bug323 (missed optimization)

Average Error: 6.8% → 10.4%

Time: 12.5s

Precision: binary64

Cost: 136896.00

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

Math

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

↓

\[\begin{array}{l} t_0 := \sin^{-1} \left(1 - x\right)\\ \frac{\frac{\left({\pi}^{6} \cdot {\pi}^{6}\right) \cdot 0.000244140625 - {t_0}^{12}}{\mathsf{fma}\left({\pi}^{3}, 0.125, {t_0}^{3}\right) \cdot \left({\pi}^{6} \cdot 0.015625 + {t_0}^{6}\right)}}{\left(\pi \cdot \pi\right) \cdot 0.25 + t_0 \cdot \mathsf{fma}\left(\pi, 0.5, t_0\right)} \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (asin (- 1.0 x))))
   (/
    (/
     (- (* (* (pow PI 6.0) (pow PI 6.0)) 0.000244140625) (pow t_0 12.0))
     (*
      (fma (pow PI 3.0) 0.125 (pow t_0 3.0))
      (+ (* (pow PI 6.0) 0.015625) (pow t_0 6.0))))
    (+ (* (* PI PI) 0.25) (* t_0 (fma PI 0.5 t_0))))))

C

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

↓

double code(double x) {
	double t_0 = asin((1.0 - x));
	return ((((pow(((double) M_PI), 6.0) * pow(((double) M_PI), 6.0)) * 0.000244140625) - pow(t_0, 12.0)) / (fma(pow(((double) M_PI), 3.0), 0.125, pow(t_0, 3.0)) * ((pow(((double) M_PI), 6.0) * 0.015625) + pow(t_0, 6.0)))) / (((((double) M_PI) * ((double) M_PI)) * 0.25) + (t_0 * fma(((double) M_PI), 0.5, t_0)));
}

Julia

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

↓

function code(x)
	t_0 = asin(Float64(1.0 - x))
	return Float64(Float64(Float64(Float64(Float64((pi ^ 6.0) * (pi ^ 6.0)) * 0.000244140625) - (t_0 ^ 12.0)) / Float64(fma((pi ^ 3.0), 0.125, (t_0 ^ 3.0)) * Float64(Float64((pi ^ 6.0) * 0.015625) + (t_0 ^ 6.0)))) / Float64(Float64(Float64(pi * pi) * 0.25) + Float64(t_0 * fma(pi, 0.5, t_0))))
end

Wolfram

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

↓

code[x_] := Block[{t$95$0 = N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(N[(N[(N[Power[Pi, 6.0], $MachinePrecision] * N[Power[Pi, 6.0], $MachinePrecision]), $MachinePrecision] * 0.000244140625), $MachinePrecision] - N[Power[t$95$0, 12.0], $MachinePrecision]), $MachinePrecision] / N[(N[(N[Power[Pi, 3.0], $MachinePrecision] * 0.125 + N[Power[t$95$0, 3.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[Pi, 6.0], $MachinePrecision] * 0.015625), $MachinePrecision] + N[Power[t$95$0, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(Pi * Pi), $MachinePrecision] * 0.25), $MachinePrecision] + N[(t$95$0 * N[(Pi * 0.5 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	6.8%
Target	100.0%
Herbie	10.4%

