?

Average Error: 92.86% → 89.34%
Time: 10.8s
Precision: binary64
Cost: 214848

?

\[0 \leq x \land x \leq 0.5\]
\[\cos^{-1} \left(1 - x\right) \]
↓
\[\begin{array}{l} t_0 := \sin^{-1} \left(1 - x\right)\\ t_1 := {t_0}^{3}\\ t_2 := {t_0}^{9}\\ t_3 := {t_0}^{18}\\ \frac{\frac{\frac{\mathsf{fma}\left(-t_2, t_2, t_3\right) + \left({\pi}^{18} \cdot 3.814697265625 \cdot 10^{-6} - t_3\right)}{\mathsf{fma}\left(t_1, \mathsf{fma}\left(0.125, {\pi}^{3}, t_1\right), {\pi}^{6} \cdot 0.015625\right)}}{\mathsf{fma}\left({\left({\pi}^{3}\right)}^{3}, 0.001953125, t_2\right)}}{\left(\pi \cdot \pi\right) \cdot 0.25 + t_0 \cdot \mathsf{fma}\left(\pi, 0.5, t_0\right)} \end{array} \]
(FPCore (x) :precision binary64 (acos (- 1.0 x)))
↓
(FPCore (x)
 :precision binary64
 (let* ((t_0 (asin (- 1.0 x)))
        (t_1 (pow t_0 3.0))
        (t_2 (pow t_0 9.0))
        (t_3 (pow t_0 18.0)))
   (/
    (/
     (/
      (+ (fma (- t_2) t_2 t_3) (- (* (pow PI 18.0) 3.814697265625e-6) t_3))
      (fma t_1 (fma 0.125 (pow PI 3.0) t_1) (* (pow PI 6.0) 0.015625)))
     (fma (pow (pow PI 3.0) 3.0) 0.001953125 t_2))
    (+ (* (* PI PI) 0.25) (* t_0 (fma PI 0.5 t_0))))))
double code(double x) {
	return acos((1.0 - x));
}
↓
double code(double x) {
	double t_0 = asin((1.0 - x));
	double t_1 = pow(t_0, 3.0);
	double t_2 = pow(t_0, 9.0);
	double t_3 = pow(t_0, 18.0);
	return (((fma(-t_2, t_2, t_3) + ((pow(((double) M_PI), 18.0) * 3.814697265625e-6) - t_3)) / fma(t_1, fma(0.125, pow(((double) M_PI), 3.0), t_1), (pow(((double) M_PI), 6.0) * 0.015625))) / fma(pow(pow(((double) M_PI), 3.0), 3.0), 0.001953125, t_2)) / (((((double) M_PI) * ((double) M_PI)) * 0.25) + (t_0 * fma(((double) M_PI), 0.5, t_0)));
}

function code(x)
	return acos(Float64(1.0 - x))
end
↓
function code(x)
	t_0 = asin(Float64(1.0 - x))
	t_1 = t_0 ^ 3.0
	t_2 = t_0 ^ 9.0
	t_3 = t_0 ^ 18.0
	return Float64(Float64(Float64(Float64(fma(Float64(-t_2), t_2, t_3) + Float64(Float64((pi ^ 18.0) * 3.814697265625e-6) - t_3)) / fma(t_1, fma(0.125, (pi ^ 3.0), t_1), Float64((pi ^ 6.0) * 0.015625))) / fma(((pi ^ 3.0) ^ 3.0), 0.001953125, t_2)) / Float64(Float64(Float64(pi * pi) * 0.25) + Float64(t_0 * fma(pi, 0.5, t_0))))
end

code[x_] := N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]
↓
code[x_] := Block[{t$95$0 = N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Power[t$95$0, 3.0], $MachinePrecision]}, Block[{t$95$2 = N[Power[t$95$0, 9.0], $MachinePrecision]}, Block[{t$95$3 = N[Power[t$95$0, 18.0], $MachinePrecision]}, N[(N[(N[(N[(N[((-t$95$2) * t$95$2 + t$95$3), $MachinePrecision] + N[(N[(N[Power[Pi, 18.0], $MachinePrecision] * 3.814697265625e-6), $MachinePrecision] - t$95$3), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * N[(0.125 * N[Power[Pi, 3.0], $MachinePrecision] + t$95$1), $MachinePrecision] + N[(N[Power[Pi, 6.0], $MachinePrecision] * 0.015625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[Power[Pi, 3.0], $MachinePrecision], 3.0], $MachinePrecision] * 0.001953125 + t$95$2), $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]]]]]

