Ian Simplification

Average Accuracy: 6.9% → 8.4%

Time: 25.1s

Precision: binary64

Cost: 181824

Math

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

↓

\[\begin{array}{l} t_0 := \cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)\\ t_1 := {t_0}^{2}\\ \frac{0.015625 \cdot {\pi}^{6} + \left(1 - e^{\mathsf{log1p}\left(64 \cdot {t_0}^{6}\right)}\right)}{\mathsf{fma}\left(\pi, -0.5, t_0 \cdot -2\right) \cdot \left(0.0625 \cdot \left({\pi}^{2} \cdot {\pi}^{2}\right) + \left(t_1 \cdot 4\right) \cdot \mathsf{fma}\left(t_1, 4, {\pi}^{2} \cdot 0.25\right)\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- (/ PI 2.0) (* 2.0 (asin (sqrt (/ (- 1.0 x) 2.0))))))

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (acos (sqrt (fma -0.5 x 0.5)))) (t_1 (pow t_0 2.0)))
   (/
    (+ (* 0.015625 (pow PI 6.0)) (- 1.0 (exp (log1p (* 64.0 (pow t_0 6.0))))))
    (*
     (fma PI -0.5 (* t_0 -2.0))
     (+
      (* 0.0625 (* (pow PI 2.0) (pow PI 2.0)))
      (* (* t_1 4.0) (fma t_1 4.0 (* (pow PI 2.0) 0.25))))))))

C

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 = acos(sqrt(fma(-0.5, x, 0.5)));
	double t_1 = pow(t_0, 2.0);
	return ((0.015625 * pow(((double) M_PI), 6.0)) + (1.0 - exp(log1p((64.0 * pow(t_0, 6.0)))))) / (fma(((double) M_PI), -0.5, (t_0 * -2.0)) * ((0.0625 * (pow(((double) M_PI), 2.0) * pow(((double) M_PI), 2.0))) + ((t_1 * 4.0) * fma(t_1, 4.0, (pow(((double) M_PI), 2.0) * 0.25)))));
}

Julia

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 = acos(sqrt(fma(-0.5, x, 0.5)))
	t_1 = t_0 ^ 2.0
	return Float64(Float64(Float64(0.015625 * (pi ^ 6.0)) + Float64(1.0 - exp(log1p(Float64(64.0 * (t_0 ^ 6.0)))))) / Float64(fma(pi, -0.5, Float64(t_0 * -2.0)) * Float64(Float64(0.0625 * Float64((pi ^ 2.0) * (pi ^ 2.0))) + Float64(Float64(t_1 * 4.0) * fma(t_1, 4.0, Float64((pi ^ 2.0) * 0.25))))))
end

Wolfram

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[ArcCos[N[Sqrt[N[(-0.5 * x + 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Power[t$95$0, 2.0], $MachinePrecision]}, N[(N[(N[(0.015625 * N[Power[Pi, 6.0], $MachinePrecision]), $MachinePrecision] + N[(1.0 - N[Exp[N[Log[1 + N[(64.0 * N[Power[t$95$0, 6.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(Pi * -0.5 + N[(t$95$0 * -2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(0.0625 * N[(N[Power[Pi, 2.0], $MachinePrecision] * N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$1 * 4.0), $MachinePrecision] * N[(t$95$1 * 4.0 + N[(N[Power[Pi, 2.0], $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	6.9%
Target	100.0%
Herbie	8.4%

\[\sin^{-1} x \]

Derivation

1. Initial program 6.9%

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

2. Applied egg-rr 8.4%

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

   [Start] 6.9	\[ \frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
   asin-acos [=>] 8.4	\[ \frac{\pi}{2} - 2 \cdot \color{blue}{\left(\frac{\pi}{2} - \cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)} \]
   div-inv [=>] 8.4	\[ \frac{\pi}{2} - 2 \cdot \left(\color{blue}{\pi \cdot \frac{1}{2}} - \cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right) \]
   metadata-eval [=>] 8.4	\[ \frac{\pi}{2} - 2 \cdot \left(\pi \cdot \color{blue}{0.5} - \cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right) \]
   div-sub [=>] 8.4	\[ \frac{\pi}{2} - 2 \cdot \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\color{blue}{\frac{1}{2} - \frac{x}{2}}}\right)\right) \]
   metadata-eval [=>] 8.4	\[ \frac{\pi}{2} - 2 \cdot \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\color{blue}{0.5} - \frac{x}{2}}\right)\right) \]
   div-inv [=>] 8.4	\[ \frac{\pi}{2} - 2 \cdot \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{0.5 - \color{blue}{x \cdot \frac{1}{2}}}\right)\right) \]
   metadata-eval [=>] 8.4	\[ \frac{\pi}{2} - 2 \cdot \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{0.5 - x \cdot \color{blue}{0.5}}\right)\right) \]

