?

Average Accuracy: 6.8% → 8.2%
Time: 21.6s
Precision: binary64

?

\[\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
↓
\[\mathsf{fma}\left({\left({\sin^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)}^{\left(0.3333333333333333 \cdot 1\right)}\right)}^{3}, -2, 0.5 \cdot \pi\right) \]
(FPCore (x)
 :precision binary64
 (- (/ PI 2.0) (* 2.0 (asin (sqrt (/ (- 1.0 x) 2.0))))))
↓
(FPCore (x)
 :precision binary64
 (fma
  (pow (pow (asin (sqrt (fma x -0.5 0.5))) (* 0.3333333333333333 1.0)) 3.0)
  -2.0
  (* 0.5 PI)))
double code(double x) {
	return (((double) M_PI) / 2.0) - (2.0 * asin(sqrt(((1.0 - x) / 2.0))));
}
↓
double code(double x) {
	return fma(pow(pow(asin(sqrt(fma(x, -0.5, 0.5))), (0.3333333333333333 * 1.0)), 3.0), -2.0, (0.5 * ((double) M_PI)));
}

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)
	return fma(((asin(sqrt(fma(x, -0.5, 0.5))) ^ Float64(0.3333333333333333 * 1.0)) ^ 3.0), -2.0, Float64(0.5 * pi))
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_] := N[(N[Power[N[Power[N[ArcSin[N[Sqrt[N[(x * -0.5 + 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[(0.3333333333333333 * 1.0), $MachinePrecision]], $MachinePrecision], 3.0], $MachinePrecision] * -2.0 + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]

\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)
↓
\mathsf{fma}\left({\left({\sin^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)}^{\left(0.3333333333333333 \cdot 1\right)}\right)}^{3}, -2, 0.5 \cdot \pi\right)

Error?

Target

Original6.8%
Target100.0%
Herbie8.2%
\[\sin^{-1} x \]

Derivation?

  1. Initial program 6.8%

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

  2. Simplified6.8%

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

  3. Applied egg-rr8.2%

    \[\leadsto \color{blue}{\mathsf{fma}\left({\left({\sin^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)}^{\left(0.3333333333333333 \cdot 1\right)}\right)}^{3}, -2, 0.5 \cdot \pi\right)} \]

Reproduce?

herbie shell --seed 2023151 
(FPCore (x)
  :name "Ian Simplification"
  :precision binary64

  :herbie-target
  (asin x)

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