\cos^{-1} \left(1 - x\right)
↓
\begin{array}{l}
t_0 := \sin^{-1} \left(1 - x\right)\\
t_1 := {t_0}^{3}\\
t_2 := {t_0}^{9}\\
t_3 := {t_0}^{18}\\
\frac{\frac{\frac{\mathsf{fma}\left(-t_2, t_2, t_3\right) + \left({\pi}^{18} \cdot 3.814697265625 \cdot 10^{-6} - t_3\right)}{\mathsf{fma}\left(t_1, \mathsf{fma}\left(0.125, {\pi}^{3}, t_1\right), {\pi}^{6} \cdot 0.015625\right)}}{\mathsf{fma}\left({\left({\pi}^{3}\right)}^{3}, 0.001953125, t_2\right)}}{\left(\pi \cdot \pi\right) \cdot 0.25 + t_0 \cdot \mathsf{fma}\left(\pi, 0.5, t_0\right)}
\end{array}

Error?

Target

Original92.86%
Target0.02%
Herbie89.34%
\[2 \cdot \sin^{-1} \left(\sqrt{\frac{x}{2}}\right) \]

Derivation?

  1. Initial program 92.86

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

  2. Applied egg-rr92.86

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

  3. Simplified92.86

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

    [Start]92.86

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

    cube-prod [=>]92.86

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

    metadata-eval [=>]92.86

    \[ \frac{{\pi}^{3} \cdot \color{blue}{0.125} - {\sin^{-1} \left(1 - x\right)}^{3}}{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) + \left(\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right) + \left(\pi \cdot 0.5\right) \cdot \sin^{-1} \left(1 - x\right)\right)} \]

    swap-sqr [=>]92.86

    \[ \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(1 - x\right)}^{3}}{\color{blue}{\left(\pi \cdot \pi\right) \cdot \left(0.5 \cdot 0.5\right)} + \left(\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right) + \left(\pi \cdot 0.5\right) \cdot \sin^{-1} \left(1 - x\right)\right)} \]

    metadata-eval [=>]92.86

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

    distribute-rgt-out [=>]92.86

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

    +-commutative [<=]92.86

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

    fma-def [=>]92.86

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

  4. Applied egg-rr89.4

    \[\leadsto \frac{{\pi}^{3} \cdot 0.125 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\sin^{-1} \left(1 - x\right)}^{3}\right)\right)}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]

  5. Applied egg-rr92.86

    \[\leadsto \frac{\color{blue}{\frac{{\left({\pi}^{3} \cdot 0.125\right)}^{3} \cdot {\left({\pi}^{3} \cdot 0.125\right)}^{3} - {\sin^{-1} \left(1 - x\right)}^{9} \cdot {\sin^{-1} \left(1 - x\right)}^{9}}{\left({\pi}^{6} \cdot 0.015625 + {\sin^{-1} \left(1 - x\right)}^{3} \cdot \mathsf{fma}\left({\pi}^{3}, 0.125, {\sin^{-1} \left(1 - x\right)}^{3}\right)\right) \cdot \left({\left({\pi}^{3} \cdot 0.125\right)}^{3} + {\sin^{-1} \left(1 - x\right)}^{9}\right)}}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]

  6. Simplified89.37

    \[\leadsto \frac{\color{blue}{\frac{\frac{{\left({\pi}^{6}\right)}^{3} \cdot 3.814697265625 \cdot 10^{-6} - {\sin^{-1} \left(1 - x\right)}^{18}}{\mathsf{fma}\left({\sin^{-1} \left(1 - x\right)}^{3}, \mathsf{fma}\left(0.125, {\pi}^{3}, {\sin^{-1} \left(1 - x\right)}^{3}\right), {\pi}^{6} \cdot 0.015625\right)}}{\mathsf{fma}\left({\left({\pi}^{3}\right)}^{3}, 0.001953125, {\sin^{-1} \left(1 - x\right)}^{9}\right)}}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]
    Proof

    [Start]92.86

    \[ \frac{\frac{{\left({\pi}^{3} \cdot 0.125\right)}^{3} \cdot {\left({\pi}^{3} \cdot 0.125\right)}^{3} - {\sin^{-1} \left(1 - x\right)}^{9} \cdot {\sin^{-1} \left(1 - x\right)}^{9}}{\left({\pi}^{6} \cdot 0.015625 + {\sin^{-1} \left(1 - x\right)}^{3} \cdot \mathsf{fma}\left({\pi}^{3}, 0.125, {\sin^{-1} \left(1 - x\right)}^{3}\right)\right) \cdot \left({\left({\pi}^{3} \cdot 0.125\right)}^{3} + {\sin^{-1} \left(1 - x\right)}^{9}\right)}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]

