bug323 (missed optimization)

Average Error: 59.4 → 57.2

Time: 9.5s

Precision: binary64

Cost: 156352

\[0 \leq x \land x \leq 0.5\]

Math

\[\cos^{-1} \left(1 - x\right) \]

↓

\[\begin{array}{l} t_0 := \sin^{-1} \left(1 - x\right)\\ t_1 := {t_0}^{6}\\ \frac{\frac{\mathsf{fma}\left({\pi}^{6}, {\pi}^{6} \cdot 0.000244140625, t_1 \cdot \left(-t_1\right)\right)}{\mathsf{fma}\left({\pi}^{3}, 0.125, {t_0}^{3}\right) \cdot \left(t_1 + {\pi}^{6} \cdot 0.015625\right)}}{\left(\pi \cdot \pi\right) \cdot 0.25 + t_0 \cdot \mathsf{fma}\left(\pi, 0.5, t_0\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (acos (- 1.0 x)))

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (asin (- 1.0 x))) (t_1 (pow t_0 6.0)))
   (/
    (/
     (fma (pow PI 6.0) (* (pow PI 6.0) 0.000244140625) (* t_1 (- t_1)))
     (*
      (fma (pow PI 3.0) 0.125 (pow t_0 3.0))
      (+ t_1 (* (pow PI 6.0) 0.015625))))
    (+ (* (* PI PI) 0.25) (* t_0 (fma PI 0.5 t_0))))))

C

double code(double x) {
	return acos((1.0 - x));
}

↓

double code(double x) {
	double t_0 = asin((1.0 - x));
	double t_1 = pow(t_0, 6.0);
	return (fma(pow(((double) M_PI), 6.0), (pow(((double) M_PI), 6.0) * 0.000244140625), (t_1 * -t_1)) / (fma(pow(((double) M_PI), 3.0), 0.125, pow(t_0, 3.0)) * (t_1 + (pow(((double) M_PI), 6.0) * 0.015625)))) / (((((double) M_PI) * ((double) M_PI)) * 0.25) + (t_0 * fma(((double) M_PI), 0.5, t_0)));
}

Julia

function code(x)
	return acos(Float64(1.0 - x))
end

↓

function code(x)
	t_0 = asin(Float64(1.0 - x))
	t_1 = t_0 ^ 6.0
	return Float64(Float64(fma((pi ^ 6.0), Float64((pi ^ 6.0) * 0.000244140625), Float64(t_1 * Float64(-t_1))) / Float64(fma((pi ^ 3.0), 0.125, (t_0 ^ 3.0)) * Float64(t_1 + Float64((pi ^ 6.0) * 0.015625)))) / Float64(Float64(Float64(pi * pi) * 0.25) + Float64(t_0 * fma(pi, 0.5, t_0))))
end

Wolfram

code[x_] := N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]

↓

code[x_] := Block[{t$95$0 = N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Power[t$95$0, 6.0], $MachinePrecision]}, N[(N[(N[(N[Power[Pi, 6.0], $MachinePrecision] * N[(N[Power[Pi, 6.0], $MachinePrecision] * 0.000244140625), $MachinePrecision] + N[(t$95$1 * (-t$95$1)), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Power[Pi, 3.0], $MachinePrecision] * 0.125 + N[Power[t$95$0, 3.0], $MachinePrecision]), $MachinePrecision] * N[(t$95$1 + N[(N[Power[Pi, 6.0], $MachinePrecision] * 0.015625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(Pi * Pi), $MachinePrecision] * 0.25), $MachinePrecision] + N[(t$95$0 * N[(Pi * 0.5 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	59.4
Target	0.0
Herbie	57.2

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

Derivation

1. Initial program 59.4

   \[\cos^{-1} \left(1 - x\right) \]

2. Applied egg-rr 59.4

   \[\leadsto \color{blue}{\frac{{\left(\pi \cdot 0.5\right)}^{3} - {\sin^{-1} \left(1 - x\right)}^{3}}{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) + \left(\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right) + \left(\pi \cdot 0.5\right) \cdot \sin^{-1} \left(1 - x\right)\right)}} \]

3. Simplified 59.4

   \[\leadsto \color{blue}{\frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(1 - x\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)}} \]
   Proof

