bug323 (missed optimization)

Average Accuracy: 7.1% → 10.7%

Time: 11.9s

Precision: binary64

Cost: 208704

\[0 \leq x \land x \leq 0.5\]

Math

\[\cos^{-1} \left(1 - x\right) \]

↓

\[\begin{array}{l} t_0 := \left(\pi \cdot 0.5\right) \cdot \sqrt{\pi \cdot 0.5}\\ t_1 := \sin^{-1} \left(1 - x\right)\\ t_2 := {t_1}^{2}\\ t_3 := \sqrt{t_1}\\ t_4 := t_3 \cdot t_2\\ \frac{\mathsf{fma}\left(t_0, t_0, t_4 \cdot \left(-t_3\right)\right) + \mathsf{fma}\left(t_3 \cdot \left(-t_2\right), t_3, t_3 \cdot t_4\right)}{\left(\pi \cdot \pi\right) \cdot 0.25 + t_1 \cdot \mathsf{fma}\left(\pi, 0.5, t_1\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (acos (- 1.0 x)))

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* PI 0.5) (sqrt (* PI 0.5))))
        (t_1 (asin (- 1.0 x)))
        (t_2 (pow t_1 2.0))
        (t_3 (sqrt t_1))
        (t_4 (* t_3 t_2)))
   (/
    (+ (fma t_0 t_0 (* t_4 (- t_3))) (fma (* t_3 (- t_2)) t_3 (* t_3 t_4)))
    (+ (* (* PI PI) 0.25) (* t_1 (fma PI 0.5 t_1))))))

C

double code(double x) {
	return acos((1.0 - x));
}

↓

double code(double x) {
	double t_0 = (((double) M_PI) * 0.5) * sqrt((((double) M_PI) * 0.5));
	double t_1 = asin((1.0 - x));
	double t_2 = pow(t_1, 2.0);
	double t_3 = sqrt(t_1);
	double t_4 = t_3 * t_2;
	return (fma(t_0, t_0, (t_4 * -t_3)) + fma((t_3 * -t_2), t_3, (t_3 * t_4))) / (((((double) M_PI) * ((double) M_PI)) * 0.25) + (t_1 * fma(((double) M_PI), 0.5, t_1)));
}

Julia

function code(x)
	return acos(Float64(1.0 - x))
end

↓

function code(x)
	t_0 = Float64(Float64(pi * 0.5) * sqrt(Float64(pi * 0.5)))
	t_1 = asin(Float64(1.0 - x))
	t_2 = t_1 ^ 2.0
	t_3 = sqrt(t_1)
	t_4 = Float64(t_3 * t_2)
	return Float64(Float64(fma(t_0, t_0, Float64(t_4 * Float64(-t_3))) + fma(Float64(t_3 * Float64(-t_2)), t_3, Float64(t_3 * t_4))) / Float64(Float64(Float64(pi * pi) * 0.25) + Float64(t_1 * fma(pi, 0.5, t_1))))
end

Wolfram

code[x_] := N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]

↓

code[x_] := Block[{t$95$0 = N[(N[(Pi * 0.5), $MachinePrecision] * N[Sqrt[N[(Pi * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Power[t$95$1, 2.0], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[t$95$1], $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 * t$95$2), $MachinePrecision]}, N[(N[(N[(t$95$0 * t$95$0 + N[(t$95$4 * (-t$95$3)), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$3 * (-t$95$2)), $MachinePrecision] * t$95$3 + N[(t$95$3 * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(Pi * Pi), $MachinePrecision] * 0.25), $MachinePrecision] + N[(t$95$1 * N[(Pi * 0.5 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Target

Original	7.1%
Target	100.0%
Herbie	10.7%

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

Derivation

1. Initial program 7.1%

   \[\cos^{-1} \left(1 - x\right) \]

2. Applied egg-rr 7.1%

   \[\leadsto \color{blue}{\frac{{\left(\pi \cdot 0.5\right)}^{3} - {\sin^{-1} \left(1 - x\right)}^{3}}{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) + \left(\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right) + \left(\pi \cdot 0.5\right) \cdot \sin^{-1} \left(1 - x\right)\right)}} \]
   Proof

