Average Error: 0.4 → 0.3
Time: 10.8s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\mathsf{fma}\left(1 \cdot \frac{{\left(-2 \cdot \log u1\right)}^{0.5}}{6}, \cos \left(\left(2 \cdot \pi\right) \cdot u2\right), 0.5\right)\]
\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
↓
\mathsf{fma}\left(1 \cdot \frac{{\left(-2 \cdot \log u1\right)}^{0.5}}{6}, \cos \left(\left(2 \cdot \pi\right) \cdot u2\right), 0.5\right)
double f(double u1, double u2) {
        double r67453 = 1.0;
        double r67454 = 6.0;
        double r67455 = r67453 / r67454;
        double r67456 = -2.0;
        double r67457 = u1;
        double r67458 = log(r67457);
        double r67459 = r67456 * r67458;
        double r67460 = 0.5;
        double r67461 = pow(r67459, r67460);
        double r67462 = r67455 * r67461;
        double r67463 = 2.0;
        double r67464 = atan2(1.0, 0.0);
        double r67465 = r67463 * r67464;
        double r67466 = u2;
        double r67467 = r67465 * r67466;
        double r67468 = cos(r67467);
        double r67469 = r67462 * r67468;
        double r67470 = r67469 + r67460;
        return r67470;
}
↓
double f(double u1, double u2) {
        double r67471 = 1.0;
        double r67472 = -2.0;
        double r67473 = u1;
        double r67474 = log(r67473);
        double r67475 = r67472 * r67474;
        double r67476 = 0.5;
        double r67477 = pow(r67475, r67476);
        double r67478 = 6.0;
        double r67479 = r67477 / r67478;
        double r67480 = r67471 * r67479;
        double r67481 = 2.0;
        double r67482 = atan2(1.0, 0.0);
        double r67483 = r67481 * r67482;
        double r67484 = u2;
        double r67485 = r67483 * r67484;
        double r67486 = cos(r67485);
        double r67487 = fma(r67480, r67486, r67476);
        return r67487;
}

Error

Bits error versus u1

Bits error versus u2

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

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

  4. Applied div-inv0.4

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

  5. Applied associate-*l*0.4

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

  6. Simplified0.3

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

  7. Final simplification0.3

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

Reproduce

herbie shell --seed 2019362 +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))