   [Start] 59.4	\[ \frac{{\left(\pi \cdot 0.5\right)}^{3} - {\sin^{-1} \left(1 - x\right)}^{3}}{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) + \left(\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right) + \left(\pi \cdot 0.5\right) \cdot \sin^{-1} \left(1 - x\right)\right)} \]
   cube-prod [=>] 59.4	\[ \frac{\color{blue}{{\pi}^{3} \cdot {0.5}^{3}} - {\sin^{-1} \left(1 - x\right)}^{3}}{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) + \left(\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right) + \left(\pi \cdot 0.5\right) \cdot \sin^{-1} \left(1 - x\right)\right)} \]
   metadata-eval [=>] 59.4	\[ \frac{{\pi}^{3} \cdot \color{blue}{0.125} - {\sin^{-1} \left(1 - x\right)}^{3}}{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) + \left(\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right) + \left(\pi \cdot 0.5\right) \cdot \sin^{-1} \left(1 - x\right)\right)} \]
   swap-sqr [=>] 59.4	\[ \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(1 - x\right)}^{3}}{\color{blue}{\left(\pi \cdot \pi\right) \cdot \left(0.5 \cdot 0.5\right)} + \left(\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right) + \left(\pi \cdot 0.5\right) \cdot \sin^{-1} \left(1 - x\right)\right)} \]
   metadata-eval [=>] 59.4	\[ \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(1 - x\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot \color{blue}{0.25} + \left(\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right) + \left(\pi \cdot 0.5\right) \cdot \sin^{-1} \left(1 - x\right)\right)} \]
   distribute-rgt-out [=>] 59.4	\[ \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(1 - x\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \color{blue}{\sin^{-1} \left(1 - x\right) \cdot \left(\sin^{-1} \left(1 - x\right) + \pi \cdot 0.5\right)}} \]
   +-commutative [<=] 59.4	\[ \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(1 - x\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \color{blue}{\left(\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)\right)}} \]
   fma-def [=>] 59.4	\[ \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(1 - x\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \color{blue}{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)}} \]

4. Applied egg-rr 59.4

   \[\leadsto \frac{\color{blue}{\frac{\left({\pi}^{6} \cdot 0.015625\right) \cdot \left({\pi}^{6} \cdot 0.015625\right) - {\sin^{-1} \left(1 - x\right)}^{6} \cdot {\sin^{-1} \left(1 - x\right)}^{6}}{\mathsf{fma}\left({\pi}^{3}, 0.125, {\sin^{-1} \left(1 - x\right)}^{3}\right) \cdot \left({\sin^{-1} \left(1 - x\right)}^{6} + {\pi}^{6} \cdot 0.015625\right)}}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]

5. Simplified 57.2

   \[\leadsto \frac{\color{blue}{\frac{\left({\pi}^{6} \cdot {\pi}^{6}\right) \cdot 0.000244140625 - {\sin^{-1} \left(1 - x\right)}^{12}}{\mathsf{fma}\left({\pi}^{3}, 0.125, {\sin^{-1} \left(1 - x\right)}^{3}\right) \cdot \left({\pi}^{6} \cdot 0.015625 + {\sin^{-1} \left(1 - x\right)}^{6}\right)}}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]
   Proof