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

Derivation

1. Initial program 6.8

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

2. Applied egg-rr 6.8

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

   [Start] 6.8	\[ \cos^{-1} \left(1 - x\right) \]
   acos-asin [=>] 6.8	\[ \color{blue}{\frac{\pi}{2} - \sin^{-1} \left(1 - x\right)} \]
   flip3-- [=>] 6.8	\[ \color{blue}{\frac{{\left(\frac{\pi}{2}\right)}^{3} - {\sin^{-1} \left(1 - x\right)}^{3}}{\frac{\pi}{2} \cdot \frac{\pi}{2} + \left(\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right) + \frac{\pi}{2} \cdot \sin^{-1} \left(1 - x\right)\right)}} \]
   div-inv [=>] 6.8	\[ \frac{{\color{blue}{\left(\pi \cdot \frac{1}{2}\right)}}^{3} - {\sin^{-1} \left(1 - x\right)}^{3}}{\frac{\pi}{2} \cdot \frac{\pi}{2} + \left(\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right) + \frac{\pi}{2} \cdot \sin^{-1} \left(1 - x\right)\right)} \]
   metadata-eval [=>] 6.8	\[ \frac{{\left(\pi \cdot \color{blue}{0.5}\right)}^{3} - {\sin^{-1} \left(1 - x\right)}^{3}}{\frac{\pi}{2} \cdot \frac{\pi}{2} + \left(\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right) + \frac{\pi}{2} \cdot \sin^{-1} \left(1 - x\right)\right)} \]
   div-inv [=>] 6.8	\[ \frac{{\left(\pi \cdot 0.5\right)}^{3} - {\sin^{-1} \left(1 - x\right)}^{3}}{\color{blue}{\left(\pi \cdot \frac{1}{2}\right)} \cdot \frac{\pi}{2} + \left(\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right) + \frac{\pi}{2} \cdot \sin^{-1} \left(1 - x\right)\right)} \]
   metadata-eval [=>] 6.8	\[ \frac{{\left(\pi \cdot 0.5\right)}^{3} - {\sin^{-1} \left(1 - x\right)}^{3}}{\left(\pi \cdot \color{blue}{0.5}\right) \cdot \frac{\pi}{2} + \left(\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right) + \frac{\pi}{2} \cdot \sin^{-1} \left(1 - x\right)\right)} \]
   div-inv [=>] 6.8	\[ \frac{{\left(\pi \cdot 0.5\right)}^{3} - {\sin^{-1} \left(1 - x\right)}^{3}}{\left(\pi \cdot 0.5\right) \cdot \color{blue}{\left(\pi \cdot \frac{1}{2}\right)} + \left(\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right) + \frac{\pi}{2} \cdot \sin^{-1} \left(1 - x\right)\right)} \]
   metadata-eval [=>] 6.8	\[ \frac{{\left(\pi \cdot 0.5\right)}^{3} - {\sin^{-1} \left(1 - x\right)}^{3}}{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot \color{blue}{0.5}\right) + \left(\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right) + \frac{\pi}{2} \cdot \sin^{-1} \left(1 - x\right)\right)} \]
   div-inv [=>] 6.8	\[ \frac{{\left(\pi \cdot 0.5\right)}^{3} - {\sin^{-1} \left(1 - x\right)}^{3}}{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) + \left(\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right) + \color{blue}{\left(\pi \cdot \frac{1}{2}\right)} \cdot \sin^{-1} \left(1 - x\right)\right)} \]
   metadata-eval [=>] 6.8	\[ \frac{{\left(\pi \cdot 0.5\right)}^{3} - {\sin^{-1} \left(1 - x\right)}^{3}}{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) + \left(\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right) + \left(\pi \cdot \color{blue}{0.5}\right) \cdot \sin^{-1} \left(1 - x\right)\right)} \]

3. Simplified 6.8

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

   [Start] 6.8	\[ \frac{{\left(\pi \cdot 0.5\right)}^{3} - {\sin^{-1} \left(1 - x\right)}^{3}}{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) + \left(\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right) + \left(\pi \cdot 0.5\right) \cdot \sin^{-1} \left(1 - x\right)\right)} \]
   cube-prod [=>] 6.8	\[ \frac{\color{blue}{{\pi}^{3} \cdot {0.5}^{3}} - {\sin^{-1} \left(1 - x\right)}^{3}}{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) + \left(\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right) + \left(\pi \cdot 0.5\right) \cdot \sin^{-1} \left(1 - x\right)\right)} \]
   metadata-eval [=>] 6.8	\[ \frac{{\pi}^{3} \cdot \color{blue}{0.125} - {\sin^{-1} \left(1 - x\right)}^{3}}{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) + \left(\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right) + \left(\pi \cdot 0.5\right) \cdot \sin^{-1} \left(1 - x\right)\right)} \]
   swap-sqr [=>] 6.8	\[ \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(1 - x\right)}^{3}}{\color{blue}{\left(\pi \cdot \pi\right) \cdot \left(0.5 \cdot 0.5\right)} + \left(\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right) + \left(\pi \cdot 0.5\right) \cdot \sin^{-1} \left(1 - x\right)\right)} \]
   metadata-eval [=>] 6.8	\[ \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(1 - x\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot \color{blue}{0.25} + \left(\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right) + \left(\pi \cdot 0.5\right) \cdot \sin^{-1} \left(1 - x\right)\right)} \]
   distribute-rgt-out [=>] 6.8	\[ \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(1 - x\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \color{blue}{\sin^{-1} \left(1 - x\right) \cdot \left(\sin^{-1} \left(1 - x\right) + \pi \cdot 0.5\right)}} \]
   +-commutative [<=] 6.8	\[ \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(1 - x\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \color{blue}{\left(\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)\right)}} \]
   fma-def [=>] 6.8	\[ \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(1 - x\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \color{blue}{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)}} \]

