Average Error: 0.4 → 0.3
Time: 25.0s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\frac{{\left(-2 \cdot \log u1\right)}^{0.5} \cdot 1}{6} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
\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{{\left(-2 \cdot \log u1\right)}^{0.5} \cdot 1}{6} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5
double f(double u1, double u2) {
        double r80151 = 1.0;
        double r80152 = 6.0;
        double r80153 = r80151 / r80152;
        double r80154 = -2.0;
        double r80155 = u1;
        double r80156 = log(r80155);
        double r80157 = r80154 * r80156;
        double r80158 = 0.5;
        double r80159 = pow(r80157, r80158);
        double r80160 = r80153 * r80159;
        double r80161 = 2.0;
        double r80162 = atan2(1.0, 0.0);
        double r80163 = r80161 * r80162;
        double r80164 = u2;
        double r80165 = r80163 * r80164;
        double r80166 = cos(r80165);
        double r80167 = r80160 * r80166;
        double r80168 = r80167 + r80158;
        return r80168;
}
↓
double f(double u1, double u2) {
        double r80169 = -2.0;
        double r80170 = u1;
        double r80171 = log(r80170);
        double r80172 = r80169 * r80171;
        double r80173 = 0.5;
        double r80174 = pow(r80172, r80173);
        double r80175 = 1.0;
        double r80176 = r80174 * r80175;
        double r80177 = 6.0;
        double r80178 = r80176 / r80177;
        double r80179 = 2.0;
        double r80180 = atan2(1.0, 0.0);
        double r80181 = r80179 * r80180;
        double r80182 = u2;
        double r80183 = r80181 * r80182;
        double r80184 = cos(r80183);
        double r80185 = r80178 * r80184;
        double r80186 = r80185 + r80173;
        return r80186;
}

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 add-sqr-sqrt0.4

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

  4. Applied associate-*l*0.3

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

  6. Applied sqrt-div0.8

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

  7. Applied associate-*l/0.5

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

  8. Applied sqrt-div1.1

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

  9. Applied frac-times0.8

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

  10. Simplified0.8

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

  11. Simplified0.3

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

  12. Final simplification0.3

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

Reproduce

herbie shell --seed 2019294 
(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))