Ian Simplification

Average Error: 59.7 → 58.7

Time: 20.7s

Precision: binary64

Cost: 377088

Math

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

↓

\[\begin{array}{l} t_0 := {\pi}^{2} \cdot -0.25\\ t_1 := 4 \cdot {\cos^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)}^{2}\\ t_2 := \cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)\\ t_3 := {t_2}^{2}\\ t_4 := 0.25 \cdot {\pi}^{2}\\ t_5 := \mathsf{fma}\left(2, t_2, 0.5 \cdot \pi\right)\\ t_6 := t_3 \cdot t_5\\ \frac{t_1 \cdot t_1 + t_4 \cdot t_0}{\frac{\mathsf{fma}\left(64, {t_6}^{3}, {\left(t_4 \cdot t_5\right)}^{3}\right)}{\mathsf{fma}\left(0.0625 \cdot {\pi}^{4}, {t_5}^{2}, \left(4 \cdot t_6\right) \cdot \left(t_5 \cdot \left(4 \cdot t_3 + t_0\right)\right)\right)}} \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (pow PI 2.0) -0.25))
        (t_1 (* 4.0 (pow (acos (sqrt (- 0.5 (* 0.5 x)))) 2.0)))
        (t_2 (acos (sqrt (fma -0.5 x 0.5))))
        (t_3 (pow t_2 2.0))
        (t_4 (* 0.25 (pow PI 2.0)))
        (t_5 (fma 2.0 t_2 (* 0.5 PI)))
        (t_6 (* t_3 t_5)))
   (/
    (+ (* t_1 t_1) (* t_4 t_0))
    (/
     (fma 64.0 (pow t_6 3.0) (pow (* t_4 t_5) 3.0))
     (fma
      (* 0.0625 (pow PI 4.0))
      (pow t_5 2.0)
      (* (* 4.0 t_6) (* t_5 (+ (* 4.0 t_3) t_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 = pow(((double) M_PI), 2.0) * -0.25;
	double t_1 = 4.0 * pow(acos(sqrt((0.5 - (0.5 * x)))), 2.0);
	double t_2 = acos(sqrt(fma(-0.5, x, 0.5)));
	double t_3 = pow(t_2, 2.0);
	double t_4 = 0.25 * pow(((double) M_PI), 2.0);
	double t_5 = fma(2.0, t_2, (0.5 * ((double) M_PI)));
	double t_6 = t_3 * t_5;
	return ((t_1 * t_1) + (t_4 * t_0)) / (fma(64.0, pow(t_6, 3.0), pow((t_4 * t_5), 3.0)) / fma((0.0625 * pow(((double) M_PI), 4.0)), pow(t_5, 2.0), ((4.0 * t_6) * (t_5 * ((4.0 * t_3) + t_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((pi ^ 2.0) * -0.25)
	t_1 = Float64(4.0 * (acos(sqrt(Float64(0.5 - Float64(0.5 * x)))) ^ 2.0))
	t_2 = acos(sqrt(fma(-0.5, x, 0.5)))
	t_3 = t_2 ^ 2.0
	t_4 = Float64(0.25 * (pi ^ 2.0))
	t_5 = fma(2.0, t_2, Float64(0.5 * pi))
	t_6 = Float64(t_3 * t_5)
	return Float64(Float64(Float64(t_1 * t_1) + Float64(t_4 * t_0)) / Float64(fma(64.0, (t_6 ^ 3.0), (Float64(t_4 * t_5) ^ 3.0)) / fma(Float64(0.0625 * (pi ^ 4.0)), (t_5 ^ 2.0), Float64(Float64(4.0 * t_6) * Float64(t_5 * Float64(Float64(4.0 * t_3) + t_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[(N[Power[Pi, 2.0], $MachinePrecision] * -0.25), $MachinePrecision]}, Block[{t$95$1 = N[(4.0 * N[Power[N[ArcCos[N[Sqrt[N[(0.5 - N[(0.5 * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[ArcCos[N[Sqrt[N[(-0.5 * x + 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Power[t$95$2, 2.0], $MachinePrecision]}, Block[{t$95$4 = N[(0.25 * N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(2.0 * t$95$2 + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(t$95$3 * t$95$5), $MachinePrecision]}, N[(N[(N[(t$95$1 * t$95$1), $MachinePrecision] + N[(t$95$4 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(N[(64.0 * N[Power[t$95$6, 3.0], $MachinePrecision] + N[Power[N[(t$95$4 * t$95$5), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[(0.0625 * N[Power[Pi, 4.0], $MachinePrecision]), $MachinePrecision] * N[Power[t$95$5, 2.0], $MachinePrecision] + N[(N[(4.0 * t$95$6), $MachinePrecision] * N[(t$95$5 * N[(N[(4.0 * t$95$3), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Target

