Ian Simplification

Percentage Accurate: 6.7% → 9.6%

Time: 23.5s

Precision: binary64

Cost: 26436

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{-162}:\\ \;\;\;\;\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt[3]{{\left(0.5 - 0.5 \cdot x\right)}^{1.5}}\right)\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{-162}:\\ \;\;\;\;\frac{\pi}{2} - 2 \cdot \left(\left(1 + \sin^{-1} \left(\sqrt{0.5}\right)\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\frac{1}{\sqrt{\frac{2}{1 - x}}}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= x -1.65e-162)
   (- (/ PI 2.0) (* 2.0 (asin (cbrt (pow (- 0.5 (* 0.5 x)) 1.5)))))
   (if (<= x 1.65e-162)
     (- (/ PI 2.0) (* 2.0 (+ (+ 1.0 (asin (sqrt 0.5))) -1.0)))
     (- (/ 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 <= -1.65e-162) {
		tmp = (((double) M_PI) / 2.0) - (2.0 * asin(cbrt(pow((0.5 - (0.5 * x)), 1.5))));
	} else if (x <= 1.65e-162) {
		tmp = (((double) M_PI) / 2.0) - (2.0 * ((1.0 + asin(sqrt(0.5))) + -1.0));
	} 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 <= -1.65e-162) {
		tmp = (Math.PI / 2.0) - (2.0 * Math.asin(Math.cbrt(Math.pow((0.5 - (0.5 * x)), 1.5))));
	} else if (x <= 1.65e-162) {
		tmp = (Math.PI / 2.0) - (2.0 * ((1.0 + Math.asin(Math.sqrt(0.5))) + -1.0));
	} 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 <= -1.65e-162)
		tmp = Float64(Float64(pi / 2.0) - Float64(2.0 * asin(cbrt((Float64(0.5 - Float64(0.5 * x)) ^ 1.5)))));
	elseif (x <= 1.65e-162)
		tmp = Float64(Float64(pi / 2.0) - Float64(2.0 * Float64(Float64(1.0 + asin(sqrt(0.5))) + -1.0)));
	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

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, -1.65e-162], N[(N[(Pi / 2.0), $MachinePrecision] - N[(2.0 * N[ArcSin[N[Power[N[Power[N[(0.5 - N[(0.5 * x), $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.65e-162], N[(N[(Pi / 2.0), $MachinePrecision] - N[(2.0 * N[(N[(1.0 + N[ArcSin[N[Sqrt[0.5], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + -1.0), $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	6.7%
Target	100.0%
Herbie	9.6%

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

Derivation

1. Split input into 3 regimes

2. if x < -1.65000000000000007e-162

   1. Initial program 14.7%

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

   2. Applied egg-rr 14.8%

      \[\leadsto \frac{\pi}{2} - 2 \cdot \sin^{-1} \color{blue}{\left(\sqrt[3]{{\left(0.5 - x \cdot 0.5\right)}^{1.5}}\right)} \]
      Step-by-step derivation

