Ian Simplification

Average Accuracy: 6.9% → 8.4%

Time: 29.1s

Precision: binary64

Cost: 149824

Math

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

↓

\[\begin{array}{l} t_0 := 0.25 \cdot {\pi}^{2}\\ t_1 := \cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)\\ t_2 := {t_1}^{2} \cdot 4\\ \frac{\left(t_0 \cdot t_0 - t_2 \cdot t_2\right) \cdot \frac{1}{\mathsf{fma}\left(\pi, -0.5, t_1 \cdot -2\right)}}{t_0 + t_2} \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* 0.25 (pow PI 2.0)))
        (t_1 (acos (sqrt (fma -0.5 x 0.5))))
        (t_2 (* (pow t_1 2.0) 4.0)))
   (/
    (* (- (* t_0 t_0) (* t_2 t_2)) (/ 1.0 (fma PI -0.5 (* t_1 -2.0))))
    (+ t_0 t_2))))

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 t_0 = 0.25 * pow(((double) M_PI), 2.0);
	double t_1 = acos(sqrt(fma(-0.5, x, 0.5)));
	double t_2 = pow(t_1, 2.0) * 4.0;
	return (((t_0 * t_0) - (t_2 * t_2)) * (1.0 / fma(((double) M_PI), -0.5, (t_1 * -2.0)))) / (t_0 + t_2);
}

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)
	t_0 = Float64(0.25 * (pi ^ 2.0))
	t_1 = acos(sqrt(fma(-0.5, x, 0.5)))
	t_2 = Float64((t_1 ^ 2.0) * 4.0)
	return Float64(Float64(Float64(Float64(t_0 * t_0) - Float64(t_2 * t_2)) * Float64(1.0 / fma(pi, -0.5, Float64(t_1 * -2.0)))) / Float64(t_0 + t_2))
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_] := Block[{t$95$0 = N[(0.25 * N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[ArcCos[N[Sqrt[N[(-0.5 * x + 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[t$95$1, 2.0], $MachinePrecision] * 4.0), $MachinePrecision]}, N[(N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(Pi * -0.5 + N[(t$95$1 * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 + t$95$2), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	6.9%
Target	100.0%
Herbie	8.4%

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

Derivation

1. Initial program 6.9%

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

2. Applied egg-rr 8.4%

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

   [Start] 6.9	\[ \frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
   asin-acos [=>] 8.4	\[ \frac{\pi}{2} - 2 \cdot \color{blue}{\left(\frac{\pi}{2} - \cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)} \]
   div-inv [=>] 8.4	\[ \frac{\pi}{2} - 2 \cdot \left(\color{blue}{\pi \cdot \frac{1}{2}} - \cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right) \]
   metadata-eval [=>] 8.4	\[ \frac{\pi}{2} - 2 \cdot \left(\pi \cdot \color{blue}{0.5} - \cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right) \]
   div-sub [=>] 8.4	\[ \frac{\pi}{2} - 2 \cdot \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\color{blue}{\frac{1}{2} - \frac{x}{2}}}\right)\right) \]
   metadata-eval [=>] 8.4	\[ \frac{\pi}{2} - 2 \cdot \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\color{blue}{0.5} - \frac{x}{2}}\right)\right) \]
   div-inv [=>] 8.4	\[ \frac{\pi}{2} - 2 \cdot \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{0.5 - \color{blue}{x \cdot \frac{1}{2}}}\right)\right) \]
   metadata-eval [=>] 8.4	\[ \frac{\pi}{2} - 2 \cdot \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{0.5 - x \cdot \color{blue}{0.5}}\right)\right) \]

3. Taylor expanded in x around 0 8.4%

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

4. Simplified 8.4%

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

   [Start] 8.4	\[ 0.5 \cdot \pi - 2 \cdot \left(0.5 \cdot \pi - \cos^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\right) \]
   *-commutative [<=] 8.4	\[ \color{blue}{\pi \cdot 0.5} - 2 \cdot \left(0.5 \cdot \pi - \cos^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\right) \]
   cancel-sign-sub-inv [=>] 8.4	\[ \pi \cdot 0.5 - 2 \cdot \left(0.5 \cdot \pi - \cos^{-1} \left(\sqrt{\color{blue}{0.5 + \left(-0.5\right) \cdot x}}\right)\right) \]
   metadata-eval [=>] 8.4	\[ \pi \cdot 0.5 - 2 \cdot \left(0.5 \cdot \pi - \cos^{-1} \left(\sqrt{0.5 + \color{blue}{-0.5} \cdot x}\right)\right) \]
   cancel-sign-sub-inv [=>] 8.4	\[ \color{blue}{\pi \cdot 0.5 + \left(-2\right) \cdot \left(0.5 \cdot \pi - \cos^{-1} \left(\sqrt{0.5 + -0.5 \cdot x}\right)\right)} \]
   metadata-eval [=>] 8.4	\[ \pi \cdot 0.5 + \color{blue}{-2} \cdot \left(0.5 \cdot \pi - \cos^{-1} \left(\sqrt{0.5 + -0.5 \cdot x}\right)\right) \]
   *-commutative [<=] 8.4	\[ \pi \cdot 0.5 + -2 \cdot \left(\color{blue}{\pi \cdot 0.5} - \cos^{-1} \left(\sqrt{0.5 + -0.5 \cdot x}\right)\right) \]
   metadata-eval [<=] 8.4	\[ \pi \cdot 0.5 + -2 \cdot \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{0.5 + \color{blue}{\left(-0.5\right)} \cdot x}\right)\right) \]
   cancel-sign-sub-inv [<=] 8.4	\[ \pi \cdot 0.5 + -2 \cdot \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\color{blue}{0.5 - 0.5 \cdot x}}\right)\right) \]
   cancel-sign-sub-inv [=>] 8.4	\[ \pi \cdot 0.5 + -2 \cdot \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\color{blue}{0.5 + \left(-0.5\right) \cdot x}}\right)\right) \]
   metadata-eval [=>] 8.4	\[ \pi \cdot 0.5 + -2 \cdot \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{0.5 + \color{blue}{-0.5} \cdot x}\right)\right) \]
   *-commutative [<=] 8.4	\[ \pi \cdot 0.5 + -2 \cdot \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{0.5 + \color{blue}{x \cdot -0.5}}\right)\right) \]

