Average Error: 0.4 → 0.3
Time: 10.2s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\log \left(e^{\frac{1 \cdot {\left(-2 \cdot \log u1\right)}^{0.5}}{6} \cdot \cos \left(\left(\left(2 \cdot \pi\right) \cdot \sqrt{u2}\right) \cdot \sqrt{u2}\right) + 0.5}\right)\]
\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
↓
\log \left(e^{\frac{1 \cdot {\left(-2 \cdot \log u1\right)}^{0.5}}{6} \cdot \cos \left(\left(\left(2 \cdot \pi\right) \cdot \sqrt{u2}\right) \cdot \sqrt{u2}\right) + 0.5}\right)
double f(double u1, double u2) {
        double r64 = 1.0;
        double r65 = 6.0;
        double r66 = r64 / r65;
        double r67 = -2.0;
        double r68 = u1;
        double r69 = log(r68);
        double r70 = r67 * r69;
        double r71 = 0.5;
        double r72 = pow(r70, r71);
        double r73 = r66 * r72;
        double r74 = 2.0;
        double r75 = atan2(1.0, 0.0);
        double r76 = r74 * r75;
        double r77 = u2;
        double r78 = r76 * r77;
        double r79 = cos(r78);
        double r80 = r73 * r79;
        double r81 = r80 + r71;
        return r81;
}
↓
double f(double u1, double u2) {
        double r82 = 1.0;
        double r83 = -2.0;
        double r84 = u1;
        double r85 = log(r84);
        double r86 = r83 * r85;
        double r87 = 0.5;
        double r88 = pow(r86, r87);
        double r89 = r82 * r88;
        double r90 = 6.0;
        double r91 = r89 / r90;
        double r92 = 2.0;
        double r93 = atan2(1.0, 0.0);
        double r94 = r92 * r93;
        double r95 = u2;
        double r96 = sqrt(r95);
        double r97 = r94 * r96;
        double r98 = r97 * r96;
        double r99 = cos(r98);
        double r100 = r91 * r99;
        double r101 = r100 + r87;
        double r102 = exp(r101);
        double r103 = log(r102);
        return r103;
}

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. Using strategy rm

  5. Applied add-log-exp0.3

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

  6. Applied add-log-exp0.3

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

  7. Applied sum-log0.3

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

  8. Simplified0.3

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

  10. Applied add-sqr-sqrt0.3

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

  11. Applied associate-*r*0.3

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

  12. Final simplification0.3

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

Reproduce

herbie shell --seed 2020025 
(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))