Average Error: 0.4 → 0.4
Time: 32.0s
Precision: 64
\[0 \le u1 \le 1 \land 0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\mathsf{fma}\left(\cos \left(\left(\pi \cdot 2\right) \cdot u2\right), \left(\sqrt{\frac{1}{6}} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \sqrt{\frac{1}{6}}, 0.5\right)\]
\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(\cos \left(\left(\pi \cdot 2\right) \cdot u2\right), \left(\sqrt{\frac{1}{6}} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \sqrt{\frac{1}{6}}, 0.5\right)
double f(double u1, double u2) {
        double r1020546 = 1.0;
        double r1020547 = 6.0;
        double r1020548 = r1020546 / r1020547;
        double r1020549 = -2.0;
        double r1020550 = u1;
        double r1020551 = log(r1020550);
        double r1020552 = r1020549 * r1020551;
        double r1020553 = 0.5;
        double r1020554 = pow(r1020552, r1020553);
        double r1020555 = r1020548 * r1020554;
        double r1020556 = 2.0;
        double r1020557 = atan2(1.0, 0.0);
        double r1020558 = r1020556 * r1020557;
        double r1020559 = u2;
        double r1020560 = r1020558 * r1020559;
        double r1020561 = cos(r1020560);
        double r1020562 = r1020555 * r1020561;
        double r1020563 = r1020562 + r1020553;
        return r1020563;
}
↓
double f(double u1, double u2) {
        double r1020564 = atan2(1.0, 0.0);
        double r1020565 = 2.0;
        double r1020566 = r1020564 * r1020565;
        double r1020567 = u2;
        double r1020568 = r1020566 * r1020567;
        double r1020569 = cos(r1020568);
        double r1020570 = 0.16666666666666666;
        double r1020571 = sqrt(r1020570);
        double r1020572 = -2.0;
        double r1020573 = u1;
        double r1020574 = log(r1020573);
        double r1020575 = r1020572 * r1020574;
        double r1020576 = 0.5;
        double r1020577 = pow(r1020575, r1020576);
        double r1020578 = r1020571 * r1020577;
        double r1020579 = r1020578 * r1020571;
        double r1020580 = fma(r1020569, r1020579, r1020576);
        return r1020580;
}

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

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

  4. Applied add-sqr-sqrt0.4

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

  5. Applied associate-*l*0.4

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

  6. Final simplification0.4

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

Reproduce

herbie shell --seed 2019151 +o rules:numerics
(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))