Ian Simplification

Average Error: 93.3% → 91.83%

Time: 21.2s

Precision: binary64

Cost: 38912

Math

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

↓

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

FPCore

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

↓

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

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 log(exp(((2.0 * acos(sqrt(fma(-0.5, x, 0.5)))) + (-0.5 * ((double) M_PI)))));
}

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 log(exp(Float64(Float64(2.0 * acos(sqrt(fma(-0.5, x, 0.5)))) + Float64(-0.5 * pi))))
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[Log[N[Exp[N[(N[(2.0 * N[ArcCos[N[Sqrt[N[(-0.5 * x + 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(-0.5 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]

Target

Original	93.3%
Target	0%
Herbie	91.83%

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

Derivation

1. Initial program 93.3

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

2. Applied egg-rr 93.31

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

3. Applied egg-rr 91.83

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

4. Applied egg-rr 91.91

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

5. Simplified 91.83

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

   [Start] 91.91	\[ \log \left(e^{\frac{-2}{\frac{2}{\pi}} + \left(-2 \cdot \left(-\cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)\right) + \pi \cdot 0.5\right)}\right) \]
   +-commutative [=>] 91.91	\[ \log \left(e^{\color{blue}{\left(-2 \cdot \left(-\cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)\right) + \pi \cdot 0.5\right) + \frac{-2}{\frac{2}{\pi}}}}\right) \]
   associate-+r+ [<=] 91.83	\[ \log \left(e^{\color{blue}{-2 \cdot \left(-\cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)\right) + \left(\pi \cdot 0.5 + \frac{-2}{\frac{2}{\pi}}\right)}}\right) \]
   distribute-rgt-neg-out [=>] 91.83	\[ \log \left(e^{\color{blue}{\left(--2 \cdot \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)\right)} + \left(\pi \cdot 0.5 + \frac{-2}{\frac{2}{\pi}}\right)}\right) \]
   distribute-lft-neg-in [=>] 91.83	\[ \log \left(e^{\color{blue}{\left(--2\right) \cdot \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)} + \left(\pi \cdot 0.5 + \frac{-2}{\frac{2}{\pi}}\right)}\right) \]
   metadata-eval [=>] 91.83	\[ \log \left(e^{\color{blue}{2} \cdot \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right) + \left(\pi \cdot 0.5 + \frac{-2}{\frac{2}{\pi}}\right)}\right) \]
   fma-udef [=>] 91.83	\[ \log \left(e^{2 \cdot \cos^{-1} \left(\sqrt{\color{blue}{x \cdot -0.5 + 0.5}}\right) + \left(\pi \cdot 0.5 + \frac{-2}{\frac{2}{\pi}}\right)}\right) \]
   *-commutative [<=] 91.83	\[ \log \left(e^{2 \cdot \cos^{-1} \left(\sqrt{\color{blue}{-0.5 \cdot x} + 0.5}\right) + \left(\pi \cdot 0.5 + \frac{-2}{\frac{2}{\pi}}\right)}\right) \]
   fma-def [=>] 91.83	\[ \log \left(e^{2 \cdot \cos^{-1} \left(\sqrt{\color{blue}{\mathsf{fma}\left(-0.5, x, 0.5\right)}}\right) + \left(\pi \cdot 0.5 + \frac{-2}{\frac{2}{\pi}}\right)}\right) \]
   *-commutative [=>] 91.83	\[ \log \left(e^{2 \cdot \cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right) + \left(\color{blue}{0.5 \cdot \pi} + \frac{-2}{\frac{2}{\pi}}\right)}\right) \]
   associate-/r/ [=>] 91.83	\[ \log \left(e^{2 \cdot \cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right) + \left(0.5 \cdot \pi + \color{blue}{\frac{-2}{2} \cdot \pi}\right)}\right) \]
   distribute-rgt-out [=>] 91.83	\[ \log \left(e^{2 \cdot \cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right) + \color{blue}{\pi \cdot \left(0.5 + \frac{-2}{2}\right)}}\right) \]
   metadata-eval [=>] 91.83	\[ \log \left(e^{2 \cdot \cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right) + \pi \cdot \left(0.5 + \color{blue}{-1}\right)}\right) \]
   metadata-eval [=>] 91.83	\[ \log \left(e^{2 \cdot \cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right) + \pi \cdot \color{blue}{-0.5}}\right) \]

6. Final simplification 91.83

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

Alternatives

Alternative 1
Error	91.82%
Cost	26112

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

Alternative 2
Error	93.3%
Cost	19840

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

Reproduce

herbie shell --seed 2023089 
(FPCore (x)
  :name "Ian Simplification"
  :precision binary64

  :herbie-target
  (asin x)

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