Average Error: 0.4 → 0.3
Time: 10.5s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\frac{1 \cdot {\left(-2 \cdot \log u1\right)}^{0.5}}{6} \cdot \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\left(2 \cdot \pi\right) \cdot u2\right)\right)\right) + 0.5\]
\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
↓
\frac{1 \cdot {\left(-2 \cdot \log u1\right)}^{0.5}}{6} \cdot \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\left(2 \cdot \pi\right) \cdot u2\right)\right)\right) + 0.5
double f(double u1, double u2) {
        double r63023 = 1.0;
        double r63024 = 6.0;
        double r63025 = r63023 / r63024;
        double r63026 = -2.0;
        double r63027 = u1;
        double r63028 = log(r63027);
        double r63029 = r63026 * r63028;
        double r63030 = 0.5;
        double r63031 = pow(r63029, r63030);
        double r63032 = r63025 * r63031;
        double r63033 = 2.0;
        double r63034 = atan2(1.0, 0.0);
        double r63035 = r63033 * r63034;
        double r63036 = u2;
        double r63037 = r63035 * r63036;
        double r63038 = cos(r63037);
        double r63039 = r63032 * r63038;
        double r63040 = r63039 + r63030;
        return r63040;
}
↓
double f(double u1, double u2) {
        double r63041 = 1.0;
        double r63042 = -2.0;
        double r63043 = u1;
        double r63044 = log(r63043);
        double r63045 = r63042 * r63044;
        double r63046 = 0.5;
        double r63047 = pow(r63045, r63046);
        double r63048 = r63041 * r63047;
        double r63049 = 6.0;
        double r63050 = r63048 / r63049;
        double r63051 = 2.0;
        double r63052 = atan2(1.0, 0.0);
        double r63053 = r63051 * r63052;
        double r63054 = u2;
        double r63055 = r63053 * r63054;
        double r63056 = log1p(r63055);
        double r63057 = expm1(r63056);
        double r63058 = cos(r63057);
        double r63059 = r63050 * r63058;
        double r63060 = r63059 + r63046;
        return r63060;
}

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 associate-*l/0.3

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

  5. Applied expm1-log1p-u0.3

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

  6. Final simplification0.3

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

Reproduce

herbie shell --seed 2020036 +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))