3. Taylor expanded in x around 0 8.4%

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

4. Simplified 8.4%

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

   [Start] 8.4	\[ 0.5 \cdot \pi - 2 \cdot \left(0.5 \cdot \pi - \cos^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\right) \]
   *-commutative [<=] 8.4	\[ \color{blue}{\pi \cdot 0.5} - 2 \cdot \left(0.5 \cdot \pi - \cos^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\right) \]
   cancel-sign-sub-inv [=>] 8.4	\[ \pi \cdot 0.5 - 2 \cdot \left(0.5 \cdot \pi - \cos^{-1} \left(\sqrt{\color{blue}{0.5 + \left(-0.5\right) \cdot x}}\right)\right) \]
   metadata-eval [=>] 8.4	\[ \pi \cdot 0.5 - 2 \cdot \left(0.5 \cdot \pi - \cos^{-1} \left(\sqrt{0.5 + \color{blue}{-0.5} \cdot x}\right)\right) \]
   cancel-sign-sub-inv [=>] 8.4	\[ \color{blue}{\pi \cdot 0.5 + \left(-2\right) \cdot \left(0.5 \cdot \pi - \cos^{-1} \left(\sqrt{0.5 + -0.5 \cdot x}\right)\right)} \]
   metadata-eval [=>] 8.4	\[ \pi \cdot 0.5 + \color{blue}{-2} \cdot \left(0.5 \cdot \pi - \cos^{-1} \left(\sqrt{0.5 + -0.5 \cdot x}\right)\right) \]
   *-commutative [<=] 8.4	\[ \pi \cdot 0.5 + -2 \cdot \left(\color{blue}{\pi \cdot 0.5} - \cos^{-1} \left(\sqrt{0.5 + -0.5 \cdot x}\right)\right) \]
   metadata-eval [<=] 8.4	\[ \pi \cdot 0.5 + -2 \cdot \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{0.5 + \color{blue}{\left(-0.5\right)} \cdot x}\right)\right) \]
   cancel-sign-sub-inv [<=] 8.4	\[ \pi \cdot 0.5 + -2 \cdot \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\color{blue}{0.5 - 0.5 \cdot x}}\right)\right) \]
   cancel-sign-sub-inv [=>] 8.4	\[ \pi \cdot 0.5 + -2 \cdot \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\color{blue}{0.5 + \left(-0.5\right) \cdot x}}\right)\right) \]
   metadata-eval [=>] 8.4	\[ \pi \cdot 0.5 + -2 \cdot \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{0.5 + \color{blue}{-0.5} \cdot x}\right)\right) \]
   *-commutative [<=] 8.4	\[ \pi \cdot 0.5 + -2 \cdot \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{0.5 + \color{blue}{x \cdot -0.5}}\right)\right) \]

5. Applied egg-rr 8.4%

   \[\leadsto \color{blue}{\frac{{\left(0.25 \cdot {\pi}^{2}\right)}^{3} - {\left({\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{2} \cdot 4\right)}^{3}}{\mathsf{fma}\left(\pi, -0.5, -2 \cdot \cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)\right) \cdot \left(\left(0.25 \cdot {\pi}^{2}\right) \cdot \left(0.25 \cdot {\pi}^{2}\right) + \left(\left({\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{2} \cdot 4\right) \cdot \left({\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{2} \cdot 4\right) + \left(0.25 \cdot {\pi}^{2}\right) \cdot \left({\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{2} \cdot 4\right)\right)\right)}} \]
   Proof