      [Start] 14.7	\[ \frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
      add-cbrt-cube [=>] 14.7	\[ \frac{\pi}{2} - 2 \cdot \sin^{-1} \color{blue}{\left(\sqrt[3]{\left(\sqrt{\frac{1 - x}{2}} \cdot \sqrt{\frac{1 - x}{2}}\right) \cdot \sqrt{\frac{1 - x}{2}}}\right)} \]
      add-sqr-sqrt [<=] 14.7	\[ \frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt[3]{\color{blue}{\frac{1 - x}{2}} \cdot \sqrt{\frac{1 - x}{2}}}\right) \]
      pow1 [=>] 14.7	\[ \frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt[3]{\color{blue}{{\left(\frac{1 - x}{2}\right)}^{1}} \cdot \sqrt{\frac{1 - x}{2}}}\right) \]
      pow1/2 [=>] 14.7	\[ \frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt[3]{{\left(\frac{1 - x}{2}\right)}^{1} \cdot \color{blue}{{\left(\frac{1 - x}{2}\right)}^{0.5}}}\right) \]
      pow-prod-up [=>] 14.8	\[ \frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt[3]{\color{blue}{{\left(\frac{1 - x}{2}\right)}^{\left(1 + 0.5\right)}}}\right) \]
      div-sub [=>] 14.8	\[ \frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt[3]{{\color{blue}{\left(\frac{1}{2} - \frac{x}{2}\right)}}^{\left(1 + 0.5\right)}}\right) \]
      metadata-eval [=>] 14.8	\[ \frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt[3]{{\left(\color{blue}{0.5} - \frac{x}{2}\right)}^{\left(1 + 0.5\right)}}\right) \]
      div-inv [=>] 14.8	\[ \frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt[3]{{\left(0.5 - \color{blue}{x \cdot \frac{1}{2}}\right)}^{\left(1 + 0.5\right)}}\right) \]
      metadata-eval [=>] 14.8	\[ \frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt[3]{{\left(0.5 - x \cdot \color{blue}{0.5}\right)}^{\left(1 + 0.5\right)}}\right) \]
      metadata-eval [=>] 14.8	\[ \frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt[3]{{\left(0.5 - x \cdot 0.5\right)}^{\color{blue}{1.5}}}\right) \]

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

   1. Initial program 3.3%

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

   2. Taylor expanded in x around 0 3.3%

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

   3. Applied egg-rr 7.0%

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

      [Start] 3.3	\[ \frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{0.5}\right) \]
      expm1-log1p-u [=>] 3.3	\[ \frac{\pi}{2} - 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(\sqrt{0.5}\right)\right)\right)} \]
      expm1-udef [=>] 3.3	\[ \frac{\pi}{2} - 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sin^{-1} \left(\sqrt{0.5}\right)\right)} - 1\right)} \]
      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 6.8%

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

   2. Applied egg-rr 11.3%

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

      [Start] 6.8	\[ \frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
      clear-num [=>] 6.8	\[ \frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\color{blue}{\frac{1}{\frac{2}{1 - x}}}}\right) \]
      sqrt-div [=>] 11.3	\[ \frac{\pi}{2} - 2 \cdot \sin^{-1} \color{blue}{\left(\frac{\sqrt{1}}{\sqrt{\frac{2}{1 - x}}}\right)} \]
      metadata-eval [=>] 11.3	\[ \frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\frac{\color{blue}{1}}{\sqrt{\frac{2}{1 - x}}}\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 10.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{-162}:\\ \;\;\;\;\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt[3]{{\left(0.5 - 0.5 \cdot x\right)}^{1.5}}\right)\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{-162}:\\ \;\;\;\;\frac{\pi}{2} - 2 \cdot \left(\left(1 + \sin^{-1} \left(\sqrt{0.5}\right)\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\frac{1}{\sqrt{\frac{2}{1 - x}}}\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	8.2%
Cost	26432

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

Alternative 2
Accuracy	9.7%
Cost	20232

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

Alternative 3
Accuracy	8.2%
Cost	20105

\[\begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-310} \lor \neg \left(x \leq 10^{-17}\right):\\ \;\;\;\;\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\frac{1}{\sqrt{2}}\right)\\ \end{array} \]

Alternative 4
Accuracy	8.9%
Cost	20105

\[\begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{-162} \lor \neg \left(x \leq 5.5 \cdot 10^{-17}\right):\\ \;\;\;\;\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{2} - 2 \cdot \left(\left(1 + \sin^{-1} \left(\sqrt{0.5}\right)\right) + -1\right)\\ \end{array} \]

Alternative 5
Accuracy	5.9%
Cost	19844

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

Alternative 6
Accuracy	4.1%
Cost	19584

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

Reproduce

herbie shell --seed 2023159 
(FPCore (x)
  :name "Ian Simplification"
  :precision binary64

  :herbie-target
  (asin x)

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