5. Applied egg-rr 8.4%

   \[\leadsto \color{blue}{\frac{\left(\left(0.25 \cdot {\pi}^{2}\right) \cdot \left(0.25 \cdot {\pi}^{2}\right) - \left({\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{2} \cdot 4\right) \cdot \left({\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{2} \cdot 4\right)\right) \cdot \frac{1}{\mathsf{fma}\left(\pi, -0.5, -2 \cdot \cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)\right)}}{{\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{2} \cdot 4 + 0.25 \cdot {\pi}^{2}}} \]
   Proof

   [Start] 8.4	\[ \pi \cdot -0.5 - -2 \cdot \cos^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right) \]
   flip-- [=>] 8.4	\[ \color{blue}{\frac{\left(\pi \cdot -0.5\right) \cdot \left(\pi \cdot -0.5\right) - \left(-2 \cdot \cos^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)\right) \cdot \left(-2 \cdot \cos^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)\right)}{\pi \cdot -0.5 + -2 \cdot \cos^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)}} \]
   div-inv [=>] 8.4	\[ \color{blue}{\left(\left(\pi \cdot -0.5\right) \cdot \left(\pi \cdot -0.5\right) - \left(-2 \cdot \cos^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)\right) \cdot \left(-2 \cdot \cos^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)\right)\right) \cdot \frac{1}{\pi \cdot -0.5 + -2 \cdot \cos^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)}} \]
   flip-- [=>] 8.4	\[ \color{blue}{\frac{\left(\left(\pi \cdot -0.5\right) \cdot \left(\pi \cdot -0.5\right)\right) \cdot \left(\left(\pi \cdot -0.5\right) \cdot \left(\pi \cdot -0.5\right)\right) - \left(\left(-2 \cdot \cos^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)\right) \cdot \left(-2 \cdot \cos^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)\right)\right) \cdot \left(\left(-2 \cdot \cos^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)\right) \cdot \left(-2 \cdot \cos^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)\right)\right)}{\left(\pi \cdot -0.5\right) \cdot \left(\pi \cdot -0.5\right) + \left(-2 \cdot \cos^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)\right) \cdot \left(-2 \cdot \cos^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)\right)}} \cdot \frac{1}{\pi \cdot -0.5 + -2 \cdot \cos^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)} \]

6. Final simplification 8.4%

   \[\leadsto \frac{\left(\left(0.25 \cdot {\pi}^{2}\right) \cdot \left(0.25 \cdot {\pi}^{2}\right) - \left({\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{2} \cdot 4\right) \cdot \left({\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{2} \cdot 4\right)\right) \cdot \frac{1}{\mathsf{fma}\left(\pi, -0.5, \cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right) \cdot -2\right)}}{0.25 \cdot {\pi}^{2} + {\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{2} \cdot 4} \]

Alternatives

Alternative 1
Accuracy	8.4%
Cost	149696

\[\begin{array}{l} t_0 := 0.25 \cdot {\pi}^{2}\\ t_1 := \cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)\\ t_2 := {t_1}^{2} \cdot 4\\ \frac{t_0 \cdot t_0 - t_2 \cdot t_2}{\mathsf{fma}\left(\pi, -0.5, t_1 \cdot -2\right) \cdot \left(t_0 + t_2\right)} \end{array} \]

Alternative 2
Accuracy	8.4%
Cost	71616

\[\begin{array}{l} t_0 := \cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)\\ \frac{1}{\mathsf{fma}\left(\pi, -0.5, t_0 \cdot -2\right)} \cdot \left(0.25 \cdot {\pi}^{2} + {t_0}^{2} \cdot -4\right) \end{array} \]

Alternative 3
Accuracy	8.4%
Cost	71488

\[\begin{array}{l} t_0 := \cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)\\ \frac{0.25 \cdot {\pi}^{2} + {t_0}^{2} \cdot -4}{\mathsf{fma}\left(\pi, -0.5, t_0 \cdot -2\right)} \end{array} \]

Alternative 4
Accuracy	8.4%
Cost	45248

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

Alternative 5
Accuracy	8.4%
Cost	19840

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

Alternative 6
Accuracy	5.4%
Cost	19584

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

Reproduce

herbie shell --seed 2023131 
(FPCore (x)
  :name "Ian Simplification"
  :precision binary64

  :herbie-target
  (asin x)

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