Ian Simplification

Average Error: 59.8 → 58.8

Time: 20.6s

Precision: binary64

Cost: 161728

Math

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

↓

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

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)))))
   (/
    (*
     (pow (cbrt (fma PI -0.5 (* t_0 2.0))) 2.0)
     (cbrt
      (fma
       (cbrt (* (pow PI 4.0) 0.0625))
       (cbrt (* 0.25 (pow PI 2.0)))
       (* (pow t_0 2.0) -4.0))))
    (cbrt (fma PI -0.5 (* t_0 -2.0))))))

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)));
	return (pow(cbrt(fma(((double) M_PI), -0.5, (t_0 * 2.0))), 2.0) * cbrt(fma(cbrt((pow(((double) M_PI), 4.0) * 0.0625)), cbrt((0.25 * pow(((double) M_PI), 2.0))), (pow(t_0, 2.0) * -4.0)))) / cbrt(fma(((double) M_PI), -0.5, (t_0 * -2.0)));
}

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)))
	return Float64(Float64((cbrt(fma(pi, -0.5, Float64(t_0 * 2.0))) ^ 2.0) * cbrt(fma(cbrt(Float64((pi ^ 4.0) * 0.0625)), cbrt(Float64(0.25 * (pi ^ 2.0))), Float64((t_0 ^ 2.0) * -4.0)))) / cbrt(fma(pi, -0.5, Float64(t_0 * -2.0))))
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]}, N[(N[(N[Power[N[Power[N[(Pi * -0.5 + N[(t$95$0 * 2.0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision] * N[Power[N[(N[Power[N[(N[Power[Pi, 4.0], $MachinePrecision] * 0.0625), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(0.25 * N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[(N[Power[t$95$0, 2.0], $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] / N[Power[N[(Pi * -0.5 + N[(t$95$0 * -2.0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]

Target

Original	59.8
Target	0
Herbie	58.8

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

Derivation

1. Initial program 59.8

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

2. Applied egg-rr 58.8

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

3. Taylor expanded in x around 0 58.8

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

4. Simplified 58.8

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

   [Start] 58.8	\[ 0.5 \cdot \pi - 2 \cdot \left(0.5 \cdot \pi - \cos^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\right) \]
   *-commutative [<=] 58.8	\[ \color{blue}{\pi \cdot 0.5} - 2 \cdot \left(0.5 \cdot \pi - \cos^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\right) \]
   cancel-sign-sub-inv [=>] 58.8	\[ \pi \cdot 0.5 - 2 \cdot \left(0.5 \cdot \pi - \cos^{-1} \left(\sqrt{\color{blue}{0.5 + \left(-0.5\right) \cdot x}}\right)\right) \]
   metadata-eval [=>] 58.8	\[ \pi \cdot 0.5 - 2 \cdot \left(0.5 \cdot \pi - \cos^{-1} \left(\sqrt{0.5 + \color{blue}{-0.5} \cdot x}\right)\right) \]
   cancel-sign-sub-inv [=>] 58.8	\[ \color{blue}{\pi \cdot 0.5 + \left(-2\right) \cdot \left(0.5 \cdot \pi - \cos^{-1} \left(\sqrt{0.5 + -0.5 \cdot x}\right)\right)} \]
   metadata-eval [=>] 58.8	\[ \pi \cdot 0.5 + \color{blue}{-2} \cdot \left(0.5 \cdot \pi - \cos^{-1} \left(\sqrt{0.5 + -0.5 \cdot x}\right)\right) \]
   *-commutative [<=] 58.8	\[ \pi \cdot 0.5 + -2 \cdot \left(\color{blue}{\pi \cdot 0.5} - \cos^{-1} \left(\sqrt{0.5 + -0.5 \cdot x}\right)\right) \]
   metadata-eval [<=] 58.8	\[ \pi \cdot 0.5 + -2 \cdot \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{0.5 + \color{blue}{\left(-0.5\right)} \cdot x}\right)\right) \]
   cancel-sign-sub-inv [<=] 58.8	\[ \pi \cdot 0.5 + -2 \cdot \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\color{blue}{0.5 - 0.5 \cdot x}}\right)\right) \]
   cancel-sign-sub-inv [=>] 58.8	\[ \pi \cdot 0.5 + -2 \cdot \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\color{blue}{0.5 + \left(-0.5\right) \cdot x}}\right)\right) \]
   metadata-eval [=>] 58.8	\[ \pi \cdot 0.5 + -2 \cdot \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{0.5 + \color{blue}{-0.5} \cdot x}\right)\right) \]
   *-commutative [<=] 58.8	\[ \pi \cdot 0.5 + -2 \cdot \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{0.5 + \color{blue}{x \cdot -0.5}}\right)\right) \]

5. Applied egg-rr 58.8

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

6. Applied egg-rr 58.8

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

7. Final simplification 58.8

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

Alternatives

Alternative 1
Error	58.8
Cost	123328

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

Alternative 2
Error	58.8
Cost	19840

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

Alternative 3
Error	60.6
Cost	19584

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

Reproduce

herbie shell --seed 2023045 
(FPCore (x)
  :name "Ian Simplification"
  :precision binary64

  :herbie-target
  (asin x)

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