Average Error: 0.4 → 0.3
Time: 10.2s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\left(1 \cdot \frac{1}{\frac{6}{{\left(-2 \cdot \log u1\right)}^{0.5}}}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
\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(1 \cdot \frac{1}{\frac{6}{{\left(-2 \cdot \log u1\right)}^{0.5}}}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5
double f(double u1, double u2) {
        double r64057 = 1.0;
        double r64058 = 6.0;
        double r64059 = r64057 / r64058;
        double r64060 = -2.0;
        double r64061 = u1;
        double r64062 = log(r64061);
        double r64063 = r64060 * r64062;
        double r64064 = 0.5;
        double r64065 = pow(r64063, r64064);
        double r64066 = r64059 * r64065;
        double r64067 = 2.0;
        double r64068 = atan2(1.0, 0.0);
        double r64069 = r64067 * r64068;
        double r64070 = u2;
        double r64071 = r64069 * r64070;
        double r64072 = cos(r64071);
        double r64073 = r64066 * r64072;
        double r64074 = r64073 + r64064;
        return r64074;
}
↓
double f(double u1, double u2) {
        double r64075 = 1.0;
        double r64076 = 1.0;
        double r64077 = 6.0;
        double r64078 = -2.0;
        double r64079 = u1;
        double r64080 = log(r64079);
        double r64081 = r64078 * r64080;
        double r64082 = 0.5;
        double r64083 = pow(r64081, r64082);
        double r64084 = r64077 / r64083;
        double r64085 = r64076 / r64084;
        double r64086 = r64075 * r64085;
        double r64087 = 2.0;
        double r64088 = atan2(1.0, 0.0);
        double r64089 = r64087 * r64088;
        double r64090 = u2;
        double r64091 = r64089 * r64090;
        double r64092 = cos(r64091);
        double r64093 = r64086 * r64092;
        double r64094 = r64093 + r64082;
        return r64094;
}

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 div-inv0.4

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

  4. Applied associate-*l*0.4

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

  5. Simplified0.3

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

  7. Applied clear-num0.3

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

  8. Final simplification0.3

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

Reproduce

herbie shell --seed 2020018 
(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))