Average Error: 0.4 → 0.3
Time: 27.0s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\mathsf{fma}\left(\frac{1 \cdot {\left(-2 \cdot \log u1\right)}^{0.5}}{6}, \cos \left(\left(2 \cdot \pi\right) \cdot u2\right), 0.5\right)\]
\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 \cdot {\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 r74027 = 1.0;
        double r74028 = 6.0;
        double r74029 = r74027 / r74028;
        double r74030 = -2.0;
        double r74031 = u1;
        double r74032 = log(r74031);
        double r74033 = r74030 * r74032;
        double r74034 = 0.5;
        double r74035 = pow(r74033, r74034);
        double r74036 = r74029 * r74035;
        double r74037 = 2.0;
        double r74038 = atan2(1.0, 0.0);
        double r74039 = r74037 * r74038;
        double r74040 = u2;
        double r74041 = r74039 * r74040;
        double r74042 = cos(r74041);
        double r74043 = r74036 * r74042;
        double r74044 = r74043 + r74034;
        return r74044;
}
↓
double f(double u1, double u2) {
        double r74045 = 1.0;
        double r74046 = -2.0;
        double r74047 = u1;
        double r74048 = log(r74047);
        double r74049 = r74046 * r74048;
        double r74050 = 0.5;
        double r74051 = pow(r74049, r74050);
        double r74052 = r74045 * r74051;
        double r74053 = 6.0;
        double r74054 = r74052 / r74053;
        double r74055 = 2.0;
        double r74056 = atan2(1.0, 0.0);
        double r74057 = r74055 * r74056;
        double r74058 = u2;
        double r74059 = r74057 * r74058;
        double r74060 = cos(r74059);
        double r74061 = fma(r74054, r74060, r74050);
        return r74061;
}

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 *-un-lft-identity0.4

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

  5. Applied *-un-lft-identity0.4

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

  6. Applied times-frac0.4

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

  7. Applied associate-*l*0.4

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

  8. Simplified0.3

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

  9. Final simplification0.3

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

Reproduce

herbie shell --seed 2019212 +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))