?

Average Accuracy: 99.4% → 99.5%
Time: 2.0min
Precision: binary64

?

\[\left(0 \leq u1 \land u1 \leq 1\right) \land \left(0 \leq u2 \land u2 \leq 1\right)\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
↓
\[\frac{1}{\pi} \cdot \mathsf{fma}\left(\pi, \sqrt{-0.05555555555555555 \cdot \log u1} \cdot \cos \left(\left(\pi + \pi\right) \cdot u2\right), \frac{\pi}{2}\right) \]
(FPCore (u1 u2)
 :precision binary64
 (+
  (* (* (/ 1.0 6.0) (pow (* -2.0 (log u1)) 0.5)) (cos (* (* 2.0 PI) u2)))
  0.5))
↓
(FPCore (u1 u2)
 :precision binary64
 (*
  (/ 1.0 PI)
  (fma
   PI
   (* (sqrt (* -0.05555555555555555 (log u1))) (cos (* (+ PI PI) u2)))
   (/ PI 2.0))))
double code(double u1, double u2) {
	return (((1.0 / 6.0) * pow((-2.0 * log(u1)), 0.5)) * cos(((2.0 * ((double) M_PI)) * u2))) + 0.5;
}
↓
double code(double u1, double u2) {
	return (1.0 / ((double) M_PI)) * fma(((double) M_PI), (sqrt((-0.05555555555555555 * log(u1))) * cos(((((double) M_PI) + ((double) M_PI)) * u2))), (((double) M_PI) / 2.0));
}

function code(u1, u2)
	return Float64(Float64(Float64(Float64(1.0 / 6.0) * (Float64(-2.0 * log(u1)) ^ 0.5)) * cos(Float64(Float64(2.0 * pi) * u2))) + 0.5)
end
↓
function code(u1, u2)
	return Float64(Float64(1.0 / pi) * fma(pi, Float64(sqrt(Float64(-0.05555555555555555 * log(u1))) * cos(Float64(Float64(pi + pi) * u2))), Float64(pi / 2.0)))
end

code[u1_, u2_] := N[(N[(N[(N[(1.0 / 6.0), $MachinePrecision] * N[Power[N[(-2.0 * N[Log[u1], $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(2.0 * Pi), $MachinePrecision] * u2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]
↓
code[u1_, u2_] := N[(N[(1.0 / Pi), $MachinePrecision] * N[(Pi * N[(N[Sqrt[N[(-0.05555555555555555 * N[Log[u1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(N[(Pi + Pi), $MachinePrecision] * u2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5
↓
\frac{1}{\pi} \cdot \mathsf{fma}\left(\pi, \sqrt{-0.05555555555555555 \cdot \log u1} \cdot \cos \left(\left(\pi + \pi\right) \cdot u2\right), \frac{\pi}{2}\right)

Error?

Derivation?

  1. Initial program 99.4%

    \[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

  2. Simplified99.4%

    \[\leadsto \color{blue}{\mathsf{fma}\left(0.16666666666666666, \sqrt{-2 \cdot \log u1} \cdot \cos \left(\left(\pi + \pi\right) \cdot u2\right), 0.5\right)} \]
    Proof

  3. Applied egg-rr99.5%

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{0.027777777777777776 \cdot \left(-2 \cdot \log u1\right)}, \cos \left(\left(\pi + \pi\right) \cdot u2\right), 0.5\right) \cdot \left(\pi \cdot \pi\right)}{\pi \cdot \pi}} \]

  4. Applied egg-rr99.5%

    \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\color{blue}{-0.05555555555555555 \cdot \log \left(\sqrt[3]{u1}\right) + -0.05555555555555555 \cdot \log \left({\left(\sqrt[3]{u1}\right)}^{2}\right)}}, \cos \left(\left(\pi + \pi\right) \cdot u2\right), 0.5\right) \cdot \left(\pi \cdot \pi\right)}{\pi \cdot \pi} \]

  5. Simplified99.5%

    \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\color{blue}{-0.05555555555555555 \cdot \left(\log \left(\sqrt[3]{u1}\right) + \log \left({\left(\sqrt[3]{u1}\right)}^{2}\right)\right)}}, \cos \left(\left(\pi + \pi\right) \cdot u2\right), 0.5\right) \cdot \left(\pi \cdot \pi\right)}{\pi \cdot \pi} \]
    Proof

  6. Applied egg-rr99.5%

    \[\leadsto \color{blue}{\frac{1}{\pi} \cdot \left(\pi \cdot \mathsf{fma}\left(\sqrt{-0.05555555555555555 \cdot \log u1}, \cos \left(\left(\pi + \pi\right) \cdot u2\right), 0.5\right)\right)} \]

  7. Applied egg-rr99.5%

    \[\leadsto \color{blue}{\frac{1}{\pi} \cdot \left(\pi \cdot \left(\sqrt{-0.05555555555555555 \cdot \log u1} \cdot \cos \left(\left(\pi + \pi\right) \cdot u2\right)\right)\right) + \frac{1}{\pi} \cdot \frac{\pi}{2}} \]

  8. Simplified99.5%

    \[\leadsto \color{blue}{\frac{1}{\pi} \cdot \mathsf{fma}\left(\pi, \sqrt{-0.05555555555555555 \cdot \log u1} \cdot \cos \left(\left(\pi + \pi\right) \cdot u2\right), \frac{\pi}{2}\right)} \]
    Proof

Reproduce?

herbie shell --seed 2023237 
(FPCore (u1 u2)
  :name "normal distribution"
  :precision binary64
  :pre (and (and (<= 0.0 u1) (<= u1 1.0)) (and (<= 0.0 u2) (<= u2 1.0)))
  (+ (* (* (/ 1.0 6.0) (pow (* -2.0 (log u1)) 0.5)) (cos (* (* 2.0 PI) u2))) 0.5))