Average Error: 0.4 → 0.3
Time: 11.3s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\frac{1 \cdot {\left(-2 \cdot \log u1\right)}^{0.5}}{6} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
\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{1 \cdot {\left(-2 \cdot \log u1\right)}^{0.5}}{6} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5
double f(double u1, double u2) {
        double r69476 = 1.0;
        double r69477 = 6.0;
        double r69478 = r69476 / r69477;
        double r69479 = -2.0;
        double r69480 = u1;
        double r69481 = log(r69480);
        double r69482 = r69479 * r69481;
        double r69483 = 0.5;
        double r69484 = pow(r69482, r69483);
        double r69485 = r69478 * r69484;
        double r69486 = 2.0;
        double r69487 = atan2(1.0, 0.0);
        double r69488 = r69486 * r69487;
        double r69489 = u2;
        double r69490 = r69488 * r69489;
        double r69491 = cos(r69490);
        double r69492 = r69485 * r69491;
        double r69493 = r69492 + r69483;
        return r69493;
}
↓
double f(double u1, double u2) {
        double r69494 = 1.0;
        double r69495 = -2.0;
        double r69496 = u1;
        double r69497 = log(r69496);
        double r69498 = r69495 * r69497;
        double r69499 = 0.5;
        double r69500 = pow(r69498, r69499);
        double r69501 = r69494 * r69500;
        double r69502 = 6.0;
        double r69503 = r69501 / r69502;
        double r69504 = 2.0;
        double r69505 = atan2(1.0, 0.0);
        double r69506 = r69504 * r69505;
        double r69507 = u2;
        double r69508 = r69506 * r69507;
        double r69509 = cos(r69508);
        double r69510 = r69503 * r69509;
        double r69511 = r69510 + r69499;
        return r69511;
}

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. Final simplification0.3

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

Reproduce

herbie shell --seed 2020056 +o rules:numerics
(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))