    *-commutative [=>]92.86

    \[ \frac{\frac{{\left({\pi}^{3} \cdot 0.125\right)}^{3} \cdot {\left({\pi}^{3} \cdot 0.125\right)}^{3} - {\sin^{-1} \left(1 - x\right)}^{9} \cdot {\sin^{-1} \left(1 - x\right)}^{9}}{\color{blue}{\left({\left({\pi}^{3} \cdot 0.125\right)}^{3} + {\sin^{-1} \left(1 - x\right)}^{9}\right) \cdot \left({\pi}^{6} \cdot 0.015625 + {\sin^{-1} \left(1 - x\right)}^{3} \cdot \mathsf{fma}\left({\pi}^{3}, 0.125, {\sin^{-1} \left(1 - x\right)}^{3}\right)\right)}}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]

    /-rgt-identity [<=]92.86

    \[ \frac{\frac{{\left({\pi}^{3} \cdot 0.125\right)}^{3} \cdot {\left({\pi}^{3} \cdot 0.125\right)}^{3} - {\sin^{-1} \left(1 - x\right)}^{9} \cdot {\sin^{-1} \left(1 - x\right)}^{9}}{\left({\left({\pi}^{3} \cdot 0.125\right)}^{3} + {\sin^{-1} \left(1 - x\right)}^{9}\right) \cdot \color{blue}{\frac{{\pi}^{6} \cdot 0.015625 + {\sin^{-1} \left(1 - x\right)}^{3} \cdot \mathsf{fma}\left({\pi}^{3}, 0.125, {\sin^{-1} \left(1 - x\right)}^{3}\right)}{1}}}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]

    associate-*r/ [=>]92.86

    \[ \frac{\frac{{\left({\pi}^{3} \cdot 0.125\right)}^{3} \cdot {\left({\pi}^{3} \cdot 0.125\right)}^{3} - {\sin^{-1} \left(1 - x\right)}^{9} \cdot {\sin^{-1} \left(1 - x\right)}^{9}}{\color{blue}{\frac{\left({\left({\pi}^{3} \cdot 0.125\right)}^{3} + {\sin^{-1} \left(1 - x\right)}^{9}\right) \cdot \left({\pi}^{6} \cdot 0.015625 + {\sin^{-1} \left(1 - x\right)}^{3} \cdot \mathsf{fma}\left({\pi}^{3}, 0.125, {\sin^{-1} \left(1 - x\right)}^{3}\right)\right)}{1}}}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]

    associate-*l/ [<=]92.86

    \[ \frac{\frac{{\left({\pi}^{3} \cdot 0.125\right)}^{3} \cdot {\left({\pi}^{3} \cdot 0.125\right)}^{3} - {\sin^{-1} \left(1 - x\right)}^{9} \cdot {\sin^{-1} \left(1 - x\right)}^{9}}{\color{blue}{\frac{{\left({\pi}^{3} \cdot 0.125\right)}^{3} + {\sin^{-1} \left(1 - x\right)}^{9}}{1} \cdot \left({\pi}^{6} \cdot 0.015625 + {\sin^{-1} \left(1 - x\right)}^{3} \cdot \mathsf{fma}\left({\pi}^{3}, 0.125, {\sin^{-1} \left(1 - x\right)}^{3}\right)\right)}}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]

  7. Applied egg-rr89.34

    \[\leadsto \frac{\frac{\frac{\color{blue}{\mathsf{fma}\left(-{\sin^{-1} \left(1 - x\right)}^{9}, {\sin^{-1} \left(1 - x\right)}^{9}, {\sin^{-1} \left(1 - x\right)}^{18}\right) + \left({\pi}^{18} \cdot 3.814697265625 \cdot 10^{-6} - {\sin^{-1} \left(1 - x\right)}^{18}\right)}}{\mathsf{fma}\left({\sin^{-1} \left(1 - x\right)}^{3}, \mathsf{fma}\left(0.125, {\pi}^{3}, {\sin^{-1} \left(1 - x\right)}^{3}\right), {\pi}^{6} \cdot 0.015625\right)}}{\mathsf{fma}\left({\left({\pi}^{3}\right)}^{3}, 0.001953125, {\sin^{-1} \left(1 - x\right)}^{9}\right)}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]

  8. Final simplification89.34

    \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(-{\sin^{-1} \left(1 - x\right)}^{9}, {\sin^{-1} \left(1 - x\right)}^{9}, {\sin^{-1} \left(1 - x\right)}^{18}\right) + \left({\pi}^{18} \cdot 3.814697265625 \cdot 10^{-6} - {\sin^{-1} \left(1 - x\right)}^{18}\right)}{\mathsf{fma}\left({\sin^{-1} \left(1 - x\right)}^{3}, \mathsf{fma}\left(0.125, {\pi}^{3}, {\sin^{-1} \left(1 - x\right)}^{3}\right), {\pi}^{6} \cdot 0.015625\right)}}{\mathsf{fma}\left({\left({\pi}^{3}\right)}^{3}, 0.001953125, {\sin^{-1} \left(1 - x\right)}^{9}\right)}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]

