Average Error: 0.4 → 0.3
Time: 27.5s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\mathsf{fma}\left(\sqrt{\frac{1}{6}} \cdot \left(\sqrt{\frac{1}{6}} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right), \cos \left(\left(2 \cdot \pi\right) \cdot u2\right), 0.5\right)\]
\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(\sqrt{\frac{1}{6}} \cdot \left(\sqrt{\frac{1}{6}} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right), \cos \left(\left(2 \cdot \pi\right) \cdot u2\right), 0.5\right)
double f(double u1, double u2) {
        double r73877 = 1.0;
        double r73878 = 6.0;
        double r73879 = r73877 / r73878;
        double r73880 = -2.0;
        double r73881 = u1;
        double r73882 = log(r73881);
        double r73883 = r73880 * r73882;
        double r73884 = 0.5;
        double r73885 = pow(r73883, r73884);
        double r73886 = r73879 * r73885;
        double r73887 = 2.0;
        double r73888 = atan2(1.0, 0.0);
        double r73889 = r73887 * r73888;
        double r73890 = u2;
        double r73891 = r73889 * r73890;
        double r73892 = cos(r73891);
        double r73893 = r73886 * r73892;
        double r73894 = r73893 + r73884;
        return r73894;
}
↓
double f(double u1, double u2) {
        double r73895 = 1.0;
        double r73896 = 6.0;
        double r73897 = r73895 / r73896;
        double r73898 = sqrt(r73897);
        double r73899 = -2.0;
        double r73900 = u1;
        double r73901 = log(r73900);
        double r73902 = r73899 * r73901;
        double r73903 = 0.5;
        double r73904 = pow(r73902, r73903);
        double r73905 = r73898 * r73904;
        double r73906 = r73898 * r73905;
        double r73907 = 2.0;
        double r73908 = atan2(1.0, 0.0);
        double r73909 = r73907 * r73908;
        double r73910 = u2;
        double r73911 = r73909 * r73910;
        double r73912 = cos(r73911);
        double r73913 = fma(r73906, r73912, r73903);
        return r73913;
}

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

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

  4. Applied add-sqr-sqrt0.4

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

  5. Applied associate-*l*0.3

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

  6. Final simplification0.3

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

Reproduce

herbie shell --seed 2019304 +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))