Average Error: 0.4 → 0.3
Time: 5.7s
Precision: binary64
\[\left(0 \leq u1 \land u1 \leq 1\right) \land \left(0 \leq u2 \land u2 \leq 1\right)\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
↓
\[\mathsf{fma}\left(\left(0.16666666666666666 \cdot \sqrt{-\log u1}\right) \cdot \sqrt{2}, \cos \log \left(e^{\left(2 \cdot \pi\right) \cdot u2}\right), 0.5\right) \]
\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
↓
\mathsf{fma}\left(\left(0.16666666666666666 \cdot \sqrt{-\log u1}\right) \cdot \sqrt{2}, \cos \log \left(e^{\left(2 \cdot \pi\right) \cdot u2}\right), 0.5\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
 (fma
  (* (* 0.16666666666666666 (sqrt (- (log u1)))) (sqrt 2.0))
  (cos (log (exp (* (* 2.0 PI) u2))))
  0.5))
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 fma(((0.16666666666666666 * sqrt(-log(u1))) * sqrt(2.0)), cos(log(exp((2.0 * ((double) M_PI)) * u2))), 0.5);
}

Error

Bits error versus u1

Bits error versus u2

Derivation

  1. Initial program 0.4

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

  2. Simplified0.4

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

  3. Taylor expanded in u1 around inf 0.3

    \[\leadsto \mathsf{fma}\left(\color{blue}{0.16666666666666666 \cdot \left(\sqrt{\log \left(\frac{1}{u1}\right)} \cdot \sqrt{2}\right)}, \cos \left(\left(2 \cdot \pi\right) \cdot u2\right), 0.5\right) \]

  4. Simplified0.3

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

  5. Applied associate-*r*_binary640.3

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

  6. Applied add-log-exp_binary640.3

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

  7. Final simplification0.3

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

Reproduce

herbie shell --seed 2022067 
(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))