Average Error: 0.4 → 0.4
Time: 31.7s
Precision: 64
\[0 \le u1 \le 1 \land 0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\left(\sqrt{\frac{\cos \left(\left(\pi \cdot u2\right) \cdot 2\right)}{6}} \cdot {\left(\log u1 \cdot -2\right)}^{0.5}\right) \cdot \sqrt{\frac{\cos \left(\left(\pi \cdot u2\right) \cdot 2\right)}{6}} + 0.5\]
\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(\sqrt{\frac{\cos \left(\left(\pi \cdot u2\right) \cdot 2\right)}{6}} \cdot {\left(\log u1 \cdot -2\right)}^{0.5}\right) \cdot \sqrt{\frac{\cos \left(\left(\pi \cdot u2\right) \cdot 2\right)}{6}} + 0.5
double f(double u1, double u2) {
        double r1829002 = 1.0;
        double r1829003 = 6.0;
        double r1829004 = r1829002 / r1829003;
        double r1829005 = -2.0;
        double r1829006 = u1;
        double r1829007 = log(r1829006);
        double r1829008 = r1829005 * r1829007;
        double r1829009 = 0.5;
        double r1829010 = pow(r1829008, r1829009);
        double r1829011 = r1829004 * r1829010;
        double r1829012 = 2.0;
        double r1829013 = atan2(1.0, 0.0);
        double r1829014 = r1829012 * r1829013;
        double r1829015 = u2;
        double r1829016 = r1829014 * r1829015;
        double r1829017 = cos(r1829016);
        double r1829018 = r1829011 * r1829017;
        double r1829019 = r1829018 + r1829009;
        return r1829019;
}
↓
double f(double u1, double u2) {
        double r1829020 = atan2(1.0, 0.0);
        double r1829021 = u2;
        double r1829022 = r1829020 * r1829021;
        double r1829023 = 2.0;
        double r1829024 = r1829022 * r1829023;
        double r1829025 = cos(r1829024);
        double r1829026 = 6.0;
        double r1829027 = r1829025 / r1829026;
        double r1829028 = sqrt(r1829027);
        double r1829029 = u1;
        double r1829030 = log(r1829029);
        double r1829031 = -2.0;
        double r1829032 = r1829030 * r1829031;
        double r1829033 = 0.5;
        double r1829034 = pow(r1829032, r1829033);
        double r1829035 = r1829028 * r1829034;
        double r1829036 = r1829035 * r1829028;
        double r1829037 = r1829036 + r1829033;
        return r1829037;
}

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. Simplified0.4

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

  4. Applied add-sqr-sqrt0.4

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

  5. Applied associate-*r*0.4

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

  6. Final simplification0.4

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

Reproduce

herbie shell --seed 2019134 
(FPCore (u1 u2)
  :name "normal distribution"
  :pre (and (<= 0 u1 1) (<= 0 u2 1))
  (+ (* (* (/ 1 6) (pow (* -2 (log u1)) 0.5)) (cos (* (* 2 PI) u2))) 0.5))