Original	59.7
Target	0
Herbie	58.7

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

Derivation

1. Initial program 59.7

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

2. Applied egg-rr 58.7

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

3. Taylor expanded in x around 0 58.7

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

4. Simplified 58.7

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

   [Start] 58.7	\[ 0.5 \cdot \pi - 2 \cdot \left(0.5 \cdot \pi - \cos^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\right) \]
   *-commutative [<=] 58.7	\[ \color{blue}{\pi \cdot 0.5} - 2 \cdot \left(0.5 \cdot \pi - \cos^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\right) \]
   cancel-sign-sub-inv [=>] 58.7	\[ \pi \cdot 0.5 - 2 \cdot \left(0.5 \cdot \pi - \cos^{-1} \left(\sqrt{\color{blue}{0.5 + \left(-0.5\right) \cdot x}}\right)\right) \]
   metadata-eval [=>] 58.7	\[ \pi \cdot 0.5 - 2 \cdot \left(0.5 \cdot \pi - \cos^{-1} \left(\sqrt{0.5 + \color{blue}{-0.5} \cdot x}\right)\right) \]
   cancel-sign-sub-inv [=>] 58.7	\[ \color{blue}{\pi \cdot 0.5 + \left(-2\right) \cdot \left(0.5 \cdot \pi - \cos^{-1} \left(\sqrt{0.5 + -0.5 \cdot x}\right)\right)} \]
   metadata-eval [=>] 58.7	\[ \pi \cdot 0.5 + \color{blue}{-2} \cdot \left(0.5 \cdot \pi - \cos^{-1} \left(\sqrt{0.5 + -0.5 \cdot x}\right)\right) \]
   *-commutative [<=] 58.7	\[ \pi \cdot 0.5 + -2 \cdot \left(\color{blue}{\pi \cdot 0.5} - \cos^{-1} \left(\sqrt{0.5 + -0.5 \cdot x}\right)\right) \]
   metadata-eval [<=] 58.7	\[ \pi \cdot 0.5 + -2 \cdot \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{0.5 + \color{blue}{\left(-0.5\right)} \cdot x}\right)\right) \]
   cancel-sign-sub-inv [<=] 58.7	\[ \pi \cdot 0.5 + -2 \cdot \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\color{blue}{0.5 - 0.5 \cdot x}}\right)\right) \]
   cancel-sign-sub-inv [=>] 58.7	\[ \pi \cdot 0.5 + -2 \cdot \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\color{blue}{0.5 + \left(-0.5\right) \cdot x}}\right)\right) \]
   metadata-eval [=>] 58.7	\[ \pi \cdot 0.5 + -2 \cdot \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{0.5 + \color{blue}{-0.5} \cdot x}\right)\right) \]
   *-commutative [<=] 58.7	\[ \pi \cdot 0.5 + -2 \cdot \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{0.5 + \color{blue}{x \cdot -0.5}}\right)\right) \]

5. Applied egg-rr 58.7

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

6. Applied egg-rr 58.7

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

7. Simplified 58.7

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

   [Start] 58.7

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

8. Final simplification 58.7

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

Alternatives

Alternative 1
Error	58.7
Cost	117568

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

Alternative 2
Error	58.8
Cost	52672

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

Alternative 3
Error	58.7
Cost	38912

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

Alternative 4
Error	58.7
Cost	19840

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

Alternative 5
Error	60.6
Cost	19584

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

Reproduce

herbie shell --seed 2023060 
(FPCore (x)
  :name "Ian Simplification"
  :precision binary64

  :herbie-target
  (asin x)

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