bug323 (missed optimization)

Average Error: 93.15% → 89.61%

Time: 12.1s

Precision: binary64

Cost: 188160

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

Math

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

↓

\[\begin{array}{l} t_0 := \sin^{-1} \left(1 - x\right)\\ t_1 := \sqrt[3]{\log \pi}\\ \frac{\frac{\left({\pi}^{6} \cdot {\pi}^{6}\right) \cdot 0.000244140625 - {t_0}^{12}}{\mathsf{fma}\left({\pi}^{3}, 0.125, {t_0}^{3}\right) \cdot \left({\left({\left(e^{6}\right)}^{\left({t_1}^{2}\right)}\right)}^{t_1} \cdot 0.015625 + {t_0}^{6}\right)}}{\left(\pi \cdot \pi\right) \cdot 0.25 + t_0 \cdot \mathsf{fma}\left(\pi, 0.5, t_0\right)} \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (asin (- 1.0 x))) (t_1 (cbrt (log PI))))
   (/
    (/
     (- (* (* (pow PI 6.0) (pow PI 6.0)) 0.000244140625) (pow t_0 12.0))
     (*
      (fma (pow PI 3.0) 0.125 (pow t_0 3.0))
      (+ (* (pow (pow (exp 6.0) (pow t_1 2.0)) t_1) 0.015625) (pow t_0 6.0))))
    (+ (* (* 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 = cbrt(log(((double) M_PI)));
	return ((((pow(((double) M_PI), 6.0) * pow(((double) M_PI), 6.0)) * 0.000244140625) - pow(t_0, 12.0)) / (fma(pow(((double) M_PI), 3.0), 0.125, pow(t_0, 3.0)) * ((pow(pow(exp(6.0), pow(t_1, 2.0)), t_1) * 0.015625) + pow(t_0, 6.0)))) / (((((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 = cbrt(log(pi))
	return Float64(Float64(Float64(Float64(Float64((pi ^ 6.0) * (pi ^ 6.0)) * 0.000244140625) - (t_0 ^ 12.0)) / Float64(fma((pi ^ 3.0), 0.125, (t_0 ^ 3.0)) * Float64(Float64(((exp(6.0) ^ (t_1 ^ 2.0)) ^ t_1) * 0.015625) + (t_0 ^ 6.0)))) / 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[N[Log[Pi], $MachinePrecision], 1/3], $MachinePrecision]}, N[(N[(N[(N[(N[(N[Power[Pi, 6.0], $MachinePrecision] * N[Power[Pi, 6.0], $MachinePrecision]), $MachinePrecision] * 0.000244140625), $MachinePrecision] - N[Power[t$95$0, 12.0], $MachinePrecision]), $MachinePrecision] / N[(N[(N[Power[Pi, 3.0], $MachinePrecision] * 0.125 + N[Power[t$95$0, 3.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[N[Power[N[Exp[6.0], $MachinePrecision], N[Power[t$95$1, 2.0], $MachinePrecision]], $MachinePrecision], t$95$1], $MachinePrecision] * 0.015625), $MachinePrecision] + N[Power[t$95$0, 6.0], $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	93.15%
Target	0.02%
Herbie	89.61%