4. Applied egg-rr 6.8

   \[\leadsto \frac{\color{blue}{\frac{\left({\pi}^{6} \cdot 0.015625\right) \cdot \left({\pi}^{6} \cdot 0.015625\right) - {\sin^{-1} \left(1 - x\right)}^{6} \cdot {\sin^{-1} \left(1 - x\right)}^{6}}{\mathsf{fma}\left({\pi}^{3}, 0.125, {\sin^{-1} \left(1 - x\right)}^{3}\right) \cdot \left({\sin^{-1} \left(1 - x\right)}^{6} + {\pi}^{6} \cdot 0.015625\right)}}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]
   Proof

   [Start] 6.8	\[ \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(1 - x\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]
   flip-- [=>] 6.8	\[ \frac{\color{blue}{\frac{\left({\pi}^{3} \cdot 0.125\right) \cdot \left({\pi}^{3} \cdot 0.125\right) - {\sin^{-1} \left(1 - x\right)}^{3} \cdot {\sin^{-1} \left(1 - x\right)}^{3}}{{\pi}^{3} \cdot 0.125 + {\sin^{-1} \left(1 - x\right)}^{3}}}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]
   flip-- [=>] 6.8	\[ \frac{\frac{\color{blue}{\frac{\left(\left({\pi}^{3} \cdot 0.125\right) \cdot \left({\pi}^{3} \cdot 0.125\right)\right) \cdot \left(\left({\pi}^{3} \cdot 0.125\right) \cdot \left({\pi}^{3} \cdot 0.125\right)\right) - \left({\sin^{-1} \left(1 - x\right)}^{3} \cdot {\sin^{-1} \left(1 - x\right)}^{3}\right) \cdot \left({\sin^{-1} \left(1 - x\right)}^{3} \cdot {\sin^{-1} \left(1 - x\right)}^{3}\right)}{\left({\pi}^{3} \cdot 0.125\right) \cdot \left({\pi}^{3} \cdot 0.125\right) + {\sin^{-1} \left(1 - x\right)}^{3} \cdot {\sin^{-1} \left(1 - x\right)}^{3}}}}{{\pi}^{3} \cdot 0.125 + {\sin^{-1} \left(1 - x\right)}^{3}}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]
   associate-/l/ [=>] 6.8	\[ \frac{\color{blue}{\frac{\left(\left({\pi}^{3} \cdot 0.125\right) \cdot \left({\pi}^{3} \cdot 0.125\right)\right) \cdot \left(\left({\pi}^{3} \cdot 0.125\right) \cdot \left({\pi}^{3} \cdot 0.125\right)\right) - \left({\sin^{-1} \left(1 - x\right)}^{3} \cdot {\sin^{-1} \left(1 - x\right)}^{3}\right) \cdot \left({\sin^{-1} \left(1 - x\right)}^{3} \cdot {\sin^{-1} \left(1 - x\right)}^{3}\right)}{\left({\pi}^{3} \cdot 0.125 + {\sin^{-1} \left(1 - x\right)}^{3}\right) \cdot \left(\left({\pi}^{3} \cdot 0.125\right) \cdot \left({\pi}^{3} \cdot 0.125\right) + {\sin^{-1} \left(1 - x\right)}^{3} \cdot {\sin^{-1} \left(1 - x\right)}^{3}\right)}}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]

5. Simplified 10.4

   \[\leadsto \frac{\color{blue}{\frac{\left({\pi}^{6} \cdot {\pi}^{6}\right) \cdot 0.000244140625 - {\sin^{-1} \left(1 - x\right)}^{12}}{\mathsf{fma}\left({\pi}^{3}, 0.125, {\sin^{-1} \left(1 - x\right)}^{3}\right) \cdot \left({\pi}^{6} \cdot 0.015625 + {\sin^{-1} \left(1 - x\right)}^{6}\right)}}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]
   Proof