   [Start] 59.4	\[ \frac{\frac{\left({\pi}^{6} \cdot 0.015625\right) \cdot \left({\pi}^{6} \cdot 0.015625\right) - {\sin^{-1} \left(1 - x\right)}^{6} \cdot {\sin^{-1} \left(1 - x\right)}^{6}}{\mathsf{fma}\left({\pi}^{3}, 0.125, {\sin^{-1} \left(1 - x\right)}^{3}\right) \cdot \left({\sin^{-1} \left(1 - x\right)}^{6} + {\pi}^{6} \cdot 0.015625\right)}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]
   swap-sqr [=>] 59.4	\[ \frac{\frac{\color{blue}{\left({\pi}^{6} \cdot {\pi}^{6}\right) \cdot \left(0.015625 \cdot 0.015625\right)} - {\sin^{-1} \left(1 - x\right)}^{6} \cdot {\sin^{-1} \left(1 - x\right)}^{6}}{\mathsf{fma}\left({\pi}^{3}, 0.125, {\sin^{-1} \left(1 - x\right)}^{3}\right) \cdot \left({\sin^{-1} \left(1 - x\right)}^{6} + {\pi}^{6} \cdot 0.015625\right)}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]
   metadata-eval [=>] 59.4	\[ \frac{\frac{\left({\pi}^{6} \cdot {\pi}^{6}\right) \cdot \color{blue}{0.000244140625} - {\sin^{-1} \left(1 - x\right)}^{6} \cdot {\sin^{-1} \left(1 - x\right)}^{6}}{\mathsf{fma}\left({\pi}^{3}, 0.125, {\sin^{-1} \left(1 - x\right)}^{3}\right) \cdot \left({\sin^{-1} \left(1 - x\right)}^{6} + {\pi}^{6} \cdot 0.015625\right)}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]
   pow-sqr [=>] 57.2	\[ \frac{\frac{\left({\pi}^{6} \cdot {\pi}^{6}\right) \cdot 0.000244140625 - \color{blue}{{\sin^{-1} \left(1 - x\right)}^{\left(2 \cdot 6\right)}}}{\mathsf{fma}\left({\pi}^{3}, 0.125, {\sin^{-1} \left(1 - x\right)}^{3}\right) \cdot \left({\sin^{-1} \left(1 - x\right)}^{6} + {\pi}^{6} \cdot 0.015625\right)}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]
   metadata-eval [=>] 57.2	\[ \frac{\frac{\left({\pi}^{6} \cdot {\pi}^{6}\right) \cdot 0.000244140625 - {\sin^{-1} \left(1 - x\right)}^{\color{blue}{12}}}{\mathsf{fma}\left({\pi}^{3}, 0.125, {\sin^{-1} \left(1 - x\right)}^{3}\right) \cdot \left({\sin^{-1} \left(1 - x\right)}^{6} + {\pi}^{6} \cdot 0.015625\right)}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]
   +-commutative [=>] 57.2	\[ \frac{\frac{\left({\pi}^{6} \cdot {\pi}^{6}\right) \cdot 0.000244140625 - {\sin^{-1} \left(1 - x\right)}^{12}}{\mathsf{fma}\left({\pi}^{3}, 0.125, {\sin^{-1} \left(1 - x\right)}^{3}\right) \cdot \color{blue}{\left({\pi}^{6} \cdot 0.015625 + {\sin^{-1} \left(1 - x\right)}^{6}\right)}}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]

6. Applied egg-rr 57.2

   \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left({\pi}^{6}, {\pi}^{6} \cdot 0.000244140625, -{\sin^{-1} \left(1 - x\right)}^{12}\right)}}{\mathsf{fma}\left({\pi}^{3}, 0.125, {\sin^{-1} \left(1 - x\right)}^{3}\right) \cdot \left({\pi}^{6} \cdot 0.015625 + {\sin^{-1} \left(1 - x\right)}^{6}\right)}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]

7. Applied egg-rr 57.2

   \[\leadsto \frac{\frac{\mathsf{fma}\left({\pi}^{6}, {\pi}^{6} \cdot 0.000244140625, -\color{blue}{{\sin^{-1} \left(1 - x\right)}^{6} \cdot {\sin^{-1} \left(1 - x\right)}^{6}}\right)}{\mathsf{fma}\left({\pi}^{3}, 0.125, {\sin^{-1} \left(1 - x\right)}^{3}\right) \cdot \left({\pi}^{6} \cdot 0.015625 + {\sin^{-1} \left(1 - x\right)}^{6}\right)}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]

8. Final simplification 57.2

   \[\leadsto \frac{\frac{\mathsf{fma}\left({\pi}^{6}, {\pi}^{6} \cdot 0.000244140625, {\sin^{-1} \left(1 - x\right)}^{6} \cdot \left(-{\sin^{-1} \left(1 - x\right)}^{6}\right)\right)}{\mathsf{fma}\left({\pi}^{3}, 0.125, {\sin^{-1} \left(1 - x\right)}^{3}\right) \cdot \left({\sin^{-1} \left(1 - x\right)}^{6} + {\pi}^{6} \cdot 0.015625\right)}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]

Alternatives

Alternative 1
Error	57.2
Cost	149632

\[\begin{array}{l} t_0 := \sin^{-1} \left(1 - x\right)\\ \frac{\frac{\mathsf{fma}\left({\pi}^{6}, 0.000244140625 \cdot \sqrt{{\pi}^{12}}, -{t_0}^{12}\right)}{\mathsf{fma}\left({\pi}^{3}, 0.125, {t_0}^{3}\right) \cdot \left({t_0}^{6} + {\pi}^{6} \cdot 0.015625\right)}}{\left(\pi \cdot \pi\right) \cdot 0.25 + t_0 \cdot \mathsf{fma}\left(\pi, 0.5, t_0\right)} \end{array} \]

