Average Error: 0.4 → 0.4
Time: 11.2s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\mathsf{fma}\left(\frac{1}{6 \cdot {\left(\frac{1}{{\left(\log u1\right)}^{1} \cdot {-2}^{1}}\right)}^{0.5}}, \cos \left(\left(2 \cdot \pi\right) \cdot u2\right), 0.5\right)\]
\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(\frac{1}{6 \cdot {\left(\frac{1}{{\left(\log u1\right)}^{1} \cdot {-2}^{1}}\right)}^{0.5}}, \cos \left(\left(2 \cdot \pi\right) \cdot u2\right), 0.5\right)
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((1.0 / (6.0 * pow((1.0 / (pow(log(u1), 1.0) * pow(-2.0, 1.0))), 0.5))), cos(((2.0 * ((double) M_PI)) * u2)), 0.5);
}

Error

Bits error versus u1

Bits error versus u2

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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

    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}, \cos \left(\left(2 \cdot \pi\right) \cdot u2\right), 0.5\right)}\]
  3. Using strategy rm

  4. Applied associate-*l/0.3

    \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1 \cdot {\left(-2 \cdot \log u1\right)}^{0.5}}{6}}, \cos \left(\left(2 \cdot \pi\right) \cdot u2\right), 0.5\right)\]
  5. Using strategy rm

  6. Applied associate-/l*0.4

    \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{\frac{6}{{\left(-2 \cdot \log u1\right)}^{0.5}}}}, \cos \left(\left(2 \cdot \pi\right) \cdot u2\right), 0.5\right)\]

  7. Taylor expanded around 0 0.4

    \[\leadsto \mathsf{fma}\left(\frac{1}{\color{blue}{6 \cdot {\left(\frac{1}{{\left(\log u1\right)}^{1} \cdot {-2}^{1}}\right)}^{0.5}}}, \cos \left(\left(2 \cdot \pi\right) \cdot u2\right), 0.5\right)\]

  8. Final simplification0.4

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

Reproduce

herbie shell --seed 2020053 +o rules:numerics
(FPCore (u1 u2)
  :name "normal distribution"
  :precision binary64
  :pre (and (<= 0.0 u1 1) (<= 0.0 u2 1))
  (+ (* (* (/ 1 6) (pow (* -2 (log u1)) 0.5)) (cos (* (* 2 PI) u2))) 0.5))