Average Error: 0.4 → 0.3
Time: 11.5s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\left(1 \cdot \frac{{\left(-2 \cdot \log u1\right)}^{0.5}}{6}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
\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{{\left(-2 \cdot \log u1\right)}^{0.5}}{6}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5
double f(double u1, double u2) {
        double r67136 = 1.0;
        double r67137 = 6.0;
        double r67138 = r67136 / r67137;
        double r67139 = -2.0;
        double r67140 = u1;
        double r67141 = log(r67140);
        double r67142 = r67139 * r67141;
        double r67143 = 0.5;
        double r67144 = pow(r67142, r67143);
        double r67145 = r67138 * r67144;
        double r67146 = 2.0;
        double r67147 = atan2(1.0, 0.0);
        double r67148 = r67146 * r67147;
        double r67149 = u2;
        double r67150 = r67148 * r67149;
        double r67151 = cos(r67150);
        double r67152 = r67145 * r67151;
        double r67153 = r67152 + r67143;
        return r67153;
}
↓
double f(double u1, double u2) {
        double r67154 = 1.0;
        double r67155 = -2.0;
        double r67156 = u1;
        double r67157 = log(r67156);
        double r67158 = r67155 * r67157;
        double r67159 = 0.5;
        double r67160 = pow(r67158, r67159);
        double r67161 = 6.0;
        double r67162 = r67160 / r67161;
        double r67163 = r67154 * r67162;
        double r67164 = 2.0;
        double r67165 = atan2(1.0, 0.0);
        double r67166 = r67164 * r67165;
        double r67167 = u2;
        double r67168 = r67166 * r67167;
        double r67169 = cos(r67168);
        double r67170 = r67163 * r67169;
        double r67171 = r67170 + r67159;
        return r67171;
}

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. Final simplification0.3

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

Reproduce

herbie shell --seed 2019344 
(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))