   [Start] 8.4	\[ \pi \cdot -0.5 - -2 \cdot \cos^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right) \]
   flip-- [=>] 8.4	\[ \color{blue}{\frac{\left(\pi \cdot -0.5\right) \cdot \left(\pi \cdot -0.5\right) - \left(-2 \cdot \cos^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)\right) \cdot \left(-2 \cdot \cos^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)\right)}{\pi \cdot -0.5 + -2 \cdot \cos^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)}} \]
   flip3-- [=>] 8.4	\[ \frac{\color{blue}{\frac{{\left(\left(\pi \cdot -0.5\right) \cdot \left(\pi \cdot -0.5\right)\right)}^{3} - {\left(\left(-2 \cdot \cos^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)\right) \cdot \left(-2 \cdot \cos^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)\right)\right)}^{3}}{\left(\left(\pi \cdot -0.5\right) \cdot \left(\pi \cdot -0.5\right)\right) \cdot \left(\left(\pi \cdot -0.5\right) \cdot \left(\pi \cdot -0.5\right)\right) + \left(\left(\left(-2 \cdot \cos^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)\right) \cdot \left(-2 \cdot \cos^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)\right)\right) \cdot \left(\left(-2 \cdot \cos^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)\right) \cdot \left(-2 \cdot \cos^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)\right)\right) + \left(\left(\pi \cdot -0.5\right) \cdot \left(\pi \cdot -0.5\right)\right) \cdot \left(\left(-2 \cdot \cos^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)\right) \cdot \left(-2 \cdot \cos^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)\right)\right)\right)}}}{\pi \cdot -0.5 + -2 \cdot \cos^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)} \]

6. Simplified 8.4%

   \[\leadsto \color{blue}{\frac{0.015625 \cdot {\left({\pi}^{2}\right)}^{3} - {\left({\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{2}\right)}^{3} \cdot 64}{\mathsf{fma}\left(\pi, -0.5, -2 \cdot \cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)\right) \cdot \left(0.0625 \cdot \left({\pi}^{2} \cdot {\pi}^{2}\right) + \left({\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{2} \cdot 4\right) \cdot \mathsf{fma}\left({\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{2}, 4, 0.25 \cdot {\pi}^{2}\right)\right)}} \]
   Proof

   [Start] 8.4	\[ \frac{{\left(0.25 \cdot {\pi}^{2}\right)}^{3} - {\left({\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{2} \cdot 4\right)}^{3}}{\mathsf{fma}\left(\pi, -0.5, -2 \cdot \cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)\right) \cdot \left(\left(0.25 \cdot {\pi}^{2}\right) \cdot \left(0.25 \cdot {\pi}^{2}\right) + \left(\left({\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{2} \cdot 4\right) \cdot \left({\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{2} \cdot 4\right) + \left(0.25 \cdot {\pi}^{2}\right) \cdot \left({\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{2} \cdot 4\right)\right)\right)} \]
   cube-prod [=>] 8.4	\[ \frac{\color{blue}{{0.25}^{3} \cdot {\left({\pi}^{2}\right)}^{3}} - {\left({\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{2} \cdot 4\right)}^{3}}{\mathsf{fma}\left(\pi, -0.5, -2 \cdot \cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)\right) \cdot \left(\left(0.25 \cdot {\pi}^{2}\right) \cdot \left(0.25 \cdot {\pi}^{2}\right) + \left(\left({\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{2} \cdot 4\right) \cdot \left({\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{2} \cdot 4\right) + \left(0.25 \cdot {\pi}^{2}\right) \cdot \left({\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{2} \cdot 4\right)\right)\right)} \]
   metadata-eval [=>] 8.4	\[ \frac{\color{blue}{0.015625} \cdot {\left({\pi}^{2}\right)}^{3} - {\left({\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{2} \cdot 4\right)}^{3}}{\mathsf{fma}\left(\pi, -0.5, -2 \cdot \cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)\right) \cdot \left(\left(0.25 \cdot {\pi}^{2}\right) \cdot \left(0.25 \cdot {\pi}^{2}\right) + \left(\left({\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{2} \cdot 4\right) \cdot \left({\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{2} \cdot 4\right) + \left(0.25 \cdot {\pi}^{2}\right) \cdot \left({\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{2} \cdot 4\right)\right)\right)} \]
   cube-prod [=>] 8.4	\[ \frac{0.015625 \cdot {\left({\pi}^{2}\right)}^{3} - \color{blue}{{\left({\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{2}\right)}^{3} \cdot {4}^{3}}}{\mathsf{fma}\left(\pi, -0.5, -2 \cdot \cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)\right) \cdot \left(\left(0.25 \cdot {\pi}^{2}\right) \cdot \left(0.25 \cdot {\pi}^{2}\right) + \left(\left({\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{2} \cdot 4\right) \cdot \left({\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{2} \cdot 4\right) + \left(0.25 \cdot {\pi}^{2}\right) \cdot \left({\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{2} \cdot 4\right)\right)\right)} \]
   metadata-eval [=>] 8.4	\[ \frac{0.015625 \cdot {\left({\pi}^{2}\right)}^{3} - {\left({\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{2}\right)}^{3} \cdot \color{blue}{64}}{\mathsf{fma}\left(\pi, -0.5, -2 \cdot \cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)\right) \cdot \left(\left(0.25 \cdot {\pi}^{2}\right) \cdot \left(0.25 \cdot {\pi}^{2}\right) + \left(\left({\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{2} \cdot 4\right) \cdot \left({\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{2} \cdot 4\right) + \left(0.25 \cdot {\pi}^{2}\right) \cdot \left({\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{2} \cdot 4\right)\right)\right)} \]

