Ian Simplification

Percentage Accurate: 6.8% → 8.2%

Time: 24.8s

Alternatives: 6

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 6.8% 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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 8.1%

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

   1. add-log-exp 8.1%

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

   2. div-inv 8.1%

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

   3. metadata-eval 8.1%

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

   4. fma-neg 8.1%

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

   5. *-commutative 8.1%

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

   6. distribute-rgt-neg-in 8.1%

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

   7. div-sub 8.1%

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

   8. metadata-eval 8.1%

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

   9. div-inv 8.1%

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

   10. metadata-eval 8.1%

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

   11. metadata-eval 8.1%

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

4. Applied egg-rr 8.1%

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

   1. asin-acos 9.4%

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

   2. div-inv 9.4%

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

   3. metadata-eval 9.4%

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

   4. sub-neg 9.4%

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

   5. distribute-rgt-neg-in 9.4%

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

   6. metadata-eval 9.4%

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

6. Applied egg-rr 9.4%

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

7. Taylor expanded in x around inf 9.4%

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

8. Final simplification 9.4%

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

Alternative 2: 8.2% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 8.1%

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

   1. asin-acos 9.4%

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

   2. div-inv 9.4%

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

   3. metadata-eval 9.4%

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

   4. div-sub 9.4%

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

   5. metadata-eval 9.4%

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

   6. div-inv 9.4%

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

   7. metadata-eval 9.4%

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

4. Applied egg-rr 9.4%

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

5. Final simplification 9.4%

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

Alternative 3: 5.6% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.32e-300)
   (- (/ PI 2.0) (* 2.0 (asin (sqrt 0.5))))
   (+ (* 0.5 PI) (* 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 = (0.5 * ((double) M_PI)) + (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 = (0.5 * Math.PI) + (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 = (0.5 * math.pi) + (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(0.5 * pi) + 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 = (0.5 * pi) + (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[(0.5 * Pi), $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 7.3%

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

   3. Taylor expanded in x around 0 5.8%

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

   if 1.32e-300 < x

   1. Initial program 9.1%

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

      1. asin-acos 11.5%

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

      2. div-inv 11.5%

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

      3. metadata-eval 11.5%

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

      4. div-sub 11.5%

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

      5. metadata-eval 11.5%

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

      6. div-inv 11.5%

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

      7. metadata-eval 11.5%

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

   4. Applied egg-rr 11.5%

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

      1. cancel-sign-sub-inv 11.5%

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

      2. metadata-eval 11.5%

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

      3. metadata-eval 11.5%

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

      4. div-inv 11.5%

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

      5. asin-acos 9.1%

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

      6. *-commutative 9.1%

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

      7. add-sqr-sqrt 0.0%

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

      8. sqrt-unprod 5.9%

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

      9. *-commutative 5.9%

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

      10. *-commutative 5.9%

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

      11. swap-sqr 5.9%

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

      12. metadata-eval 5.9%

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

      13. metadata-eval 5.9%

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

   6. Applied egg-rr 5.9%

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

4. Final simplification 5.8%

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

Alternative 4: 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 8.1%

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

3. Final simplification 8.1%

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

Alternative 5: 5.6% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* 2.0 (asin (sqrt 0.5)))))
   (if (<= x 1.32e-300) (- (/ PI 2.0) t_0) (+ (* 0.5 PI) 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 = (0.5 * ((double) M_PI)) + 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 = (0.5 * Math.PI) + 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 = (0.5 * math.pi) + 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(0.5 * pi) + 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 = (0.5 * pi) + 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[(0.5 * Pi), $MachinePrecision] + t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.32e-300

   1. Initial program 7.3%

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

   3. Taylor expanded in x around 0 5.8%

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

   if 1.32e-300 < x

   1. Initial program 9.1%

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

      1. asin-acos 11.5%

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

      2. div-inv 11.5%

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

      3. metadata-eval 11.5%

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

      4. div-sub 11.5%

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

      5. metadata-eval 11.5%

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

      6. div-inv 11.5%

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

      7. metadata-eval 11.5%

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

   4. Applied egg-rr 11.5%

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

      1. cancel-sign-sub-inv 11.5%

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

      2. metadata-eval 11.5%

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

      3. metadata-eval 11.5%

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

      4. div-inv 11.5%

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

      5. asin-acos 9.1%

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

      6. *-commutative 9.1%

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

      7. add-sqr-sqrt 0.0%

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

      8. sqrt-unprod 5.9%

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

      9. *-commutative 5.9%

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

      10. *-commutative 5.9%

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

      11. swap-sqr 5.9%

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

      12. metadata-eval 5.9%

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

      13. metadata-eval 5.9%

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

   6. Applied egg-rr 5.9%

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

   7. Taylor expanded in x around 0 5.9%

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

4. Final simplification 5.8%

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

Alternative 6: 3.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 8.1%

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

   1. asin-acos 9.4%

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

   2. div-inv 9.4%

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

   3. metadata-eval 9.4%

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

   4. div-sub 9.4%

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

   5. metadata-eval 9.4%

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

   6. div-inv 9.4%

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

   7. metadata-eval 9.4%

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

4. Applied egg-rr 9.4%

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

   1. cancel-sign-sub-inv 9.4%

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

   2. metadata-eval 9.4%

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

   3. metadata-eval 9.4%

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

   4. div-inv 9.4%

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

   5. asin-acos 8.1%

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

   6. *-commutative 8.1%

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

   7. add-sqr-sqrt 0.0%

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

   8. sqrt-unprod 3.9%

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

   9. *-commutative 3.9%

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

   10. *-commutative 3.9%

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

   11. swap-sqr 3.9%

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

   12. metadata-eval 3.9%

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

   13. metadata-eval 3.9%

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

6. Applied egg-rr 3.9%

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

7. Taylor expanded in x around 0 3.9%

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

8. Final simplification 3.9%

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

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 2024010 
(FPCore (x)
  :name "Ian Simplification"
  :precision binary64

  :herbie-target
  (asin x)

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