Ian Simplification

Average Error: 59.6 → 58.6

Time: 19.3s

Precision: binary64

Cost: 123904

Math

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

↓

\[\begin{array}{l} t_0 := 0.25 \cdot {\pi}^{2}\\ t_1 := \cos^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)\\ \frac{\mathsf{expm1}\left(\mathsf{log1p}\left(16 \cdot {\cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)}^{4}\right)\right) + t_0 \cdot \left({\pi}^{2} \cdot -0.25\right)}{\mathsf{fma}\left(2, t_1, 0.5 \cdot \pi\right) \cdot \left(t_0 + 4 \cdot {t_1}^{2}\right)} \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* 0.25 (pow PI 2.0))) (t_1 (acos (sqrt (+ 0.5 (* x -0.5))))))
   (/
    (+
     (expm1 (log1p (* 16.0 (pow (acos (sqrt (fma x -0.5 0.5))) 4.0))))
     (* t_0 (* (pow PI 2.0) -0.25)))
    (* (fma 2.0 t_1 (* 0.5 PI)) (+ t_0 (* 4.0 (pow t_1 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 = 0.25 * pow(((double) M_PI), 2.0);
	double t_1 = acos(sqrt((0.5 + (x * -0.5))));
	return (expm1(log1p((16.0 * pow(acos(sqrt(fma(x, -0.5, 0.5))), 4.0)))) + (t_0 * (pow(((double) M_PI), 2.0) * -0.25))) / (fma(2.0, t_1, (0.5 * ((double) M_PI))) * (t_0 + (4.0 * pow(t_1, 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 = Float64(0.25 * (pi ^ 2.0))
	t_1 = acos(sqrt(Float64(0.5 + Float64(x * -0.5))))
	return Float64(Float64(expm1(log1p(Float64(16.0 * (acos(sqrt(fma(x, -0.5, 0.5))) ^ 4.0)))) + Float64(t_0 * Float64((pi ^ 2.0) * -0.25))) / Float64(fma(2.0, t_1, Float64(0.5 * pi)) * Float64(t_0 + Float64(4.0 * (t_1 ^ 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[(0.25 * N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[ArcCos[N[Sqrt[N[(0.5 + N[(x * -0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(Exp[N[Log[1 + N[(16.0 * N[Power[N[ArcCos[N[Sqrt[N[(x * -0.5 + 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision] + N[(t$95$0 * N[(N[Power[Pi, 2.0], $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(2.0 * t$95$1 + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 + N[(4.0 * N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	59.6
Target	0
Herbie	58.6

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

Derivation

1. Initial program 59.6

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

2. Applied egg-rr 58.6

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

3. Taylor expanded in x around 0 58.6

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

4. Simplified 58.6

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

   [Start] 58.6	\[ 0.5 \cdot \pi - 2 \cdot \left(0.5 \cdot \pi - \cos^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\right) \]
   *-commutative [<=] 58.6	\[ \color{blue}{\pi \cdot 0.5} - 2 \cdot \left(0.5 \cdot \pi - \cos^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\right) \]
   cancel-sign-sub-inv [=>] 58.6	\[ \pi \cdot 0.5 - 2 \cdot \left(0.5 \cdot \pi - \cos^{-1} \left(\sqrt{\color{blue}{0.5 + \left(-0.5\right) \cdot x}}\right)\right) \]
   metadata-eval [=>] 58.6	\[ \pi \cdot 0.5 - 2 \cdot \left(0.5 \cdot \pi - \cos^{-1} \left(\sqrt{0.5 + \color{blue}{-0.5} \cdot x}\right)\right) \]
   cancel-sign-sub-inv [=>] 58.6	\[ \color{blue}{\pi \cdot 0.5 + \left(-2\right) \cdot \left(0.5 \cdot \pi - \cos^{-1} \left(\sqrt{0.5 + -0.5 \cdot x}\right)\right)} \]
   metadata-eval [=>] 58.6	\[ \pi \cdot 0.5 + \color{blue}{-2} \cdot \left(0.5 \cdot \pi - \cos^{-1} \left(\sqrt{0.5 + -0.5 \cdot x}\right)\right) \]
   *-commutative [<=] 58.6	\[ \pi \cdot 0.5 + -2 \cdot \left(\color{blue}{\pi \cdot 0.5} - \cos^{-1} \left(\sqrt{0.5 + -0.5 \cdot x}\right)\right) \]
   metadata-eval [<=] 58.6	\[ \pi \cdot 0.5 + -2 \cdot \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{0.5 + \color{blue}{\left(-0.5\right)} \cdot x}\right)\right) \]
   cancel-sign-sub-inv [<=] 58.6	\[ \pi \cdot 0.5 + -2 \cdot \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\color{blue}{0.5 - 0.5 \cdot x}}\right)\right) \]
   cancel-sign-sub-inv [=>] 58.6	\[ \pi \cdot 0.5 + -2 \cdot \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\color{blue}{0.5 + \left(-0.5\right) \cdot x}}\right)\right) \]
   metadata-eval [=>] 58.6	\[ \pi \cdot 0.5 + -2 \cdot \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{0.5 + \color{blue}{-0.5} \cdot x}\right)\right) \]
   *-commutative [<=] 58.6	\[ \pi \cdot 0.5 + -2 \cdot \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{0.5 + \color{blue}{x \cdot -0.5}}\right)\right) \]

5. Applied egg-rr 58.6

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

6. Applied egg-rr 58.6

   \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(16 \cdot {\cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)}^{4}\right)\right)} - \left(0.25 \cdot {\pi}^{2}\right) \cdot \left(0.25 \cdot {\pi}^{2}\right)}{\mathsf{fma}\left(2, \cos^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right), -\pi \cdot -0.5\right) \cdot \left(0.25 \cdot {\pi}^{2} + 4 \cdot {\cos^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)}^{2}\right)} \]

7. Final simplification 58.6

   \[\leadsto \frac{\mathsf{expm1}\left(\mathsf{log1p}\left(16 \cdot {\cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)}^{4}\right)\right) + \left(0.25 \cdot {\pi}^{2}\right) \cdot \left({\pi}^{2} \cdot -0.25\right)}{\mathsf{fma}\left(2, \cos^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right), 0.5 \cdot \pi\right) \cdot \left(0.25 \cdot {\pi}^{2} + 4 \cdot {\cos^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)}^{2}\right)} \]

Alternatives

Alternative 1
Error	58.6
Cost	117568

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

Alternative 2
Error	58.6
Cost	26368

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

Alternative 3
Error	58.6
Cost	19840

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

Alternative 4
Error	60.6
Cost	19584

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

Reproduce

herbie shell --seed 2023039 
(FPCore (x)
  :name "Ian Simplification"
  :precision binary64

  :herbie-target
  (asin x)

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