Ian Simplification

Average Error: 59.7 → 58.8

Time: 24.0s

Precision: binary64

Cost: 208000

Math

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

↓

\[\begin{array}{l} t_0 := \cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)\\ t_1 := {t_0}^{4}\\ \frac{\frac{{t_1}^{3} \cdot -4096 + 0.000244140625 \cdot {\left({\pi}^{4}\right)}^{3}}{\left(0.00390625 \cdot {\pi}^{8} + 256 \cdot {t_0}^{8}\right) + \left(t_1 \cdot -16\right) \cdot \left({\pi}^{4} \cdot -0.0625\right)}}{\mathsf{fma}\left(-0.5, \pi, t_0 \cdot -2\right) \cdot \mathsf{fma}\left(0.25, {\pi}^{2}, 4 \cdot {t_0}^{2}\right)} \end{array} \]

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)))) (t_1 (pow t_0 4.0)))
   (/
    (/
     (+ (* (pow t_1 3.0) -4096.0) (* 0.000244140625 (pow (pow PI 4.0) 3.0)))
     (+
      (+ (* 0.00390625 (pow PI 8.0)) (* 256.0 (pow t_0 8.0)))
      (* (* t_1 -16.0) (* (pow PI 4.0) -0.0625))))
    (*
     (fma -0.5 PI (* t_0 -2.0))
     (fma 0.25 (pow PI 2.0) (* 4.0 (pow 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)));
	double t_1 = pow(t_0, 4.0);
	return (((pow(t_1, 3.0) * -4096.0) + (0.000244140625 * pow(pow(((double) M_PI), 4.0), 3.0))) / (((0.00390625 * pow(((double) M_PI), 8.0)) + (256.0 * pow(t_0, 8.0))) + ((t_1 * -16.0) * (pow(((double) M_PI), 4.0) * -0.0625)))) / (fma(-0.5, ((double) M_PI), (t_0 * -2.0)) * fma(0.25, pow(((double) M_PI), 2.0), (4.0 * pow(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)))
	t_1 = t_0 ^ 4.0
	return Float64(Float64(Float64(Float64((t_1 ^ 3.0) * -4096.0) + Float64(0.000244140625 * ((pi ^ 4.0) ^ 3.0))) / Float64(Float64(Float64(0.00390625 * (pi ^ 8.0)) + Float64(256.0 * (t_0 ^ 8.0))) + Float64(Float64(t_1 * -16.0) * Float64((pi ^ 4.0) * -0.0625)))) / Float64(fma(-0.5, pi, Float64(t_0 * -2.0)) * fma(0.25, (pi ^ 2.0), Float64(4.0 * (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]}, Block[{t$95$1 = N[Power[t$95$0, 4.0], $MachinePrecision]}, N[(N[(N[(N[(N[Power[t$95$1, 3.0], $MachinePrecision] * -4096.0), $MachinePrecision] + N[(0.000244140625 * N[Power[N[Power[Pi, 4.0], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(0.00390625 * N[Power[Pi, 8.0], $MachinePrecision]), $MachinePrecision] + N[(256.0 * N[Power[t$95$0, 8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$1 * -16.0), $MachinePrecision] * N[(N[Power[Pi, 4.0], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(-0.5 * Pi + N[(t$95$0 * -2.0), $MachinePrecision]), $MachinePrecision] * N[(0.25 * N[Power[Pi, 2.0], $MachinePrecision] + N[(4.0 * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	59.7
Target	0.0
Herbie	58.8

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

Derivation

1. Initial program 59.7

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

2. Applied egg-rr 58.8

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

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

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

6. Simplified 58.8

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.0625, {\pi}^{4}, {\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{4} \cdot -16\right)}{\mathsf{fma}\left(-0.5, \pi, -2 \cdot \cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)\right) \cdot \mathsf{fma}\left(0.25, {\pi}^{2}, {\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{2} \cdot 4\right)}} \]
   Proof

   [Start] 58.8

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

7. Applied egg-rr 58.8

   \[\leadsto \frac{\color{blue}{\frac{{\left({\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{4} \cdot -16\right)}^{3} + {\left(0.0625 \cdot {\pi}^{4}\right)}^{3}}{\left(\left(0.0625 \cdot {\pi}^{4}\right) \cdot \left(0.0625 \cdot {\pi}^{4}\right) + 256 \cdot {\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{8}\right) - \left(0.0625 \cdot {\pi}^{4}\right) \cdot \left({\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{4} \cdot -16\right)}}}{\mathsf{fma}\left(-0.5, \pi, -2 \cdot \cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)\right) \cdot \mathsf{fma}\left(0.25, {\pi}^{2}, {\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{2} \cdot 4\right)} \]

