Average Error: 0.4 → 0.4
Time: 17.6s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\mathsf{fma}\left({\left({\left(\log u1\right)}^{1} \cdot {-2}^{1}\right)}^{0.5} \cdot 0.166666666666666657, \cos \left(\left(2 \cdot \pi\right) \cdot u2\right), 0.5\right)\]
\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({\left({\left(\log u1\right)}^{1} \cdot {-2}^{1}\right)}^{0.5} \cdot 0.166666666666666657, \cos \left(\left(2 \cdot \pi\right) \cdot u2\right), 0.5\right)
double f(double u1, double u2) {
        double r73542 = 1.0;
        double r73543 = 6.0;
        double r73544 = r73542 / r73543;
        double r73545 = -2.0;
        double r73546 = u1;
        double r73547 = log(r73546);
        double r73548 = r73545 * r73547;
        double r73549 = 0.5;
        double r73550 = pow(r73548, r73549);
        double r73551 = r73544 * r73550;
        double r73552 = 2.0;
        double r73553 = atan2(1.0, 0.0);
        double r73554 = r73552 * r73553;
        double r73555 = u2;
        double r73556 = r73554 * r73555;
        double r73557 = cos(r73556);
        double r73558 = r73551 * r73557;
        double r73559 = r73558 + r73549;
        return r73559;
}
↓
double f(double u1, double u2) {
        double r73560 = u1;
        double r73561 = log(r73560);
        double r73562 = 1.0;
        double r73563 = pow(r73561, r73562);
        double r73564 = -2.0;
        double r73565 = pow(r73564, r73562);
        double r73566 = r73563 * r73565;
        double r73567 = 0.5;
        double r73568 = pow(r73566, r73567);
        double r73569 = 0.16666666666666666;
        double r73570 = r73568 * r73569;
        double r73571 = 2.0;
        double r73572 = atan2(1.0, 0.0);
        double r73573 = r73571 * r73572;
        double r73574 = u2;
        double r73575 = r73573 * r73574;
        double r73576 = cos(r73575);
        double r73577 = fma(r73570, r73576, r73567);
        return r73577;
}

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 add-sqr-sqrt0.4

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

  5. Applied associate-*l*0.3

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

  6. Taylor expanded around 0 0.4

    \[\leadsto \mathsf{fma}\left(\color{blue}{{\left({\left(\log u1\right)}^{1} \cdot {-2}^{1}\right)}^{0.5} \cdot {\left(\sqrt{0.166666666666666657}\right)}^{2}}, \cos \left(\left(2 \cdot \pi\right) \cdot u2\right), 0.5\right)\]

  7. Simplified0.4

    \[\leadsto \mathsf{fma}\left(\color{blue}{{\left({\left(\log u1\right)}^{1} \cdot {-2}^{1}\right)}^{0.5} \cdot 0.166666666666666657}, \cos \left(\left(2 \cdot \pi\right) \cdot u2\right), 0.5\right)\]

  8. Final simplification0.4

    \[\leadsto \mathsf{fma}\left({\left({\left(\log u1\right)}^{1} \cdot {-2}^{1}\right)}^{0.5} \cdot 0.166666666666666657, \cos \left(\left(2 \cdot \pi\right) \cdot u2\right), 0.5\right)\]

Reproduce

herbie shell --seed 2020042 +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))