Ian Simplification

Average Accuracy: 6.6% → 8.1%

Time: 16.2s

Precision: binary64

Cost: 45568

Math

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

↓

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

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (cbrt
  (pow
   (+ (* 0.5 PI) (* -2.0 (- (* 0.5 PI) (acos (sqrt (fma x -0.5 0.5))))))
   3.0)))

C

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

↓

double code(double x) {
	return cbrt(pow(((0.5 * ((double) M_PI)) + (-2.0 * ((0.5 * ((double) M_PI)) - acos(sqrt(fma(x, -0.5, 0.5)))))), 3.0));
}

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)
	return cbrt((Float64(Float64(0.5 * pi) + Float64(-2.0 * Float64(Float64(0.5 * pi) - acos(sqrt(fma(x, -0.5, 0.5)))))) ^ 3.0))
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]

↓

code[x_] := N[Power[N[Power[N[(N[(0.5 * Pi), $MachinePrecision] + N[(-2.0 * N[(N[(0.5 * Pi), $MachinePrecision] - N[ArcCos[N[Sqrt[N[(x * -0.5 + 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision]

Target

Original	6.6%
Target	100.0%
Herbie	8.1%

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

Derivation

1. Initial program 6.1%

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

2. Applied egg-rr 6.1%

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

   [Start] 6.1	\[ \frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
   add-cbrt-cube [=>] 6.1	\[ \color{blue}{\sqrt[3]{\left(\left(\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right) \cdot \left(\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right) \cdot \left(\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)}} \]
   pow3 [=>] 6.1	\[ \sqrt[3]{\color{blue}{{\left(\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)}^{3}}} \]

3. Applied egg-rr 7.8%

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

   [Start] 6.1	\[ \sqrt[3]{{\left(\mathsf{fma}\left(\sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), -2, \pi \cdot 0.5\right)\right)}^{3}} \]
   asin-acos [=>] 7.8	\[ \sqrt[3]{{\left(\mathsf{fma}\left(\color{blue}{\frac{\pi}{2} - \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}, -2, \pi \cdot 0.5\right)\right)}^{3}} \]
   div-inv [=>] 7.8	\[ \sqrt[3]{{\left(\mathsf{fma}\left(\color{blue}{\pi \cdot \frac{1}{2}} - \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), -2, \pi \cdot 0.5\right)\right)}^{3}} \]
   metadata-eval [=>] 7.8	\[ \sqrt[3]{{\left(\mathsf{fma}\left(\pi \cdot \color{blue}{0.5} - \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), -2, \pi \cdot 0.5\right)\right)}^{3}} \]
   sub-neg [=>] 7.8	\[ \sqrt[3]{{\left(\mathsf{fma}\left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\color{blue}{0.5 + \left(-x \cdot 0.5\right)}}\right), -2, \pi \cdot 0.5\right)\right)}^{3}} \]
   +-commutative [=>] 7.8	\[ \sqrt[3]{{\left(\mathsf{fma}\left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\color{blue}{\left(-x \cdot 0.5\right) + 0.5}}\right), -2, \pi \cdot 0.5\right)\right)}^{3}} \]
   distribute-rgt-neg-in [=>] 7.8	\[ \sqrt[3]{{\left(\mathsf{fma}\left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\color{blue}{x \cdot \left(-0.5\right)} + 0.5}\right), -2, \pi \cdot 0.5\right)\right)}^{3}} \]
   metadata-eval [=>] 7.8	\[ \sqrt[3]{{\left(\mathsf{fma}\left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{x \cdot \color{blue}{-0.5} + 0.5}\right), -2, \pi \cdot 0.5\right)\right)}^{3}} \]
   metadata-eval [<=] 7.8	\[ \sqrt[3]{{\left(\mathsf{fma}\left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{x \cdot \color{blue}{\frac{1}{-2}} + 0.5}\right), -2, \pi \cdot 0.5\right)\right)}^{3}} \]
   metadata-eval [<=] 7.8	\[ \sqrt[3]{{\left(\mathsf{fma}\left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{x \cdot \frac{1}{\color{blue}{-2}} + 0.5}\right), -2, \pi \cdot 0.5\right)\right)}^{3}} \]
   fma-def [=>] 7.8	\[ \sqrt[3]{{\left(\mathsf{fma}\left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\color{blue}{\mathsf{fma}\left(x, \frac{1}{-2}, 0.5\right)}}\right), -2, \pi \cdot 0.5\right)\right)}^{3}} \]
   metadata-eval [=>] 7.8	\[ \sqrt[3]{{\left(\mathsf{fma}\left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, \frac{1}{\color{blue}{-2}}, 0.5\right)}\right), -2, \pi \cdot 0.5\right)\right)}^{3}} \]
   metadata-eval [=>] 7.8	\[ \sqrt[3]{{\left(\mathsf{fma}\left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, \color{blue}{-0.5}, 0.5\right)}\right), -2, \pi \cdot 0.5\right)\right)}^{3}} \]

4. Taylor expanded in x around 0 7.8%

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

5. Final simplification 7.8%

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

Alternatives

Alternative 1
Accuracy	8.1%
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 2
Accuracy	6.6%
Cost	19840

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

Reproduce

herbie shell --seed 2023157 
(FPCore (x)
  :name "Ian Simplification"
  :precision binary64

  :herbie-target
  (asin x)

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