   [Start] 6.8	\[ \frac{\frac{\left({\pi}^{6} \cdot 0.015625\right) \cdot \left({\pi}^{6} \cdot 0.015625\right) - {\sin^{-1} \left(1 - x\right)}^{6} \cdot {\sin^{-1} \left(1 - x\right)}^{6}}{\mathsf{fma}\left({\pi}^{3}, 0.125, {\sin^{-1} \left(1 - x\right)}^{3}\right) \cdot \left({\sin^{-1} \left(1 - x\right)}^{6} + {\pi}^{6} \cdot 0.015625\right)}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]
   swap-sqr [=>] 6.8	\[ \frac{\frac{\color{blue}{\left({\pi}^{6} \cdot {\pi}^{6}\right) \cdot \left(0.015625 \cdot 0.015625\right)} - {\sin^{-1} \left(1 - x\right)}^{6} \cdot {\sin^{-1} \left(1 - x\right)}^{6}}{\mathsf{fma}\left({\pi}^{3}, 0.125, {\sin^{-1} \left(1 - x\right)}^{3}\right) \cdot \left({\sin^{-1} \left(1 - x\right)}^{6} + {\pi}^{6} \cdot 0.015625\right)}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]
   metadata-eval [=>] 6.8	\[ \frac{\frac{\left({\pi}^{6} \cdot {\pi}^{6}\right) \cdot \color{blue}{0.000244140625} - {\sin^{-1} \left(1 - x\right)}^{6} \cdot {\sin^{-1} \left(1 - x\right)}^{6}}{\mathsf{fma}\left({\pi}^{3}, 0.125, {\sin^{-1} \left(1 - x\right)}^{3}\right) \cdot \left({\sin^{-1} \left(1 - x\right)}^{6} + {\pi}^{6} \cdot 0.015625\right)}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]
   pow-sqr [=>] 10.4	\[ \frac{\frac{\left({\pi}^{6} \cdot {\pi}^{6}\right) \cdot 0.000244140625 - \color{blue}{{\sin^{-1} \left(1 - x\right)}^{\left(2 \cdot 6\right)}}}{\mathsf{fma}\left({\pi}^{3}, 0.125, {\sin^{-1} \left(1 - x\right)}^{3}\right) \cdot \left({\sin^{-1} \left(1 - x\right)}^{6} + {\pi}^{6} \cdot 0.015625\right)}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]
   metadata-eval [=>] 10.4	\[ \frac{\frac{\left({\pi}^{6} \cdot {\pi}^{6}\right) \cdot 0.000244140625 - {\sin^{-1} \left(1 - x\right)}^{\color{blue}{12}}}{\mathsf{fma}\left({\pi}^{3}, 0.125, {\sin^{-1} \left(1 - x\right)}^{3}\right) \cdot \left({\sin^{-1} \left(1 - x\right)}^{6} + {\pi}^{6} \cdot 0.015625\right)}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]
   +-commutative [=>] 10.4	\[ \frac{\frac{\left({\pi}^{6} \cdot {\pi}^{6}\right) \cdot 0.000244140625 - {\sin^{-1} \left(1 - x\right)}^{12}}{\mathsf{fma}\left({\pi}^{3}, 0.125, {\sin^{-1} \left(1 - x\right)}^{3}\right) \cdot \color{blue}{\left({\pi}^{6} \cdot 0.015625 + {\sin^{-1} \left(1 - x\right)}^{6}\right)}}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]

6. Final simplification 10.4

   \[\leadsto \frac{\frac{\left({\pi}^{6} \cdot {\pi}^{6}\right) \cdot 0.000244140625 - {\sin^{-1} \left(1 - x\right)}^{12}}{\mathsf{fma}\left({\pi}^{3}, 0.125, {\sin^{-1} \left(1 - x\right)}^{3}\right) \cdot \left({\pi}^{6} \cdot 0.015625 + {\sin^{-1} \left(1 - x\right)}^{6}\right)}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]

Alternatives

Alternative 1
Error	10.4%
Cost	78272.00

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

Alternative 2
Error	10.3%
Cost	45696.00

\[\begin{array}{l} t_0 := \sin^{-1} \left(1 - x\right)\\ t_1 := \sqrt{t_0}\\ \mathsf{fma}\left(-t_1, t_1, t_0\right) + \cos^{-1} \left(1 - x\right) \end{array} \]

Alternative 3
Error	10.3%
Cost	26048.00

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

Alternative 4
Error	6.8%
Cost	19972.00

\[\begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right) + -1\\ \mathbf{if}\;1 - x \leq 1:\\ \;\;\;\;1 + {\left(\sqrt[3]{t_0}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;1 + \left|t_0\right|\\ \end{array} \]

Alternative 5
Error	9.4%
Cost	19908.00

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

Alternative 6
Error	10.3%
Cost	19840.00

\[\left({\left(\sqrt[3]{2 + \cos^{-1} \left(1 - x\right)}\right)}^{3} + -1\right) + -1 \]

Alternative 7
Error	10.3%
Cost	19840.00

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

Alternative 8
Error	6.8%
Cost	13184.00

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

Alternative 9
Error	6.8%
Cost	6848.00

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

Alternative 10
Error	6.8%
Cost	6848.00

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

Alternative 11
Error	6.8%
Cost	6592.00

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

Reproduce

herbie shell --seed 2023104 
(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)))