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

Derivation

1. Initial program 93.15

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

2. Applied egg-rr 93.16

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

3. Simplified 93.16

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

   [Start] 93.16	\[ \frac{{\left(\pi \cdot 0.5\right)}^{3} - {\sin^{-1} \left(1 - x\right)}^{3}}{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) + \left(\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right) + \left(\pi \cdot 0.5\right) \cdot \sin^{-1} \left(1 - x\right)\right)} \]
   cube-prod [=>] 93.16	\[ \frac{\color{blue}{{\pi}^{3} \cdot {0.5}^{3}} - {\sin^{-1} \left(1 - x\right)}^{3}}{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) + \left(\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right) + \left(\pi \cdot 0.5\right) \cdot \sin^{-1} \left(1 - x\right)\right)} \]
   metadata-eval [=>] 93.16	\[ \frac{{\pi}^{3} \cdot \color{blue}{0.125} - {\sin^{-1} \left(1 - x\right)}^{3}}{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) + \left(\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right) + \left(\pi \cdot 0.5\right) \cdot \sin^{-1} \left(1 - x\right)\right)} \]
   swap-sqr [=>] 93.16	\[ \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(1 - x\right)}^{3}}{\color{blue}{\left(\pi \cdot \pi\right) \cdot \left(0.5 \cdot 0.5\right)} + \left(\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right) + \left(\pi \cdot 0.5\right) \cdot \sin^{-1} \left(1 - x\right)\right)} \]
   metadata-eval [=>] 93.16	\[ \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(1 - x\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot \color{blue}{0.25} + \left(\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right) + \left(\pi \cdot 0.5\right) \cdot \sin^{-1} \left(1 - x\right)\right)} \]
   distribute-rgt-out [=>] 93.16	\[ \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(1 - x\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \color{blue}{\sin^{-1} \left(1 - x\right) \cdot \left(\sin^{-1} \left(1 - x\right) + \pi \cdot 0.5\right)}} \]
   +-commutative [<=] 93.16	\[ \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(1 - x\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \color{blue}{\left(\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)\right)}} \]
   fma-def [=>] 93.16	\[ \frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(1 - x\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \color{blue}{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)}} \]

4. Applied egg-rr 93.16

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

5. Simplified 89.58

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

   [Start] 93.16	\[ \frac{\frac{\left({\pi}^{6} \cdot 0.015625\right) \cdot \left({\pi}^{6} \cdot 0.015625\right) - {\sin^{-1} \left(1 - x\right)}^{6} \cdot {\sin^{-1} \left(1 - x\right)}^{6}}{\mathsf{fma}\left({\pi}^{3}, 0.125, {\sin^{-1} \left(1 - x\right)}^{3}\right) \cdot \left({\sin^{-1} \left(1 - x\right)}^{6} + {\pi}^{6} \cdot 0.015625\right)}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]
   swap-sqr [=>] 93.16	\[ \frac{\frac{\color{blue}{\left({\pi}^{6} \cdot {\pi}^{6}\right) \cdot \left(0.015625 \cdot 0.015625\right)} - {\sin^{-1} \left(1 - x\right)}^{6} \cdot {\sin^{-1} \left(1 - x\right)}^{6}}{\mathsf{fma}\left({\pi}^{3}, 0.125, {\sin^{-1} \left(1 - x\right)}^{3}\right) \cdot \left({\sin^{-1} \left(1 - x\right)}^{6} + {\pi}^{6} \cdot 0.015625\right)}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]
   metadata-eval [=>] 93.16	\[ \frac{\frac{\left({\pi}^{6} \cdot {\pi}^{6}\right) \cdot \color{blue}{0.000244140625} - {\sin^{-1} \left(1 - x\right)}^{6} \cdot {\sin^{-1} \left(1 - x\right)}^{6}}{\mathsf{fma}\left({\pi}^{3}, 0.125, {\sin^{-1} \left(1 - x\right)}^{3}\right) \cdot \left({\sin^{-1} \left(1 - x\right)}^{6} + {\pi}^{6} \cdot 0.015625\right)}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]
   pow-sqr [=>] 89.58	\[ \frac{\frac{\left({\pi}^{6} \cdot {\pi}^{6}\right) \cdot 0.000244140625 - \color{blue}{{\sin^{-1} \left(1 - x\right)}^{\left(2 \cdot 6\right)}}}{\mathsf{fma}\left({\pi}^{3}, 0.125, {\sin^{-1} \left(1 - x\right)}^{3}\right) \cdot \left({\sin^{-1} \left(1 - x\right)}^{6} + {\pi}^{6} \cdot 0.015625\right)}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]
   metadata-eval [=>] 89.58	\[ \frac{\frac{\left({\pi}^{6} \cdot {\pi}^{6}\right) \cdot 0.000244140625 - {\sin^{-1} \left(1 - x\right)}^{\color{blue}{12}}}{\mathsf{fma}\left({\pi}^{3}, 0.125, {\sin^{-1} \left(1 - x\right)}^{3}\right) \cdot \left({\sin^{-1} \left(1 - x\right)}^{6} + {\pi}^{6} \cdot 0.015625\right)}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]
   +-commutative [=>] 89.58	\[ \frac{\frac{\left({\pi}^{6} \cdot {\pi}^{6}\right) \cdot 0.000244140625 - {\sin^{-1} \left(1 - x\right)}^{12}}{\mathsf{fma}\left({\pi}^{3}, 0.125, {\sin^{-1} \left(1 - x\right)}^{3}\right) \cdot \color{blue}{\left({\pi}^{6} \cdot 0.015625 + {\sin^{-1} \left(1 - x\right)}^{6}\right)}}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]

