Average Error: 0.4 → 0.3
Time: 28.4s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(1 \cdot \frac{{\left(-2 \cdot \log u1\right)}^{0.5}}{6}, \cos \left(\left(2 \cdot \pi\right) \cdot u2\right), 0.5\right)\right)\right)\]
\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{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(1 \cdot \frac{{\left(-2 \cdot \log u1\right)}^{0.5}}{6}, \cos \left(\left(2 \cdot \pi\right) \cdot u2\right), 0.5\right)\right)\right)
double f(double u1, double u2) {
        double r83024 = 1.0;
        double r83025 = 6.0;
        double r83026 = r83024 / r83025;
        double r83027 = -2.0;
        double r83028 = u1;
        double r83029 = log(r83028);
        double r83030 = r83027 * r83029;
        double r83031 = 0.5;
        double r83032 = pow(r83030, r83031);
        double r83033 = r83026 * r83032;
        double r83034 = 2.0;
        double r83035 = atan2(1.0, 0.0);
        double r83036 = r83034 * r83035;
        double r83037 = u2;
        double r83038 = r83036 * r83037;
        double r83039 = cos(r83038);
        double r83040 = r83033 * r83039;
        double r83041 = r83040 + r83031;
        return r83041;
}
↓
double f(double u1, double u2) {
        double r83042 = 1.0;
        double r83043 = -2.0;
        double r83044 = u1;
        double r83045 = log(r83044);
        double r83046 = r83043 * r83045;
        double r83047 = 0.5;
        double r83048 = pow(r83046, r83047);
        double r83049 = 6.0;
        double r83050 = r83048 / r83049;
        double r83051 = r83042 * r83050;
        double r83052 = 2.0;
        double r83053 = atan2(1.0, 0.0);
        double r83054 = r83052 * r83053;
        double r83055 = u2;
        double r83056 = r83054 * r83055;
        double r83057 = cos(r83056);
        double r83058 = fma(r83051, r83057, r83047);
        double r83059 = expm1(r83058);
        double r83060 = log1p(r83059);
        return r83060;
}

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 div-inv0.4

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

  5. Applied associate-*l*0.4

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

  6. Simplified0.3

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

  8. Applied log1p-expm1-u0.3

    \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(1 \cdot \frac{{\left(-2 \cdot \log u1\right)}^{0.5}}{6}, \cos \left(\left(2 \cdot \pi\right) \cdot u2\right), 0.5\right)\right)\right)}\]

  9. Final simplification0.3

    \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(1 \cdot \frac{{\left(-2 \cdot \log u1\right)}^{0.5}}{6}, \cos \left(\left(2 \cdot \pi\right) \cdot u2\right), 0.5\right)\right)\right)\]

Reproduce

herbie shell --seed 2019235 +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))