Average Error: 0.4 → 0.3
Time: 17.6s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\left(1 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{{\left(-2 \cdot \log u1\right)}^{0.5}}{6}\right)\right)\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
\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
↓
\left(1 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{{\left(-2 \cdot \log u1\right)}^{0.5}}{6}\right)\right)\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5
double f(double u1, double u2) {
        double r69091 = 1.0;
        double r69092 = 6.0;
        double r69093 = r69091 / r69092;
        double r69094 = -2.0;
        double r69095 = u1;
        double r69096 = log(r69095);
        double r69097 = r69094 * r69096;
        double r69098 = 0.5;
        double r69099 = pow(r69097, r69098);
        double r69100 = r69093 * r69099;
        double r69101 = 2.0;
        double r69102 = atan2(1.0, 0.0);
        double r69103 = r69101 * r69102;
        double r69104 = u2;
        double r69105 = r69103 * r69104;
        double r69106 = cos(r69105);
        double r69107 = r69100 * r69106;
        double r69108 = r69107 + r69098;
        return r69108;
}
↓
double f(double u1, double u2) {
        double r69109 = 1.0;
        double r69110 = -2.0;
        double r69111 = u1;
        double r69112 = log(r69111);
        double r69113 = r69110 * r69112;
        double r69114 = 0.5;
        double r69115 = pow(r69113, r69114);
        double r69116 = 6.0;
        double r69117 = r69115 / r69116;
        double r69118 = expm1(r69117);
        double r69119 = log1p(r69118);
        double r69120 = r69109 * r69119;
        double r69121 = 2.0;
        double r69122 = atan2(1.0, 0.0);
        double r69123 = r69121 * r69122;
        double r69124 = u2;
        double r69125 = r69123 * r69124;
        double r69126 = cos(r69125);
        double r69127 = r69120 * r69126;
        double r69128 = r69127 + r69114;
        return r69128;
}

Error

Bits error versus u1

Bits error versus u2

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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. Using strategy rm

  3. Applied div-inv0.4

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

  4. Applied associate-*l*0.4

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

  5. Simplified0.3

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

  7. Applied log1p-expm1-u0.3

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

  8. Final simplification0.3

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

Reproduce

herbie shell --seed 2020047 +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))