Average Error: 0.4 → 0.4
Time: 47.7s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right)\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
\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
↓
\mathsf{log1p}\left(\mathsf{expm1}\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
double f(double u1, double u2) {
        double r6142361 = 1.0;
        double r6142362 = 6.0;
        double r6142363 = r6142361 / r6142362;
        double r6142364 = -2.0;
        double r6142365 = u1;
        double r6142366 = log(r6142365);
        double r6142367 = r6142364 * r6142366;
        double r6142368 = 0.5;
        double r6142369 = pow(r6142367, r6142368);
        double r6142370 = r6142363 * r6142369;
        double r6142371 = 2.0;
        double r6142372 = atan2(1.0, 0.0);
        double r6142373 = r6142371 * r6142372;
        double r6142374 = u2;
        double r6142375 = r6142373 * r6142374;
        double r6142376 = cos(r6142375);
        double r6142377 = r6142370 * r6142376;
        double r6142378 = r6142377 + r6142368;
        return r6142378;
}
↓
double f(double u1, double u2) {
        double r6142379 = 1.0;
        double r6142380 = 6.0;
        double r6142381 = r6142379 / r6142380;
        double r6142382 = -2.0;
        double r6142383 = u1;
        double r6142384 = log(r6142383);
        double r6142385 = r6142382 * r6142384;
        double r6142386 = 0.5;
        double r6142387 = pow(r6142385, r6142386);
        double r6142388 = r6142381 * r6142387;
        double r6142389 = expm1(r6142388);
        double r6142390 = log1p(r6142389);
        double r6142391 = 2.0;
        double r6142392 = atan2(1.0, 0.0);
        double r6142393 = r6142391 * r6142392;
        double r6142394 = u2;
        double r6142395 = r6142393 * r6142394;
        double r6142396 = cos(r6142395);
        double r6142397 = r6142390 * r6142396;
        double r6142398 = r6142397 + r6142386;
        return r6142398;
}

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 log1p-expm1-u0.4

    \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right)\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]

  4. Final simplification0.4

    \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right)\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]

Reproduce

herbie shell --seed 2019173 +o rules:numerics
(FPCore (u1 u2)
  :name "normal distribution"
  :pre (and (<= 0.0 u1 1.0) (<= 0.0 u2 1.0))
  (+ (* (* (/ 1.0 6.0) (pow (* -2.0 (log u1)) 0.5)) (cos (* (* 2.0 PI) u2))) 0.5))