Average Error: 0.4 → 0.3
Time: 11.8s
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 r69071 = 1.0;
        double r69072 = 6.0;
        double r69073 = r69071 / r69072;
        double r69074 = -2.0;
        double r69075 = u1;
        double r69076 = log(r69075);
        double r69077 = r69074 * r69076;
        double r69078 = 0.5;
        double r69079 = pow(r69077, r69078);
        double r69080 = r69073 * r69079;
        double r69081 = 2.0;
        double r69082 = atan2(1.0, 0.0);
        double r69083 = r69081 * r69082;
        double r69084 = u2;
        double r69085 = r69083 * r69084;
        double r69086 = cos(r69085);
        double r69087 = r69080 * r69086;
        double r69088 = r69087 + r69078;
        return r69088;
}
↓
double f(double u1, double u2) {
        double r69089 = 1.0;
        double r69090 = -2.0;
        double r69091 = u1;
        double r69092 = log(r69091);
        double r69093 = r69090 * r69092;
        double r69094 = 0.5;
        double r69095 = pow(r69093, r69094);
        double r69096 = r69089 * r69095;
        double r69097 = 6.0;
        double r69098 = r69096 / r69097;
        double r69099 = 2.0;
        double r69100 = atan2(1.0, 0.0);
        double r69101 = r69099 * r69100;
        double r69102 = u2;
        double r69103 = r69101 * r69102;
        double r69104 = cos(r69103);
        double r69105 = r69098 * r69104;
        double r69106 = r69105 + r69094;
        return r69106;
}

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 2020034 +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))