?

Average Error: 93.18% → 91.72%
Time: 17.5s
Precision: binary64
Cost: 65600

?

\[\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
↓
\[\begin{array}{l} t_0 := \sqrt{0.5 + x \cdot -0.5}\\ {\left(\sqrt[3]{0.5 \cdot \pi + -2 \cdot \left(0.5 \cdot \pi - \cos^{-1} t_0\right)}\right)}^{2} \cdot \sqrt[3]{0.5 \cdot \pi + -2 \cdot \sin^{-1} t_0} \end{array} \]
(FPCore (x)
 :precision binary64
 (- (/ PI 2.0) (* 2.0 (asin (sqrt (/ (- 1.0 x) 2.0))))))
↓
(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ 0.5 (* x -0.5)))))
   (*
    (pow (cbrt (+ (* 0.5 PI) (* -2.0 (- (* 0.5 PI) (acos t_0))))) 2.0)
    (cbrt (+ (* 0.5 PI) (* -2.0 (asin t_0)))))))
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 = sqrt((0.5 + (x * -0.5)));
	return pow(cbrt(((0.5 * ((double) M_PI)) + (-2.0 * ((0.5 * ((double) M_PI)) - acos(t_0))))), 2.0) * cbrt(((0.5 * ((double) M_PI)) + (-2.0 * asin(t_0))));
}

public static double code(double x) {
	return (Math.PI / 2.0) - (2.0 * Math.asin(Math.sqrt(((1.0 - x) / 2.0))));
}
↓
public static double code(double x) {
	double t_0 = Math.sqrt((0.5 + (x * -0.5)));
	return Math.pow(Math.cbrt(((0.5 * Math.PI) + (-2.0 * ((0.5 * Math.PI) - Math.acos(t_0))))), 2.0) * Math.cbrt(((0.5 * Math.PI) + (-2.0 * Math.asin(t_0))));
}

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 = sqrt(Float64(0.5 + Float64(x * -0.5)))
	return Float64((cbrt(Float64(Float64(0.5 * pi) + Float64(-2.0 * Float64(Float64(0.5 * pi) - acos(t_0))))) ^ 2.0) * cbrt(Float64(Float64(0.5 * pi) + Float64(-2.0 * asin(t_0)))))
end

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[Sqrt[N[(0.5 + N[(x * -0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(N[Power[N[Power[N[(N[(0.5 * Pi), $MachinePrecision] + N[(-2.0 * N[(N[(0.5 * Pi), $MachinePrecision] - N[ArcCos[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision] * N[Power[N[(N[(0.5 * Pi), $MachinePrecision] + N[(-2.0 * N[ArcSin[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]

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

Error?

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original93.18%
Target0%
Herbie91.72%
\[\sin^{-1} x \]

Derivation?

  1. Initial program 93.18

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

  2. Applied egg-rr93.18

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

  3. Applied egg-rr91.72

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

  4. Applied egg-rr91.72

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

  5. Taylor expanded in x around inf 91.72

    \[\leadsto {\left(\sqrt[3]{\color{blue}{-2 \cdot \left(0.5 \cdot \pi - \cos^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\right) + 0.5 \cdot \pi}}\right)}^{2} \cdot \sqrt[3]{\sin^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right) \cdot -2 + \pi \cdot 0.5} \]

  6. Final simplification91.72

    \[\leadsto {\left(\sqrt[3]{0.5 \cdot \pi + -2 \cdot \left(0.5 \cdot \pi - \cos^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)\right)}\right)}^{2} \cdot \sqrt[3]{0.5 \cdot \pi + -2 \cdot \sin^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)} \]

Alternatives

Alternative 1
Error91.73%
Cost19840
\[\pi \cdot -0.5 + 2 \cdot \cos^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right) \]
Alternative 2
Error94.62%
Cost19584
\[\pi \cdot -0.5 + 2 \cdot \cos^{-1} \left(\sqrt{0.5}\right) \]

Error

Reproduce?

herbie shell --seed 2023090 
(FPCore (x)
  :name "Ian Simplification"
  :precision binary64

  :herbie-target
  (asin x)

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