Alternatives

Alternative 1
Error89.37%
Cost175616
\[\begin{array}{l} t_0 := \sin^{-1} \left(1 - x\right)\\ t_1 := {t_0}^{3}\\ \frac{\frac{\frac{3.814697265625 \cdot 10^{-6} \cdot {\left({\pi}^{6}\right)}^{3} - {t_0}^{18}}{\mathsf{fma}\left(t_1, \mathsf{fma}\left(0.125, {\pi}^{3}, t_1\right), {\pi}^{6} \cdot 0.015625\right)}}{\mathsf{fma}\left({\left({\pi}^{3}\right)}^{3}, 0.001953125, {t_0}^{9}\right)}}{\left(\pi \cdot \pi\right) \cdot 0.25 + t_0 \cdot \mathsf{fma}\left(\pi, 0.5, t_0\right)} \end{array} \]
Alternative 2
Error89.4%
Cost78144
\[\begin{array}{l} t_0 := \sin^{-1} \left(1 - x\right)\\ \frac{0.125 \cdot {\pi}^{3} - \mathsf{expm1}\left(\mathsf{log1p}\left({t_0}^{3}\right)\right)}{\left(\pi \cdot \pi\right) \cdot 0.25 + t_0 \cdot \mathsf{fma}\left(\pi, 0.5, t_0\right)} \end{array} \]
Alternative 3
Error89.42%
Cost71808
\[\begin{array}{l} t_0 := \sin^{-1} \left(1 - x\right)\\ \frac{0.125 \cdot e^{3 \cdot \log \pi} - {t_0}^{3}}{\left(\pi \cdot \pi\right) \cdot 0.25 + t_0 \cdot \mathsf{fma}\left(\pi, 0.5, t_0\right)} \end{array} \]
Alternative 4
Error92.86%
Cost52228
\[\begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right)\\ \mathbf{if}\;1 - x \leq 1:\\ \;\;\;\;\mathsf{fma}\left({\left(\sqrt{e^{\mathsf{log1p}\left(t_0\right) \cdot 0.6666666666666666}}\right)}^{2}, \sqrt[3]{1 + t_0}, -1\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left|t_0 + -1\right|\\ \end{array} \]
Alternative 5
Error92.86%
Cost39364
\[\begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right)\\ \mathbf{if}\;1 - x \leq 1:\\ \;\;\;\;\mathsf{fma}\left(e^{\mathsf{log1p}\left(t_0\right) \cdot 0.6666666666666666}, \sqrt[3]{1 + t_0}, -1\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left|t_0 + -1\right|\\ \end{array} \]
Alternative 6
Error90.34%
Cost39236
\[\begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right)\\ t_1 := t_0 + -1\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;1 + \left|t_1\right|\\ \mathbf{else}:\\ \;\;\;\;1 + {\left({\left(\sqrt[3]{\sqrt[3]{t_1}}\right)}^{3}\right)}^{3}\\ \end{array} \]
Alternative 7
Error90.33%
Cost26372
\[\begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right)\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;1 + \left|t_0 + -1\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\frac{1}{\sqrt[3]{t_0}}\right)}^{3}}\\ \end{array} \]
Alternative 8
Error90.32%
Cost26116
\[\begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right)\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;1 + \left|t_0 + -1\right|\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{t_0}\right)}^{3}\\ \end{array} \]
Alternative 9
Error92.86%
Cost19908
\[\begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right) + -1\\ \mathbf{if}\;1 - x \leq 1:\\ \;\;\;\;1 + \mathsf{log1p}\left(\mathsf{expm1}\left(t_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left|t_0\right|\\ \end{array} \]
Alternative 10
Error92.86%
Cost13508
\[\begin{array}{l} \mathbf{if}\;1 - x \leq 1:\\ \;\;\;\;\pi \cdot 0.5 - \sin^{-1} \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left|\cos^{-1} \left(1 - x\right) + -1\right|\\ \end{array} \]
Alternative 11
Error92.86%
Cost13184
\[\pi \cdot 0.5 - \sin^{-1} \left(1 - x\right) \]
Alternative 12
Error92.86%
Cost6848
\[\left(1 + \cos^{-1} \left(1 - x\right)\right) + -1 \]
Alternative 13
Error92.86%
Cost6592
\[\cos^{-1} \left(1 - x\right) \]

Error

Reproduce?

herbie shell --seed 2023121 
(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)))