Average Error: 0.4 → 0.4
Time: 3.3m
Precision: 64
\[0 \le u1 \le 1 \land 0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\left(\sqrt{\frac{\cos \left(\left(2 \cdot \pi\right) \cdot u2\right)}{6}} \cdot {\left(\log u1 \cdot -2\right)}^{0.5}\right) \cdot \sqrt{\frac{\cos \left(\left(2 \cdot \pi\right) \cdot u2\right)}{6}} + 0.5\]
\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(\sqrt{\frac{\cos \left(\left(2 \cdot \pi\right) \cdot u2\right)}{6}} \cdot {\left(\log u1 \cdot -2\right)}^{0.5}\right) \cdot \sqrt{\frac{\cos \left(\left(2 \cdot \pi\right) \cdot u2\right)}{6}} + 0.5
double f(double u1, double u2) {
        double r27990018 = 1.0;
        double r27990019 = 6.0;
        double r27990020 = r27990018 / r27990019;
        double r27990021 = -2.0;
        double r27990022 = u1;
        double r27990023 = log(r27990022);
        double r27990024 = r27990021 * r27990023;
        double r27990025 = 0.5;
        double r27990026 = pow(r27990024, r27990025);
        double r27990027 = r27990020 * r27990026;
        double r27990028 = 2.0;
        double r27990029 = atan2(1.0, 0.0);
        double r27990030 = r27990028 * r27990029;
        double r27990031 = u2;
        double r27990032 = r27990030 * r27990031;
        double r27990033 = cos(r27990032);
        double r27990034 = r27990027 * r27990033;
        double r27990035 = r27990034 + r27990025;
        return r27990035;
}
↓
double f(double u1, double u2) {
        double r27990036 = 2.0;
        double r27990037 = atan2(1.0, 0.0);
        double r27990038 = r27990036 * r27990037;
        double r27990039 = u2;
        double r27990040 = r27990038 * r27990039;
        double r27990041 = cos(r27990040);
        double r27990042 = 6.0;
        double r27990043 = r27990041 / r27990042;
        double r27990044 = sqrt(r27990043);
        double r27990045 = u1;
        double r27990046 = log(r27990045);
        double r27990047 = -2.0;
        double r27990048 = r27990046 * r27990047;
        double r27990049 = 0.5;
        double r27990050 = pow(r27990048, r27990049);
        double r27990051 = r27990044 * r27990050;
        double r27990052 = r27990051 * r27990044;
        double r27990053 = r27990052 + r27990049;
        return r27990053;
}

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. Simplified0.4

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

  4. Applied add-sqr-sqrt0.5

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

  5. Applied associate-*r*0.4

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

  6. Final simplification0.4

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

Reproduce

herbie shell --seed 2019128 
(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))