6. Applied egg-rr 89.59

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

7. Applied egg-rr 89.61

   \[\leadsto \frac{\frac{\left({\pi}^{6} \cdot {\pi}^{6}\right) \cdot 0.000244140625 - {\sin^{-1} \left(1 - x\right)}^{12}}{\mathsf{fma}\left({\pi}^{3}, 0.125, {\sin^{-1} \left(1 - x\right)}^{3}\right) \cdot \left(\color{blue}{{\left({\left(e^{6}\right)}^{\left({\left(\sqrt[3]{\log \pi}\right)}^{2}\right)}\right)}^{\left(\sqrt[3]{\log \pi}\right)}} \cdot 0.015625 + {\sin^{-1} \left(1 - x\right)}^{6}\right)}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]

8. Final simplification 89.61

   \[\leadsto \frac{\frac{\left({\pi}^{6} \cdot {\pi}^{6}\right) \cdot 0.000244140625 - {\sin^{-1} \left(1 - x\right)}^{12}}{\mathsf{fma}\left({\pi}^{3}, 0.125, {\sin^{-1} \left(1 - x\right)}^{3}\right) \cdot \left({\left({\left(e^{6}\right)}^{\left({\left(\sqrt[3]{\log \pi}\right)}^{2}\right)}\right)}^{\left(\sqrt[3]{\log \pi}\right)} \cdot 0.015625 + {\sin^{-1} \left(1 - x\right)}^{6}\right)}}{\left(\pi \cdot \pi\right) \cdot 0.25 + \sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)} \]

Alternatives

Alternative 1
Error	89.6%
Cost	181952

\[\begin{array}{l} t_0 := \sin^{-1} \left(1 - x\right)\\ \frac{\frac{\left({\pi}^{6} \cdot {\pi}^{6}\right) \cdot 0.000244140625 - {t_0}^{12}}{\mathsf{fma}\left({\pi}^{3}, 0.125, {t_0}^{3}\right) \cdot \left({t_0}^{6} + 0.015625 \cdot {\left(e^{\sqrt[3]{{\log \pi}^{2} \cdot 36}}\right)}^{\left(\sqrt[3]{6 \cdot \log \pi}\right)}\right)}}{\left(\pi \cdot \pi\right) \cdot 0.25 + t_0 \cdot \mathsf{fma}\left(\pi, 0.5, t_0\right)} \end{array} \]

Alternative 2
Error	89.59%
Cost	143360

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

Alternative 3
Error	89.58%
Cost	136896

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

Alternative 4
Error	89.58%
Cost	130944

\[\begin{array}{l} t_0 := \sin^{-1} \left(1 - x\right)\\ \frac{\frac{{\left(0.25 \cdot {\pi}^{2}\right)}^{6} - {t_0}^{12}}{\mathsf{fma}\left(0.125, {\pi}^{3}, {t_0}^{3}\right) \cdot \left({t_0}^{6} + {\left(\pi \cdot 0.5\right)}^{6}\right)}}{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) + \left(t_0 \cdot t_0 + t_0 \cdot \left(\pi \cdot 0.5\right)\right)} \end{array} \]

Alternative 5
Error	89.58%
Cost	78784

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

Alternative 6
Error	89.61%
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	89.66%
Cost	26048

\[\pi \cdot 0.5 - {\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{3} \]

Alternative 8
Error	89.63%
Cost	26048

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

Alternative 9
Error	93.16%
Cost	25920

\[2 \cdot \log \left(\sqrt{e^{\cos^{-1} \left(1 - x\right)}}\right) \]

Alternative 10
Error	93.16%
Cost	13184

\[\pi \cdot 0.5 - \sin^{-1} \left(1 - x\right) \]

Alternative 11
Error	93.15%
Cost	6848

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

Alternative 12
Error	93.15%
Cost	6592

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

Reproduce

herbie shell --seed 2023125 
(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)))