7%

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

      4. *-commutative 7.7%

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

   6. Applied egg-rr 7.7%

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

   8. Applied egg-rr 5.4%

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

4. Final simplification 5.6%

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

Alternative 3: 8.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x) {
	return (((double) M_PI) / 2.0) + (2.0 * (1.0 + (-1.0 - asin(sqrt((0.5 + (x * -0.5)))))));
}

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.5%

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

   1. add0 7.5%

      \[\leadsto \frac{\pi}{2} - 2 \cdot \color{blue}{\left(\sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) + 0\right)} \]

   2. *-un-lft-identity 7.5%

      \[\leadsto \frac{\pi}{2} - 2 \cdot \left(\color{blue}{1 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)} + 0\right) \]

   3. fma-define 7.5%

      \[\leadsto \frac{\pi}{2} - 2 \cdot \color{blue}{\mathsf{fma}\left(1, \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right), 0\right)} \]

   4. metadata-eval 7.5%

      \[\leadsto \frac{\pi}{2} - 2 \cdot \mathsf{fma}\left(1, \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right), \color{blue}{-0}\right) \]

   5. fma-neg 7.5%

      \[\leadsto \frac{\pi}{2} - 2 \cdot \color{blue}{\left(1 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) - 0\right)} \]

   6. *-un-lft-identity 7.5%

      \[\leadsto \frac{\pi}{2} - 2 \cdot \left(\color{blue}{\sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)} - 0\right) \]

   7. expm1-log1p-u 7.5%

      \[\leadsto \frac{\pi}{2} - 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) - 0\right)\right)} \]

   8. *-un-lft-identity 7.5%

      \[\leadsto \frac{\pi}{2} - 2 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{1 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)} - 0\right)\right) \]

   9. fma-neg 7.5%

      \[\leadsto \frac{\pi}{2} - 2 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(1, \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right), -0\right)}\right)\right) \]

   10. metadata-eval 7.5%

       \[\leadsto \frac{\pi}{2} - 2 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(1, \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right), \color{blue}{0}\right)\right)\right) \]

   11. fma-define 7.5%

       \[\leadsto \frac{\pi}{2} - 2 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{1 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) + 0}\right)\right) \]

   12. *-un-lft-identity 7.5%

       \[\leadsto \frac{\pi}{2} - 2 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)} + 0\right)\right) \]

   13. add0 7.5%

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

   14. div-sub 7.5%

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

   15. metadata-eval 7.5%

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

   16. div-inv 7.5%

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

   17. metadata-eval 7.5%

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

4. Applied egg-rr 7.5%

   \[\leadsto \frac{\pi}{2} - 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)\right)\right)} \]
5. Step-by-step derivation

   1. expm1-undefine 7.4%

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

   2. log1p-expm1-u 7.4%

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

   3. log1p-undefine 8.8%

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

   4. rem-exp-log 8.8%

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

   5. expm1-log1p-u 8.8%

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

   6. sub-neg 8.8%

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

   7. distribute-rgt-neg-in 8.8%

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

   8. metadata-eval 8.8%

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

6. Applied egg-rr 8.8%

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

7. Final simplification 8.8%

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

Alternative 4: 8.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Python

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

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 / 2.0))))
end

MATLAB

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

Derivation

1. Initial program 7.5%

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

   1. asin-acos 8.9%

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

   2. add-cube-cbrt 7.2%

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

   3. associate-/l* 7.2%

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

   4. fma-neg 7.2%

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

   5. pow2 7.2%

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

   6. div-sub 7.2%

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

   7. metadata-eval 7.2%

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

   8. div-inv 7.2%

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

   9. metadata-eval 7.2%

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

4. Applied egg-rr 7.2%

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

   1. fma-neg 7.2%

      \[\leadsto \frac{\pi}{2} - 2 \cdot \color{blue}{\left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot \frac{\sqrt[3]{\pi}}{2} - \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)\right)} \]

   2. associate-*r/ 7.2%

      \[\leadsto \frac{\pi}{2} - 2 \cdot \left(\color{blue}{\frac{{\left(\sqrt[3]{\pi}\right)}^{2} \cdot \sqrt[3]{\pi}}{2}} - \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)\right) \]

   3. unpow2 7.2%

      \[\leadsto \frac{\pi}{2} - 2 \cdot \left(\frac{\color{blue}{\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right)} \cdot \sqrt[3]{\pi}}{2} - \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)\right) \]

   4. rem-3cbrt-lft 8.9%

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

   5. sub-neg 8.9%

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

   6. distribute-rgt-neg-in 8.9%

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

   7. metadata-eval 8.9%

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

6. Simplified 8.9%

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

7. Final simplification 8.9%

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

Alternative 5: 6.9% 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 7.5%

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

   1. clear-num 7.5%

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

   2. sqrt-div 7.5%

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

   3. metadata-eval 7.5%

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

4. Applied egg-rr 7.5%

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

5. Final simplification 7.5%

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

Alternative 6: 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 9.9%

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

   3. Taylor expanded in x around 0 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 5.1%

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

      1. log1p-expm1-u 5.1%

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

      2. log1p-undefine 5.1%

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

      3. log1p-expm1-u 5.1%

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

      4. log1p-undefine 5.0%

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

      5. diff-log 5.1%

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

      6. div-inv 5.1%

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

      7. metadata-eval 5.1%

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

      8. div-sub 5.1%

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

      9. metadata-eval 5.1%

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

      10. div-inv 5.1%

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

      11. metadata-eval 5.1%

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

   4. Applied egg-rr 5.1%

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

      1. asin-acos 7.7%

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

      2. div-inv 7.7%

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

      3. metadata-eval 7.7%

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

      4. *-commutative 7.7%

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

   6. Applied egg-rr 7.7%

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

   8. Applied egg-rr 5.4%

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

   9. Taylor expanded in x around 0 5.4%

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

4. Final simplification 5.6%

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

Alternative 7: 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 7.5%

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

3. Final simplification 7.5%

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

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 7.5%

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

   1. log1p-expm1-u 7.5%

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

   2. log1p-undefine 7.5%

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

   3. log1p-expm1-u 7.5%

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

   4. log1p-undefine 7.4%

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

   5. diff-log 7.4%

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

   6. div-inv 7.4%

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

   7. metadata-eval 7.4%

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

   8. div-sub 7.4%

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

   9. metadata-eval 7.4%

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

   10. div-inv 7.4%

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

   11. metadata-eval 7.4%

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

4. Applied egg-rr 7.4%

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

   1. asin-acos 8.9%

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

   2. div-inv 8.9%

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

   3. metadata-eval 8.9%

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

   4. *-commutative 8.9%

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

6. Applied egg-rr 8.9%

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

8. Applied egg-rr 3.9%

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

9. Taylor expanded in x around 0 3.9%

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

10. Final simplification 3.9%

    \[\leadsto \pi \cdot 0.5 + 2 \cdot \sin^{-1} \left(\sqrt{0.5}\right) \]
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 2024044 
(FPCore (x)
  :name "Ian Simplification"
  :precision binary64

  :alt
  (asin x)

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