Average Error: 0.4 → 0.3
Time: 12.9s
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 r77001 = 1.0;
        double r77002 = 6.0;
        double r77003 = r77001 / r77002;
        double r77004 = -2.0;
        double r77005 = u1;
        double r77006 = log(r77005);
        double r77007 = r77004 * r77006;
        double r77008 = 0.5;
        double r77009 = pow(r77007, r77008);
        double r77010 = r77003 * r77009;
        double r77011 = 2.0;
        double r77012 = atan2(1.0, 0.0);
        double r77013 = r77011 * r77012;
        double r77014 = u2;
        double r77015 = r77013 * r77014;
        double r77016 = cos(r77015);
        double r77017 = r77010 * r77016;
        double r77018 = r77017 + r77008;
        return r77018;
}
↓
double f(double u1, double u2) {
        double r77019 = 1.0;
        double r77020 = -2.0;
        double r77021 = u1;
        double r77022 = log(r77021);
        double r77023 = r77020 * r77022;
        double r77024 = 0.5;
        double r77025 = pow(r77023, r77024);
        double r77026 = 6.0;
        double r77027 = r77025 / r77026;
        double r77028 = r77019 * r77027;
        double r77029 = 2.0;
        double r77030 = atan2(1.0, 0.0);
        double r77031 = r77029 * r77030;
        double r77032 = u2;
        double r77033 = r77031 * r77032;
        double r77034 = cos(r77033);
        double r77035 = r77028 * r77034;
        double r77036 = r77035 + r77024;
        return r77036;
}

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 2020089 
(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))