Ian Simplification

Average Accuracy: 6.8% → 8.3%

Time: 25.6s

Precision: binary64

Cost: 97984

Math

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

↓

\[\begin{array}{l} t_0 := \cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)\\ \frac{1}{0.25 \cdot {\pi}^{2} + -2 \cdot \left(t_0 \cdot \left(-2 \cdot t_0 + \pi \cdot -0.5\right)\right)} \cdot \left(-0.125 \cdot {\pi}^{3} - {t_0}^{3} \cdot -8\right) \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (acos (sqrt (fma -0.5 x 0.5)))))
   (*
    (/
     1.0
     (+ (* 0.25 (pow PI 2.0)) (* -2.0 (* t_0 (+ (* -2.0 t_0) (* PI -0.5))))))
    (- (* -0.125 (pow PI 3.0)) (* (pow t_0 3.0) -8.0)))))

C

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

↓

double code(double x) {
	double t_0 = acos(sqrt(fma(-0.5, x, 0.5)));
	return (1.0 / ((0.25 * pow(((double) M_PI), 2.0)) + (-2.0 * (t_0 * ((-2.0 * t_0) + (((double) M_PI) * -0.5)))))) * ((-0.125 * pow(((double) M_PI), 3.0)) - (pow(t_0, 3.0) * -8.0));
}

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)
	t_0 = acos(sqrt(fma(-0.5, x, 0.5)))
	return Float64(Float64(1.0 / Float64(Float64(0.25 * (pi ^ 2.0)) + Float64(-2.0 * Float64(t_0 * Float64(Float64(-2.0 * t_0) + Float64(pi * -0.5)))))) * Float64(Float64(-0.125 * (pi ^ 3.0)) - Float64((t_0 ^ 3.0) * -8.0)))
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_] := Block[{t$95$0 = N[ArcCos[N[Sqrt[N[(-0.5 * x + 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, N[(N[(1.0 / N[(N[(0.25 * N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(t$95$0 * N[(N[(-2.0 * t$95$0), $MachinePrecision] + N[(Pi * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(-0.125 * N[Power[Pi, 3.0], $MachinePrecision]), $MachinePrecision] - N[(N[Power[t$95$0, 3.0], $MachinePrecision] * -8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	6.8%
Target	100.0%
Herbie	8.3%

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

Derivation

1. Initial program 6.8%

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

2. Applied egg-rr 8.3%

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

   [Start] 6.8	\[ \frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
   asin-acos [=>] 8.3	\[ \frac{\pi}{2} - 2 \cdot \color{blue}{\left(\frac{\pi}{2} - \cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)} \]
   div-inv [=>] 8.3	\[ \frac{\pi}{2} - 2 \cdot \left(\color{blue}{\pi \cdot \frac{1}{2}} - \cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right) \]
   metadata-eval [=>] 8.3	\[ \frac{\pi}{2} - 2 \cdot \left(\pi \cdot \color{blue}{0.5} - \cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right) \]
   div-sub [=>] 8.3	\[ \frac{\pi}{2} - 2 \cdot \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\color{blue}{\frac{1}{2} - \frac{x}{2}}}\right)\right) \]
   metadata-eval [=>] 8.3	\[ \frac{\pi}{2} - 2 \cdot \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\color{blue}{0.5} - \frac{x}{2}}\right)\right) \]
   div-inv [=>] 8.3	\[ \frac{\pi}{2} - 2 \cdot \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{0.5 - \color{blue}{x \cdot \frac{1}{2}}}\right)\right) \]
   metadata-eval [=>] 8.3	\[ \frac{\pi}{2} - 2 \cdot \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{0.5 - x \cdot \color{blue}{0.5}}\right)\right) \]