   [Start] 7.1	\[ \cos^{-1} \left(1 - x\right) \]
   acos-asin [=>] 7.1	\[ \color{blue}{\frac{\pi}{2} - \sin^{-1} \left(1 - x\right)} \]
   flip3-- [=>] 7.1	\[ \color{blue}{\frac{{\left(\frac{\pi}{2}\right)}^{3} - {\sin^{-1} \left(1 - x\right)}^{3}}{\frac{\pi}{2} \cdot \frac{\pi}{2} + \left(\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right) + \frac{\pi}{2} \cdot \sin^{-1} \left(1 - x\right)\right)}} \]
   div-inv [=>] 7.1	\[ \frac{{\color{blue}{\left(\pi \cdot \frac{1}{2}\right)}}^{3} - {\sin^{-1} \left(1 - x\right)}^{3}}{\frac{\pi}{2} \cdot \frac{\pi}{2} + \left(\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right) + \frac{\pi}{2} \cdot \sin^{-1} \left(1 - x\right)\right)} \]
   metadata-eval [=>] 7.1	\[ \frac{{\left(\pi \cdot \color{blue}{0.5}\right)}^{3} - {\sin^{-1} \left(1 - x\right)}^{3}}{\frac{\pi}{2} \cdot \frac{\pi}{2} + \left(\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right) + \frac{\pi}{2} \cdot \sin^{-1} \left(1 - x\right)\right)} \]
   div-inv [=>] 7.1	\[ \frac{{\left(\pi \cdot 0.5\right)}^{3} - {\sin^{-1} \left(1 - x\right)}^{3}}{\color{blue}{\left(\pi \cdot \frac{1}{2}\right)} \cdot \frac{\pi}{2} + \left(\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right) + \frac{\pi}{2} \cdot \sin^{-1} \left(1 - x\right)\right)} \]
   metadata-eval [=>] 7.1	\[ \frac{{\left(\pi \cdot 0.5\right)}^{3} - {\sin^{-1} \left(1 - x\right)}^{3}}{\left(\pi \cdot \color{blue}{0.5}\right) \cdot \frac{\pi}{2} + \left(\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right) + \frac{\pi}{2} \cdot \sin^{-1} \left(1 - x\right)\right)} \]
   div-inv [=>] 7.1	\[ \frac{{\left(\pi \cdot 0.5\right)}^{3} - {\sin^{-1} \left(1 - x\right)}^{3}}{\left(\pi \cdot 0.5\right) \cdot \color{blue}{\left(\pi \cdot \frac{1}{2}\right)} + \left(\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right) + \frac{\pi}{2} \cdot \sin^{-1} \left(1 - x\right)\right)} \]
   metadata-eval [=>] 7.1	\[ \frac{{\left(\pi \cdot 0.5\right)}^{3} - {\sin^{-1} \left(1 - x\right)}^{3}}{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot \color{blue}{0.5}\right) + \left(\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right) + \frac{\pi}{2} \cdot \sin^{-1} \left(1 - x\right)\right)} \]
   div-inv [=>] 7.1	\[ \frac{{\left(\pi \cdot 0.5\right)}^{3} - {\sin^{-1} \left(1 - x\right)}^{3}}{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) + \left(\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right) + \color{blue}{\left(\pi \cdot \frac{1}{2}\right)} \cdot \sin^{-1} \left(1 - x\right)\right)} \]
   metadata-eval [=>] 7.1	\[ \frac{{\left(\pi \cdot 0.5\right)}^{3} - {\sin^{-1} \left(1 - x\right)}^{3}}{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) + \left(\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right) + \left(\pi \cdot \color{blue}{0.5}\right) \cdot \sin^{-1} \left(1 - x\right)\right)} \]

3. Simplified 7.1%

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

   [Start] 7.1	\[ \frac{{\left(\pi \cdot 0.5\right)}^{3} - {\sin^{-1} \left(1 - x\right)}^{3}}{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) + \left(\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right) + \left(\pi \cdot 0.5\right) \cdot \sin^{-1} \left(1 - x\right)\right)} \]
   swap-sqr [=>] 7.1	\[ \frac{{\left(\pi \cdot 0.5\right)}^{3} - {\sin^{-1} \left(1 - x\right)}^{3}}{\color{blue}{\left(\pi \cdot \pi\right) \cdot \left(0.5 \cdot 0.5\right)} + \left(\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right) + \left(\pi \cdot 0.5\right) \cdot \sin^{-1} \left(1 - x\right)\right)} \]
   metadata-eval [=>] 7.1	\[ \frac{{\left(\pi \cdot 0.5\right)}^{3} - {\sin^{-1} \left(1 - x\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot \color{blue}{0.25} + \left(\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right) + \left(\pi \cdot 0.5\right) \cdot \sin^{-1} \left(1 - x\right)\right)} \]
   distribute-rgt-out [=>] 7.1	\[ \frac{{\left(\pi \cdot 0.5\right)}^{3} - {\sin^{-1} \left(1 - x\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \color{blue}{\sin^{-1} \left(1 - x\right) \cdot \left(\sin^{-1} \left(1 - x\right) + \pi \cdot 0.5\right)}} \]
   +-commutative [<=] 7.1	\[ \frac{{\left(\pi \cdot 0.5\right)}^{3} - {\sin^{-1} \left(1 - x\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \color{blue}{\left(\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)\right)}} \]
   fma-def [=>] 7.1	\[ \frac{{\left(\pi \cdot 0.5\right)}^{3} - {\sin^{-1} \left(1 - x\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \color{blue}{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)}} \]