7. Applied egg-rr 8.4%

   \[\leadsto \frac{\color{blue}{\left(0.015625 \cdot {\pi}^{6} - e^{\mathsf{log1p}\left({\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{6} \cdot 64\right)}\right) + 1}}{\mathsf{fma}\left(\pi, -0.5, -2 \cdot \cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)\right) \cdot \left(0.0625 \cdot \left({\pi}^{2} \cdot {\pi}^{2}\right) + \left({\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{2} \cdot 4\right) \cdot \mathsf{fma}\left({\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{2}, 4, 0.25 \cdot {\pi}^{2}\right)\right)} \]
   Proof

   [Start] 8.4	\[ \frac{0.015625 \cdot {\left({\pi}^{2}\right)}^{3} - {\left({\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{2}\right)}^{3} \cdot 64}{\mathsf{fma}\left(\pi, -0.5, -2 \cdot \cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)\right) \cdot \left(0.0625 \cdot \left({\pi}^{2} \cdot {\pi}^{2}\right) + \left({\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{2} \cdot 4\right) \cdot \mathsf{fma}\left({\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{2}, 4, 0.25 \cdot {\pi}^{2}\right)\right)} \]
   expm1-log1p-u [=>] 8.4	\[ \frac{0.015625 \cdot {\left({\pi}^{2}\right)}^{3} - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left({\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{2}\right)}^{3} \cdot 64\right)\right)}}{\mathsf{fma}\left(\pi, -0.5, -2 \cdot \cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)\right) \cdot \left(0.0625 \cdot \left({\pi}^{2} \cdot {\pi}^{2}\right) + \left({\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{2} \cdot 4\right) \cdot \mathsf{fma}\left({\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{2}, 4, 0.25 \cdot {\pi}^{2}\right)\right)} \]
   expm1-udef [=>] 8.4	\[ \frac{0.015625 \cdot {\left({\pi}^{2}\right)}^{3} - \color{blue}{\left(e^{\mathsf{log1p}\left({\left({\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{2}\right)}^{3} \cdot 64\right)} - 1\right)}}{\mathsf{fma}\left(\pi, -0.5, -2 \cdot \cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)\right) \cdot \left(0.0625 \cdot \left({\pi}^{2} \cdot {\pi}^{2}\right) + \left({\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{2} \cdot 4\right) \cdot \mathsf{fma}\left({\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{2}, 4, 0.25 \cdot {\pi}^{2}\right)\right)} \]
   associate--r- [=>] 8.4	\[ \frac{\color{blue}{\left(0.015625 \cdot {\left({\pi}^{2}\right)}^{3} - e^{\mathsf{log1p}\left({\left({\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{2}\right)}^{3} \cdot 64\right)}\right) + 1}}{\mathsf{fma}\left(\pi, -0.5, -2 \cdot \cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)\right) \cdot \left(0.0625 \cdot \left({\pi}^{2} \cdot {\pi}^{2}\right) + \left({\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{2} \cdot 4\right) \cdot \mathsf{fma}\left({\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{2}, 4, 0.25 \cdot {\pi}^{2}\right)\right)} \]