8. Simplified 58.8

   \[\leadsto \frac{\color{blue}{\frac{{\left({\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{4}\right)}^{3} \cdot -4096 + 0.000244140625 \cdot {\left({\pi}^{4}\right)}^{3}}{\left(0.00390625 \cdot {\pi}^{8} + 256 \cdot {\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{8}\right) - \left(0.0625 \cdot {\pi}^{4}\right) \cdot \left(-16 \cdot {\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{4}\right)}}}{\mathsf{fma}\left(-0.5, \pi, -2 \cdot \cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)\right) \cdot \mathsf{fma}\left(0.25, {\pi}^{2}, {\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{2} \cdot 4\right)} \]
   Proof

   [Start] 58.8

   \[ \frac{\frac{{\left({\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{4} \cdot -16\right)}^{3} + {\left(0.0625 \cdot {\pi}^{4}\right)}^{3}}{\left(\left(0.0625 \cdot {\pi}^{4}\right) \cdot \left(0.0625 \cdot {\pi}^{4}\right) + 256 \cdot {\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{8}\right) - \left(0.0625 \cdot {\pi}^{4}\right) \cdot \left({\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{4} \cdot -16\right)}}{\mathsf{fma}\left(-0.5, \pi, -2 \cdot \cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)\right) \cdot \mathsf{fma}\left(0.25, {\pi}^{2}, {\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{2} \cdot 4\right)} \]

   *-lft-identity [<=] 58.8

   \[ \frac{\frac{\color{blue}{1 \cdot \left({\left({\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{4} \cdot -16\right)}^{3} + {\left(0.0625 \cdot {\pi}^{4}\right)}^{3}\right)}}{\left(\left(0.0625 \cdot {\pi}^{4}\right) \cdot \left(0.0625 \cdot {\pi}^{4}\right) + 256 \cdot {\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{8}\right) - \left(0.0625 \cdot {\pi}^{4}\right) \cdot \left({\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{4} \cdot -16\right)}}{\mathsf{fma}\left(-0.5, \pi, -2 \cdot \cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)\right) \cdot \mathsf{fma}\left(0.25, {\pi}^{2}, {\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{2} \cdot 4\right)} \]

9. Final simplification 58.8

   \[\leadsto \frac{\frac{{\left({\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{4}\right)}^{3} \cdot -4096 + 0.000244140625 \cdot {\left({\pi}^{4}\right)}^{3}}{\left(0.00390625 \cdot {\pi}^{8} + 256 \cdot {\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{8}\right) + \left({\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{4} \cdot -16\right) \cdot \left({\pi}^{4} \cdot -0.0625\right)}}{\mathsf{fma}\left(-0.5, \pi, \cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right) \cdot -2\right) \cdot \mathsf{fma}\left(0.25, {\pi}^{2}, 4 \cdot {\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{2}\right)} \]

Alternatives

Alternative 1
Error	58.8
Cost	135936

\[\begin{array}{l} t_0 := \cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)\\ \frac{\log \left(e^{\mathsf{fma}\left(0.0625, {\pi}^{4}, {t_0}^{4} \cdot -16\right)}\right)}{\mathsf{fma}\left(-0.5, \pi, t_0 \cdot -2\right) \cdot \mathsf{fma}\left(0.25, {\pi}^{2}, 4 \cdot {t_0}^{2}\right)} \end{array} \]

Alternative 2
Error	58.8
Cost	110592

\[\begin{array}{l} t_0 := \cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)\\ \frac{\mathsf{fma}\left(0.0625, {\pi}^{4}, {t_0}^{4} \cdot -16\right)}{\left(0.25 \cdot {\pi}^{2} + 4 \cdot {t_0}^{2}\right) \cdot \left(t_0 \cdot -2 + -0.5 \cdot \pi\right)} \end{array} \]

Alternative 3
Error	58.8
Cost	71616

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

Alternative 4
Error	58.8
Cost	71488

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

Alternative 5
Error	58.8
Cost	45248

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

Alternative 6
Error	58.8
Cost	19840

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

Alternative 7
Error	60.6
Cost	19584

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

Reproduce

herbie shell --seed 2023066 
(FPCore (x)
  :name "Ian Simplification"
  :precision binary64

  :herbie-target
  (asin x)

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