Average Error: 0.4 → 0.3
Time: 29.8s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\log \left(e^{\mathsf{fma}\left(\frac{{\left(-2 \cdot \log u1\right)}^{0.5}}{\frac{6}{1}}, \cos \left(2 \cdot \left(u2 \cdot \pi\right)\right), 0.5\right)}\right)\]
\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
↓
\log \left(e^{\mathsf{fma}\left(\frac{{\left(-2 \cdot \log u1\right)}^{0.5}}{\frac{6}{1}}, \cos \left(2 \cdot \left(u2 \cdot \pi\right)\right), 0.5\right)}\right)
double f(double u1, double u2) {
        double r84942 = 1.0;
        double r84943 = 6.0;
        double r84944 = r84942 / r84943;
        double r84945 = -2.0;
        double r84946 = u1;
        double r84947 = log(r84946);
        double r84948 = r84945 * r84947;
        double r84949 = 0.5;
        double r84950 = pow(r84948, r84949);
        double r84951 = r84944 * r84950;
        double r84952 = 2.0;
        double r84953 = atan2(1.0, 0.0);
        double r84954 = r84952 * r84953;
        double r84955 = u2;
        double r84956 = r84954 * r84955;
        double r84957 = cos(r84956);
        double r84958 = r84951 * r84957;
        double r84959 = r84958 + r84949;
        return r84959;
}
↓
double f(double u1, double u2) {
        double r84960 = -2.0;
        double r84961 = u1;
        double r84962 = log(r84961);
        double r84963 = r84960 * r84962;
        double r84964 = 0.5;
        double r84965 = pow(r84963, r84964);
        double r84966 = 6.0;
        double r84967 = 1.0;
        double r84968 = r84966 / r84967;
        double r84969 = r84965 / r84968;
        double r84970 = 2.0;
        double r84971 = u2;
        double r84972 = atan2(1.0, 0.0);
        double r84973 = r84971 * r84972;
        double r84974 = r84970 * r84973;
        double r84975 = cos(r84974);
        double r84976 = fma(r84969, r84975, r84964);
        double r84977 = exp(r84976);
        double r84978 = log(r84977);
        return r84978;
}

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. Simplified0.3

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

  4. Applied add-log-exp0.3

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

  5. Simplified0.3

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

  6. Final simplification0.3

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

Reproduce

herbie shell --seed 2019179 +o rules:numerics
(FPCore (u1 u2)
  :name "normal distribution"
  :pre (and (<= 0.0 u1 1.0) (<= 0.0 u2 1.0))
  (+ (* (* (/ 1.0 6.0) (pow (* -2.0 (log u1)) 0.5)) (cos (* (* 2.0 PI) u2))) 0.5))