3. Taylor expanded in x around 0 8.3%

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

4. Simplified 8.3%

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

   [Start] 8.3	\[ 0.5 \cdot \pi - 2 \cdot \left(0.5 \cdot \pi - \cos^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\right) \]
   *-commutative [<=] 8.3	\[ \color{blue}{\pi \cdot 0.5} - 2 \cdot \left(0.5 \cdot \pi - \cos^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\right) \]
   cancel-sign-sub-inv [=>] 8.3	\[ \pi \cdot 0.5 - 2 \cdot \left(0.5 \cdot \pi - \cos^{-1} \left(\sqrt{\color{blue}{0.5 + \left(-0.5\right) \cdot x}}\right)\right) \]
   metadata-eval [=>] 8.3	\[ \pi \cdot 0.5 - 2 \cdot \left(0.5 \cdot \pi - \cos^{-1} \left(\sqrt{0.5 + \color{blue}{-0.5} \cdot x}\right)\right) \]
   cancel-sign-sub-inv [=>] 8.3	\[ \color{blue}{\pi \cdot 0.5 + \left(-2\right) \cdot \left(0.5 \cdot \pi - \cos^{-1} \left(\sqrt{0.5 + -0.5 \cdot x}\right)\right)} \]
   metadata-eval [=>] 8.3	\[ \pi \cdot 0.5 + \color{blue}{-2} \cdot \left(0.5 \cdot \pi - \cos^{-1} \left(\sqrt{0.5 + -0.5 \cdot x}\right)\right) \]
   *-commutative [<=] 8.3	\[ \pi \cdot 0.5 + -2 \cdot \left(\color{blue}{\pi \cdot 0.5} - \cos^{-1} \left(\sqrt{0.5 + -0.5 \cdot x}\right)\right) \]
   metadata-eval [<=] 8.3	\[ \pi \cdot 0.5 + -2 \cdot \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{0.5 + \color{blue}{\left(-0.5\right)} \cdot x}\right)\right) \]
   cancel-sign-sub-inv [<=] 8.3	\[ \pi \cdot 0.5 + -2 \cdot \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\color{blue}{0.5 - 0.5 \cdot x}}\right)\right) \]
   cancel-sign-sub-inv [=>] 8.3	\[ \pi \cdot 0.5 + -2 \cdot \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\color{blue}{0.5 + \left(-0.5\right) \cdot x}}\right)\right) \]
   metadata-eval [=>] 8.3	\[ \pi \cdot 0.5 + -2 \cdot \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{0.5 + \color{blue}{-0.5} \cdot x}\right)\right) \]
   *-commutative [<=] 8.3	\[ \pi \cdot 0.5 + -2 \cdot \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{0.5 + \color{blue}{x \cdot -0.5}}\right)\right) \]

5. Applied egg-rr 8.3%

   \[\leadsto \color{blue}{\frac{1}{0.25 \cdot {\pi}^{2} + \left(-2 \cdot \cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)\right) \cdot \mathsf{fma}\left(\pi, -0.5, -2 \cdot \cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)\right)} \cdot \left(-0.125 \cdot {\pi}^{3} - {\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{3} \cdot -8\right)} \]
   Proof

   [Start] 8.3	\[ \pi \cdot -0.5 - -2 \cdot \cos^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right) \]
   flip3-- [=>] 8.3	\[ \color{blue}{\frac{{\left(\pi \cdot -0.5\right)}^{3} - {\left(-2 \cdot \cos^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)\right)}^{3}}{\left(\pi \cdot -0.5\right) \cdot \left(\pi \cdot -0.5\right) + \left(\left(-2 \cdot \cos^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)\right) \cdot \left(-2 \cdot \cos^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)\right) + \left(\pi \cdot -0.5\right) \cdot \left(-2 \cdot \cos^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)\right)\right)}} \]
   clear-num [=>] 8.3	\[ \color{blue}{\frac{1}{\frac{\left(\pi \cdot -0.5\right) \cdot \left(\pi \cdot -0.5\right) + \left(\left(-2 \cdot \cos^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)\right) \cdot \left(-2 \cdot \cos^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)\right) + \left(\pi \cdot -0.5\right) \cdot \left(-2 \cdot \cos^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)\right)\right)}{{\left(\pi \cdot -0.5\right)}^{3} - {\left(-2 \cdot \cos^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)\right)}^{3}}}} \]
   associate-/r/ [=>] 8.3	\[ \color{blue}{\frac{1}{\left(\pi \cdot -0.5\right) \cdot \left(\pi \cdot -0.5\right) + \left(\left(-2 \cdot \cos^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)\right) \cdot \left(-2 \cdot \cos^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)\right) + \left(\pi \cdot -0.5\right) \cdot \left(-2 \cdot \cos^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)\right)\right)} \cdot \left({\left(\pi \cdot -0.5\right)}^{3} - {\left(-2 \cdot \cos^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)\right)}^{3}\right)} \]

6. Taylor expanded in x around 0 8.3%

   \[\leadsto \frac{1}{\color{blue}{0.25 \cdot {\pi}^{2} + -2 \cdot \left(\left(-2 \cdot \cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right) + -0.5 \cdot \pi\right) \cdot \cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)\right)}} \cdot \left(-0.125 \cdot {\pi}^{3} - {\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{3} \cdot -8\right) \]

7. Final simplification 8.3%

   \[\leadsto \frac{1}{0.25 \cdot {\pi}^{2} + -2 \cdot \left(\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right) \cdot \left(-2 \cdot \cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right) + \pi \cdot -0.5\right)\right)} \cdot \left(-0.125 \cdot {\pi}^{3} - {\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{3} \cdot -8\right) \]

Alternatives

Alternative 1
Accuracy	8.3%
Cost	71488

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

Alternative 2
Accuracy	8.3%
Cost	45184

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

Alternative 3
Accuracy	8.3%
Cost	19840

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

Alternative 4
Accuracy	5.4%
Cost	19584

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

Reproduce

herbie shell --seed 2023130 
(FPCore (x)
  :name "Ian Simplification"
  :precision binary64

  :herbie-target
  (asin x)

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