Average Error: 0.4 → 0.4
Time: 44.4s
Precision: 64
\[0 \le u1 \le 1 \land 0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\mathsf{fma}\left(\cos \left(\left(\pi \cdot 2\right) \cdot u2\right), \frac{1}{\frac{6}{{\left(-2 \cdot \log u1\right)}^{0.5}}}, 0.5\right)\]
\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(\cos \left(\left(\pi \cdot 2\right) \cdot u2\right), \frac{1}{\frac{6}{{\left(-2 \cdot \log u1\right)}^{0.5}}}, 0.5\right)
double f(double u1, double u2) {
        double r1744070 = 1.0;
        double r1744071 = 6.0;
        double r1744072 = r1744070 / r1744071;
        double r1744073 = -2.0;
        double r1744074 = u1;
        double r1744075 = log(r1744074);
        double r1744076 = r1744073 * r1744075;
        double r1744077 = 0.5;
        double r1744078 = pow(r1744076, r1744077);
        double r1744079 = r1744072 * r1744078;
        double r1744080 = 2.0;
        double r1744081 = atan2(1.0, 0.0);
        double r1744082 = r1744080 * r1744081;
        double r1744083 = u2;
        double r1744084 = r1744082 * r1744083;
        double r1744085 = cos(r1744084);
        double r1744086 = r1744079 * r1744085;
        double r1744087 = r1744086 + r1744077;
        return r1744087;
}
↓
double f(double u1, double u2) {
        double r1744088 = atan2(1.0, 0.0);
        double r1744089 = 2.0;
        double r1744090 = r1744088 * r1744089;
        double r1744091 = u2;
        double r1744092 = r1744090 * r1744091;
        double r1744093 = cos(r1744092);
        double r1744094 = 1.0;
        double r1744095 = 6.0;
        double r1744096 = -2.0;
        double r1744097 = u1;
        double r1744098 = log(r1744097);
        double r1744099 = r1744096 * r1744098;
        double r1744100 = 0.5;
        double r1744101 = pow(r1744099, r1744100);
        double r1744102 = r1744095 / r1744101;
        double r1744103 = r1744094 / r1744102;
        double r1744104 = fma(r1744093, r1744103, r1744100);
        return r1744104;
}

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

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

  4. Applied sqr-pow0.5

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

  5. Applied associate-/l*0.5

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

  7. Applied clear-num0.6

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

  8. Simplified0.4

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

  9. Final simplification0.4

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

Reproduce

herbie shell --seed 2019134 +o rules:numerics
(FPCore (u1 u2)
  :name "normal distribution"
  :pre (and (<= 0 u1 1) (<= 0 u2 1))
  (+ (* (* (/ 1 6) (pow (* -2 (log u1)) 0.5)) (cos (* (* 2 PI) u2))) 0.5))