4. Applied egg-rr 10.6%

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

   [Start] 7.1	\[ \frac{{\left(\pi \cdot 0.5\right)}^{3} - {\sin^{-1} \left(1 - x\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]
   unpow3 [=>] 7.1	\[ \frac{\color{blue}{\left(\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right)\right) \cdot \left(\pi \cdot 0.5\right)} - {\sin^{-1} \left(1 - x\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]
   fma-neg [=>] 10.6	\[ \frac{\color{blue}{\mathsf{fma}\left(\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right), \pi \cdot 0.5, -{\sin^{-1} \left(1 - x\right)}^{3}\right)}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]
   swap-sqr [=>] 10.6	\[ \frac{\mathsf{fma}\left(\color{blue}{\left(\pi \cdot \pi\right) \cdot \left(0.5 \cdot 0.5\right)}, \pi \cdot 0.5, -{\sin^{-1} \left(1 - x\right)}^{3}\right)}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]
   metadata-eval [=>] 10.6	\[ \frac{\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \color{blue}{0.25}, \pi \cdot 0.5, -{\sin^{-1} \left(1 - x\right)}^{3}\right)}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]
   pow2 [=>] 10.6	\[ \frac{\mathsf{fma}\left(\color{blue}{{\pi}^{2}} \cdot 0.25, \pi \cdot 0.5, -{\sin^{-1} \left(1 - x\right)}^{3}\right)}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]

5. Applied egg-rr 10.7%

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

   [Start] 10.6	\[ \frac{\mathsf{fma}\left({\pi}^{2} \cdot 0.25, \pi \cdot 0.5, -{\sin^{-1} \left(1 - x\right)}^{3}\right)}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]
   fma-neg [<=] 7.1	\[ \frac{\color{blue}{\left({\pi}^{2} \cdot 0.25\right) \cdot \left(\pi \cdot 0.5\right) - {\sin^{-1} \left(1 - x\right)}^{3}}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]
   add-sqr-sqrt [=>] 5.3	\[ \frac{\left({\pi}^{2} \cdot 0.25\right) \cdot \color{blue}{\left(\sqrt{\pi \cdot 0.5} \cdot \sqrt{\pi \cdot 0.5}\right)} - {\sin^{-1} \left(1 - x\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]
   unpow2 [=>] 5.3	\[ \frac{\left(\color{blue}{\left(\pi \cdot \pi\right)} \cdot 0.25\right) \cdot \left(\sqrt{\pi \cdot 0.5} \cdot \sqrt{\pi \cdot 0.5}\right) - {\sin^{-1} \left(1 - x\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]
   metadata-eval [<=] 5.3	\[ \frac{\left(\left(\pi \cdot \pi\right) \cdot \color{blue}{\left(0.5 \cdot 0.5\right)}\right) \cdot \left(\sqrt{\pi \cdot 0.5} \cdot \sqrt{\pi \cdot 0.5}\right) - {\sin^{-1} \left(1 - x\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]
   swap-sqr [<=] 5.3	\[ \frac{\color{blue}{\left(\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right)\right)} \cdot \left(\sqrt{\pi \cdot 0.5} \cdot \sqrt{\pi \cdot 0.5}\right) - {\sin^{-1} \left(1 - x\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]
   unswap-sqr [=>] 5.3	\[ \frac{\color{blue}{\left(\left(\pi \cdot 0.5\right) \cdot \sqrt{\pi \cdot 0.5}\right) \cdot \left(\left(\pi \cdot 0.5\right) \cdot \sqrt{\pi \cdot 0.5}\right)} - {\sin^{-1} \left(1 - x\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]
   add-sqr-sqrt [=>] 5.3	\[ \frac{\left(\color{blue}{\left(\sqrt{\pi \cdot 0.5} \cdot \sqrt{\pi \cdot 0.5}\right)} \cdot \sqrt{\pi \cdot 0.5}\right) \cdot \left(\left(\pi \cdot 0.5\right) \cdot \sqrt{\pi \cdot 0.5}\right) - {\sin^{-1} \left(1 - x\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]
   add-sqr-sqrt [=>] 5.3	\[ \frac{\left(\left(\sqrt{\pi \cdot 0.5} \cdot \sqrt{\pi \cdot 0.5}\right) \cdot \sqrt{\pi \cdot 0.5}\right) \cdot \left(\color{blue}{\left(\sqrt{\pi \cdot 0.5} \cdot \sqrt{\pi \cdot 0.5}\right)} \cdot \sqrt{\pi \cdot 0.5}\right) - {\sin^{-1} \left(1 - x\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]
   cube-mult [=>] 5.3	\[ \frac{\left(\left(\sqrt{\pi \cdot 0.5} \cdot \sqrt{\pi \cdot 0.5}\right) \cdot \sqrt{\pi \cdot 0.5}\right) \cdot \left(\left(\sqrt{\pi \cdot 0.5} \cdot \sqrt{\pi \cdot 0.5}\right) \cdot \sqrt{\pi \cdot 0.5}\right) - \color{blue}{\sin^{-1} \left(1 - x\right) \cdot \left(\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)\right)}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]