   pow-pow [=>] 8.4	\[ \frac{\left(0.015625 \cdot \color{blue}{{\pi}^{\left(2 \cdot 3\right)}} - e^{\mathsf{log1p}\left({\left({\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{2}\right)}^{3} \cdot 64\right)}\right) + 1}{\mathsf{fma}\left(\pi, -0.5, -2 \cdot \cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)\right) \cdot \left(0.0625 \cdot \left({\pi}^{2} \cdot {\pi}^{2}\right) + \left({\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{2} \cdot 4\right) \cdot \mathsf{fma}\left({\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{2}, 4, 0.25 \cdot {\pi}^{2}\right)\right)} \]
   metadata-eval [=>] 8.4	\[ \frac{\left(0.015625 \cdot {\pi}^{\color{blue}{6}} - e^{\mathsf{log1p}\left({\left({\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{2}\right)}^{3} \cdot 64\right)}\right) + 1}{\mathsf{fma}\left(\pi, -0.5, -2 \cdot \cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)\right) \cdot \left(0.0625 \cdot \left({\pi}^{2} \cdot {\pi}^{2}\right) + \left({\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{2} \cdot 4\right) \cdot \mathsf{fma}\left({\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{2}, 4, 0.25 \cdot {\pi}^{2}\right)\right)} \]
   pow-pow [=>] 8.4	\[ \frac{\left(0.015625 \cdot {\pi}^{6} - e^{\mathsf{log1p}\left(\color{blue}{{\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{\left(2 \cdot 3\right)}} \cdot 64\right)}\right) + 1}{\mathsf{fma}\left(\pi, -0.5, -2 \cdot \cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)\right) \cdot \left(0.0625 \cdot \left({\pi}^{2} \cdot {\pi}^{2}\right) + \left({\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{2} \cdot 4\right) \cdot \mathsf{fma}\left({\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{2}, 4, 0.25 \cdot {\pi}^{2}\right)\right)} \]
   metadata-eval [=>] 8.4	\[ \frac{\left(0.015625 \cdot {\pi}^{6} - e^{\mathsf{log1p}\left({\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{\color{blue}{6}} \cdot 64\right)}\right) + 1}{\mathsf{fma}\left(\pi, -0.5, -2 \cdot \cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)\right) \cdot \left(0.0625 \cdot \left({\pi}^{2} \cdot {\pi}^{2}\right) + \left({\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{2} \cdot 4\right) \cdot \mathsf{fma}\left({\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{2}, 4, 0.25 \cdot {\pi}^{2}\right)\right)} \]

8. Simplified 8.4%

   \[\leadsto \frac{\color{blue}{0.015625 \cdot {\pi}^{6} - \left(e^{\mathsf{log1p}\left(64 \cdot {\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{6}\right)} - 1\right)}}{\mathsf{fma}\left(\pi, -0.5, -2 \cdot \cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)\right) \cdot \left(0.0625 \cdot \left({\pi}^{2} \cdot {\pi}^{2}\right) + \left({\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{2} \cdot 4\right) \cdot \mathsf{fma}\left({\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{2}, 4, 0.25 \cdot {\pi}^{2}\right)\right)} \]
   Proof

