Ian Simplification

Average Accuracy: 6.9% → 9.8%

Time: 16.0s

Precision: binary64

Cost: 20232

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{-162}:\\ \;\;\;\;\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{-162}:\\ \;\;\;\;\frac{\pi}{2} + 2 \cdot \left(1 + \left(-1 - \sin^{-1} \left(\sqrt{0.5}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\frac{1}{\sqrt{\frac{2}{1 - x}}}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= x -1.7e-162)
   (- (/ PI 2.0) (* 2.0 (asin (sqrt (/ (- 1.0 x) 2.0)))))
   (if (<= x 1.65e-162)
     (+ (/ PI 2.0) (* 2.0 (+ 1.0 (- -1.0 (asin (sqrt 0.5))))))
     (- (/ 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(sqrt(((1.0 - x) / 2.0))));
}

↓

double code(double x) {
	double tmp;
	if (x <= -1.7e-162) {
		tmp = (((double) M_PI) / 2.0) - (2.0 * asin(sqrt(((1.0 - x) / 2.0))));
	} else if (x <= 1.65e-162) {
		tmp = (((double) M_PI) / 2.0) + (2.0 * (1.0 + (-1.0 - asin(sqrt(0.5)))));
	} else {
		tmp = (((double) M_PI) / 2.0) - (2.0 * asin((1.0 / sqrt((2.0 / (1.0 - x))))));
	}
	return tmp;
}

Java

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

↓

public static double code(double x) {
	double tmp;
	if (x <= -1.7e-162) {
		tmp = (Math.PI / 2.0) - (2.0 * Math.asin(Math.sqrt(((1.0 - x) / 2.0))));
	} else if (x <= 1.65e-162) {
		tmp = (Math.PI / 2.0) + (2.0 * (1.0 + (-1.0 - Math.asin(Math.sqrt(0.5)))));
	} else {
		tmp = (Math.PI / 2.0) - (2.0 * Math.asin((1.0 / Math.sqrt((2.0 / (1.0 - x))))));
	}
	return tmp;
}

Python

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

↓

def code(x):
	tmp = 0
	if x <= -1.7e-162:
		tmp = (math.pi / 2.0) - (2.0 * math.asin(math.sqrt(((1.0 - x) / 2.0))))
	elif x <= 1.65e-162:
		tmp = (math.pi / 2.0) + (2.0 * (1.0 + (-1.0 - math.asin(math.sqrt(0.5)))))
	else:
		tmp = (math.pi / 2.0) - (2.0 * math.asin((1.0 / math.sqrt((2.0 / (1.0 - x))))))
	return tmp

Julia

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

↓

function code(x)
	tmp = 0.0
	if (x <= -1.7e-162)
		tmp = Float64(Float64(pi / 2.0) - Float64(2.0 * asin(sqrt(Float64(Float64(1.0 - x) / 2.0)))));
	elseif (x <= 1.65e-162)
		tmp = Float64(Float64(pi / 2.0) + Float64(2.0 * Float64(1.0 + Float64(-1.0 - asin(sqrt(0.5))))));
	else
		tmp = Float64(Float64(pi / 2.0) - Float64(2.0 * asin(Float64(1.0 / sqrt(Float64(2.0 / Float64(1.0 - x)))))));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.7e-162)
		tmp = (pi / 2.0) - (2.0 * asin(sqrt(((1.0 - x) / 2.0))));
	elseif (x <= 1.65e-162)
		tmp = (pi / 2.0) + (2.0 * (1.0 + (-1.0 - asin(sqrt(0.5)))));
	else
		tmp = (pi / 2.0) - (2.0 * asin((1.0 / sqrt((2.0 / (1.0 - x))))));
	end
	tmp_2 = tmp;
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]

↓

code[x_] := If[LessEqual[x, -1.7e-162], 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], If[LessEqual[x, 1.65e-162], N[(N[(Pi / 2.0), $MachinePrecision] + N[(2.0 * N[(1.0 + N[(-1.0 - N[ArcSin[N[Sqrt[0.5], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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]]]

Target

Original	6.9%
Target	100.0%
Herbie	9.8%

\[\sin^{-1} x \]

Derivation

1. Split input into 3 regimes

2. if x < -1.7e-162

   1. Initial program 12.2%

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

   if -1.7e-162 < x < 1.65000000000000007e-162

   1. Initial program 3.4%

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

   2. Taylor expanded in x around 0 3.4%

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

   3. Applied egg-rr 7.0%

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

      [Start] 3.4	\[ \frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{0.5}\right) \]
      expm1-log1p-u [=>] 3.4	\[ \frac{\pi}{2} - 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(\sqrt{0.5}\right)\right)\right)} \]
      expm1-udef [=>] 3.4	\[ \frac{\pi}{2} - 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sin^{-1} \left(\sqrt{0.5}\right)\right)} - 1\right)} \]
      log1p-udef [=>] 7.0	\[ \frac{\pi}{2} - 2 \cdot \left(e^{\color{blue}{\log \left(1 + \sin^{-1} \left(\sqrt{0.5}\right)\right)}} - 1\right) \]
      add-exp-log [<=] 7.0	\[ \frac{\pi}{2} - 2 \cdot \left(\color{blue}{\left(1 + \sin^{-1} \left(\sqrt{0.5}\right)\right)} - 1\right) \]

   if 1.65000000000000007e-162 < x

   1. Initial program 8.1%

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

   2. Applied egg-rr 12.5%

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

      [Start] 8.1	\[ \frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
      clear-num [=>] 8.1	\[ \frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\color{blue}{\frac{1}{\frac{2}{1 - x}}}}\right) \]
      sqrt-div [=>] 12.5	\[ \frac{\pi}{2} - 2 \cdot \sin^{-1} \color{blue}{\left(\frac{\sqrt{1}}{\sqrt{\frac{2}{1 - x}}}\right)} \]
      metadata-eval [=>] 12.5	\[ \frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\frac{\color{blue}{1}}{\sqrt{\frac{2}{1 - x}}}\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 9.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{-162}:\\ \;\;\;\;\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{-162}:\\ \;\;\;\;\frac{\pi}{2} + 2 \cdot \left(1 + \left(-1 - \sin^{-1} \left(\sqrt{0.5}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\frac{1}{\sqrt{\frac{2}{1 - x}}}\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	8.4%
Cost	45568

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

Alternative 2
Accuracy	8.4%
Cost	26432

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

Alternative 3
Accuracy	9.1%
Cost	20105

\[\begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{-162} \lor \neg \left(x \leq 5.5 \cdot 10^{-17}\right):\\ \;\;\;\;\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{2} + 2 \cdot \left(1 + \left(-1 - \sin^{-1} \left(\sqrt{0.5}\right)\right)\right)\\ \end{array} \]

Alternative 4
Accuracy	5.8%
Cost	19844

\[\begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\frac{1}{\sqrt{2}}\right)\\ \end{array} \]

Alternative 5
Accuracy	6.9%
Cost	19840

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

Alternative 6
Accuracy	4.1%
Cost	19584

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

Reproduce

herbie shell --seed 2023153 
(FPCore (x)
  :name "Ian Simplification"
  :precision binary64

  :herbie-target
  (asin x)

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