6. Final simplification 10.7%

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

Alternatives

Alternative 1
Accuracy	10.7%
Cost	150080

\[\begin{array}{l} t_0 := \sqrt{{\left(\pi \cdot 0.5\right)}^{3}}\\ t_1 := \sin^{-1} \left(1 - x\right)\\ t_2 := {t_1}^{2}\\ \frac{\mathsf{fma}\left(t_0, t_0, t_1 \cdot \left(-t_2\right)\right) + \mathsf{fma}\left(-t_1, t_2, t_1 \cdot t_2\right)}{\left(\pi \cdot \pi\right) \cdot 0.25 + t_1 \cdot \mathsf{fma}\left(\pi, 0.5, t_1\right)} \end{array} \]

Alternative 2
Accuracy	10.6%
Cost	52032

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

Alternative 3
Accuracy	10.6%
Cost	51776

\[\mathsf{fma}\left(\sqrt[3]{0.25 \cdot {\pi}^{2}}, \sqrt[3]{\pi} \cdot \sqrt[3]{0.5}, -\sin^{-1} \left(1 - x\right)\right) \]

Alternative 4
Accuracy	10.6%
Cost	51776

\[\mathsf{fma}\left(\sqrt[3]{0.25 \cdot {\pi}^{2}} \cdot \sqrt[3]{\pi}, \sqrt[3]{0.5}, -\sin^{-1} \left(1 - x\right)\right) \]

Alternative 5
Accuracy	10.6%
Cost	26496

\[\pi \cdot 0.5 - 3 \cdot \log \left(e^{3 \cdot \left(0.3333333333333333 \cdot \left(\sin^{-1} \left(1 - x\right) \cdot 0.3333333333333333\right)\right)}\right) \]

Alternative 6
Accuracy	9.7%
Cost	19844

\[\begin{array}{l} \mathbf{if}\;\cos^{-1} \left(1 - x\right) \leq 0:\\ \;\;\;\;\pi \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot 0.5 - \sin^{-1} \left(1 - x\right)\\ \end{array} \]

Alternative 7
Accuracy	10.6%
Cost	19712

\[\mathsf{fma}\left(\sin^{-1} \left(1 - x\right) \cdot 0.3333333333333333, -3, \pi \cdot 0.5\right) \]

Alternative 8
Accuracy	9.7%
Cost	13252

\[\begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right)\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;\pi \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Accuracy	7.1%
Cost	7108

\[\begin{array}{l} \mathbf{if}\;1 - x \leq 1:\\ \;\;\;\;1 + \left(\cos^{-1} \left(1 - x\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot 0.5\\ \end{array} \]

Alternative 10
Accuracy	7.0%
Cost	6528

\[\pi \cdot 0.5 \]

Reproduce

herbie shell --seed 2023153 
(FPCore (x)
  :name "bug323 (missed optimization)"
  :precision binary64
  :pre (and (<= 0.0 x) (<= x 0.5))

  :herbie-target
  (* 2.0 (asin (sqrt (/ x 2.0))))

  (acos (- 1.0 x)))