Average Error: 0.4 → 0.3
Time: 11.9s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\frac{1 \cdot {\left(-2 \cdot \log u1\right)}^{0.5}}{6} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
\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
↓
\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
double f(double u1, double u2) {
        double r73061 = 1.0;
        double r73062 = 6.0;
        double r73063 = r73061 / r73062;
        double r73064 = -2.0;
        double r73065 = u1;
        double r73066 = log(r73065);
        double r73067 = r73064 * r73066;
        double r73068 = 0.5;
        double r73069 = pow(r73067, r73068);
        double r73070 = r73063 * r73069;
        double r73071 = 2.0;
        double r73072 = atan2(1.0, 0.0);
        double r73073 = r73071 * r73072;
        double r73074 = u2;
        double r73075 = r73073 * r73074;
        double r73076 = cos(r73075);
        double r73077 = r73070 * r73076;
        double r73078 = r73077 + r73068;
        return r73078;
}
↓
double f(double u1, double u2) {
        double r73079 = 1.0;
        double r73080 = -2.0;
        double r73081 = u1;
        double r73082 = log(r73081);
        double r73083 = r73080 * r73082;
        double r73084 = 0.5;
        double r73085 = pow(r73083, r73084);
        double r73086 = r73079 * r73085;
        double r73087 = 6.0;
        double r73088 = r73086 / r73087;
        double r73089 = 2.0;
        double r73090 = atan2(1.0, 0.0);
        double r73091 = r73089 * r73090;
        double r73092 = u2;
        double r73093 = r73091 * r73092;
        double r73094 = cos(r73093);
        double r73095 = r73088 * r73094;
        double r73096 = r73095 + r73084;
        return r73096;
}

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. Final simplification0.3

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

Reproduce

herbie shell --seed 2020046 
(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))