Alternative 2
Error	57.2
Cost	143232

\[\begin{array}{l} t_0 := \sin^{-1} \left(1 - x\right)\\ \frac{\frac{\mathsf{fma}\left({\pi}^{6}, {\pi}^{6} \cdot 0.000244140625, -{t_0}^{12}\right)}{\mathsf{fma}\left({\pi}^{3}, 0.125, {t_0}^{3}\right) \cdot \left({t_0}^{6} + {\pi}^{6} \cdot 0.015625\right)}}{\left(\pi \cdot \pi\right) \cdot 0.25 + t_0 \cdot \mathsf{fma}\left(\pi, 0.5, t_0\right)} \end{array} \]

Alternative 3
Error	57.2
Cost	137024

\[\begin{array}{l} t_0 := \sin^{-1} \left(1 - x\right)\\ \frac{\frac{0.000244140625 \cdot \left({\pi}^{6} \cdot {\pi}^{6}\right) - {t_0}^{12}}{\frac{{t_0}^{6} + {\pi}^{6} \cdot 0.015625}{\frac{1}{\mathsf{fma}\left({\pi}^{3}, 0.125, {t_0}^{3}\right)}}}}{\left(\pi \cdot \pi\right) \cdot 0.25 + t_0 \cdot \mathsf{fma}\left(\pi, 0.5, t_0\right)} \end{array} \]

Alternative 4
Error	57.2
Cost	136896

\[\begin{array}{l} t_0 := \sin^{-1} \left(1 - x\right)\\ \frac{\frac{0.000244140625 \cdot \left({\pi}^{6} \cdot {\pi}^{6}\right) - {t_0}^{12}}{\mathsf{fma}\left({\pi}^{3}, 0.125, {t_0}^{3}\right) \cdot \left({t_0}^{6} + {\pi}^{6} \cdot 0.015625\right)}}{\left(\pi \cdot \pi\right) \cdot 0.25 + t_0 \cdot \mathsf{fma}\left(\pi, 0.5, t_0\right)} \end{array} \]

Alternative 5
Error	57.2
Cost	78272

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

Alternative 6
Error	57.2
Cost	78144

\[\begin{array}{l} t_0 := \sin^{-1} \left(1 - x\right)\\ \frac{{\pi}^{3} \cdot 0.125 - \mathsf{expm1}\left(\mathsf{log1p}\left({t_0}^{3}\right)\right)}{\left(\pi \cdot \pi\right) \cdot 0.25 + t_0 \cdot \mathsf{fma}\left(\pi, 0.5, t_0\right)} \end{array} \]

Alternative 7
Error	57.2
Cost	71808

\[\begin{array}{l} t_0 := \sin^{-1} \left(1 - x\right)\\ \frac{0.125 \cdot e^{3 \cdot \log \pi} - {t_0}^{3}}{\left(\pi \cdot \pi\right) \cdot 0.25 + t_0 \cdot \mathsf{fma}\left(\pi, 0.5, t_0\right)} \end{array} \]

Alternative 8
Error	57.2
Cost	71744

\[\begin{array}{l} t_0 := \sin^{-1} \left(1 - x\right)\\ \frac{{\pi}^{3} \cdot 0.125 - {\left(\sqrt[3]{t_0}\right)}^{9}}{\left(\pi \cdot \pi\right) \cdot 0.25 + t_0 \cdot \mathsf{fma}\left(\pi, 0.5, t_0\right)} \end{array} \]

Alternative 9
Error	59.4
Cost	40004

\[\begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right)\\ t_1 := t_0 + -1\\ \mathbf{if}\;1 - x \leq 1:\\ \;\;\;\;{\left({\left({t_0}^{0.08333333333333333}\right)}^{4}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \sqrt{{t_1}^{6}}}{{t_1}^{2} + \left(2 - t_0\right)}\\ \end{array} \]

Alternative 10
Error	59.4
Cost	25984

\[{\left(e^{\log \cos^{-1} \left(1 - x\right) \cdot 0.3333333333333333}\right)}^{3} \]

Alternative 11
Error	59.4
Cost	25984

\[{\left({\left({\cos^{-1} \left(1 - x\right)}^{0.08333333333333333}\right)}^{4}\right)}^{3} \]

Alternative 12
Error	59.4
Cost	19456

\[{\left(\sqrt[3]{\cos^{-1} \left(1 - x\right)}\right)}^{3} \]

Alternative 13
Error	59.4
Cost	6592

\[\cos^{-1} \left(1 - x\right) \]

Reproduce

herbie shell --seed 2023187 
(FPCore (x)
  :name "bug323 (missed optimization)"
  :precision binary64
  :pre (and (<= 0.0 x) (<= x 0.5))

  :herbie-target
  (* 2.0 (asin (sqrt (/ x 2.0))))

  (acos (- 1.0 x)))