Average Error: 0.4 → 0.3
Time: 13.0s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\mathsf{fma}\left(1 \cdot \frac{{\left(-2 \cdot \log u1\right)}^{0.5}}{6}, \cos \left(\left(2 \cdot \pi\right) \cdot u2\right), 0.5\right)\]
\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(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)
double f(double u1, double u2) {
        double r79011 = 1.0;
        double r79012 = 6.0;
        double r79013 = r79011 / r79012;
        double r79014 = -2.0;
        double r79015 = u1;
        double r79016 = log(r79015);
        double r79017 = r79014 * r79016;
        double r79018 = 0.5;
        double r79019 = pow(r79017, r79018);
        double r79020 = r79013 * r79019;
        double r79021 = 2.0;
        double r79022 = atan2(1.0, 0.0);
        double r79023 = r79021 * r79022;
        double r79024 = u2;
        double r79025 = r79023 * r79024;
        double r79026 = cos(r79025);
        double r79027 = r79020 * r79026;
        double r79028 = r79027 + r79018;
        return r79028;
}
↓
double f(double u1, double u2) {
        double r79029 = 1.0;
        double r79030 = -2.0;
        double r79031 = u1;
        double r79032 = log(r79031);
        double r79033 = r79030 * r79032;
        double r79034 = 0.5;
        double r79035 = pow(r79033, r79034);
        double r79036 = 6.0;
        double r79037 = r79035 / r79036;
        double r79038 = r79029 * r79037;
        double r79039 = 2.0;
        double r79040 = atan2(1.0, 0.0);
        double r79041 = r79039 * r79040;
        double r79042 = u2;
        double r79043 = r79041 * r79042;
        double r79044 = cos(r79043);
        double r79045 = fma(r79038, r79044, r79034);
        return r79045;
}

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. Final simplification0.3

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

Reproduce

herbie shell --seed 2019353 +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))