   [Start] 8.4	\[ \frac{\left(0.015625 \cdot {\pi}^{6} - e^{\mathsf{log1p}\left({\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{6} \cdot 64\right)}\right) + 1}{\mathsf{fma}\left(\pi, -0.5, -2 \cdot \cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)\right) \cdot \left(0.0625 \cdot \left({\pi}^{2} \cdot {\pi}^{2}\right) + \left({\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{2} \cdot 4\right) \cdot \mathsf{fma}\left({\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{2}, 4, 0.25 \cdot {\pi}^{2}\right)\right)} \]
   associate-+l- [=>] 8.4	\[ \frac{\color{blue}{0.015625 \cdot {\pi}^{6} - \left(e^{\mathsf{log1p}\left({\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{6} \cdot 64\right)} - 1\right)}}{\mathsf{fma}\left(\pi, -0.5, -2 \cdot \cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)\right) \cdot \left(0.0625 \cdot \left({\pi}^{2} \cdot {\pi}^{2}\right) + \left({\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{2} \cdot 4\right) \cdot \mathsf{fma}\left({\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{2}, 4, 0.25 \cdot {\pi}^{2}\right)\right)} \]
   *-commutative [=>] 8.4	\[ \frac{0.015625 \cdot {\pi}^{6} - \left(e^{\mathsf{log1p}\left(\color{blue}{64 \cdot {\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{6}}\right)} - 1\right)}{\mathsf{fma}\left(\pi, -0.5, -2 \cdot \cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)\right) \cdot \left(0.0625 \cdot \left({\pi}^{2} \cdot {\pi}^{2}\right) + \left({\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{2} \cdot 4\right) \cdot \mathsf{fma}\left({\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{2}, 4, 0.25 \cdot {\pi}^{2}\right)\right)} \]

9. Final simplification 8.4%

   \[\leadsto \frac{0.015625 \cdot {\pi}^{6} + \left(1 - e^{\mathsf{log1p}\left(64 \cdot {\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{6}\right)}\right)}{\mathsf{fma}\left(\pi, -0.5, \cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right) \cdot -2\right) \cdot \left(0.0625 \cdot \left({\pi}^{2} \cdot {\pi}^{2}\right) + \left({\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{2} \cdot 4\right) \cdot \mathsf{fma}\left({\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{2}, 4, {\pi}^{2} \cdot 0.25\right)\right)} \]

Alternatives

Alternative 1
Accuracy	8.4%
Cost	181696

\[\begin{array}{l} t_0 := \cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)\\ t_1 := {t_0}^{2}\\ \frac{0.015625 \cdot {\left({\pi}^{2}\right)}^{3} - 64 \cdot {t_1}^{3}}{\mathsf{fma}\left(\pi, -0.5, t_0 \cdot -2\right) \cdot \left(\left(t_1 \cdot 4\right) \cdot \mathsf{fma}\left(t_1, 4, {\pi}^{2} \cdot 0.25\right) + 0.0625 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left({\pi}^{4}\right)\right)\right)} \end{array} \]

Alternative 2
Accuracy	8.4%
Cost	168896

\[\begin{array}{l} t_0 := \cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)\\ t_1 := {t_0}^{2}\\ \frac{0.015625 \cdot {\left({\pi}^{2}\right)}^{3} - 64 \cdot {t_1}^{3}}{\mathsf{fma}\left(\pi, -0.5, t_0 \cdot -2\right) \cdot \left(\left(t_1 \cdot 4\right) \cdot \mathsf{fma}\left(t_1, 4, {\pi}^{2} \cdot 0.25\right) + 0.0625 \cdot {\pi}^{4}\right)} \end{array} \]

Alternative 3
Accuracy	8.4%
Cost	155584

\[\begin{array}{l} t_0 := \cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)\\ t_1 := \sqrt[3]{\mathsf{fma}\left(\pi, -0.5, t_0 \cdot -2\right)}\\ \frac{\frac{{\pi}^{2} \cdot 0.25 + {t_0}^{2} \cdot -4}{t_1 \cdot t_1}}{t_1} \end{array} \]

Alternative 4
Accuracy	8.4%
Cost	71488

\[\begin{array}{l} t_0 := \cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)\\ \frac{{\pi}^{2} \cdot 0.25 + {t_0}^{2} \cdot -4}{\mathsf{fma}\left(\pi, -0.5, t_0 \cdot -2\right)} \end{array} \]

Alternative 5
Accuracy	8.4%
Cost	45184

\[\log \left(e^{\mathsf{fma}\left(\pi, -0.5, \cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right) \cdot 2\right)}\right) \]

Alternative 6
Accuracy	8.4%
Cost	19840

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

Alternative 7
Accuracy	5.4%
Cost	19584

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

Reproduce

herbie shell --seed 2023137 
(FPCore (x)
  :name "Ian Simplification"
  :precision binary64

  :herbie-target
  (asin x)

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