Average Error: 0.4 → 0.4
Time: 35.2s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\left(0.1666666666666666574148081281236954964697 \cdot {\left({\left(\log u1\right)}^{1} \cdot {-2}^{1}\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
\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(0.1666666666666666574148081281236954964697 \cdot {\left({\left(\log u1\right)}^{1} \cdot {-2}^{1}\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5
double f(double u1, double u2) {
        double r87080 = 1.0;
        double r87081 = 6.0;
        double r87082 = r87080 / r87081;
        double r87083 = -2.0;
        double r87084 = u1;
        double r87085 = log(r87084);
        double r87086 = r87083 * r87085;
        double r87087 = 0.5;
        double r87088 = pow(r87086, r87087);
        double r87089 = r87082 * r87088;
        double r87090 = 2.0;
        double r87091 = atan2(1.0, 0.0);
        double r87092 = r87090 * r87091;
        double r87093 = u2;
        double r87094 = r87092 * r87093;
        double r87095 = cos(r87094);
        double r87096 = r87089 * r87095;
        double r87097 = r87096 + r87087;
        return r87097;
}
↓
double f(double u1, double u2) {
        double r87098 = 0.16666666666666666;
        double r87099 = u1;
        double r87100 = log(r87099);
        double r87101 = 1.0;
        double r87102 = pow(r87100, r87101);
        double r87103 = -2.0;
        double r87104 = pow(r87103, r87101);
        double r87105 = r87102 * r87104;
        double r87106 = 0.5;
        double r87107 = pow(r87105, r87106);
        double r87108 = r87098 * r87107;
        double r87109 = 2.0;
        double r87110 = atan2(1.0, 0.0);
        double r87111 = r87109 * r87110;
        double r87112 = u2;
        double r87113 = r87111 * r87112;
        double r87114 = cos(r87113);
        double r87115 = r87108 * r87114;
        double r87116 = r87115 + r87106;
        return r87116;
}

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 expm1-log1p-u0.5

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

  4. Taylor expanded around 0 0.4

    \[\leadsto \color{blue}{\left(0.1666666666666666574148081281236954964697 \cdot {\left({\left(\log u1\right)}^{1} \cdot {-2}^{1}\right)}^{0.5}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]

  5. Final simplification0.4

    \[\leadsto \left(0.1666666666666666574148081281236954964697 \cdot {\left({\left(\log u1\right)}^{1} \cdot {-2}^{1}\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]

Reproduce

herbie shell --seed 2019306 +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))