Average Error: 0.4 → 0.4
Time: 32.6s
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 r1332551 = 1.0;
        double r1332552 = 6.0;
        double r1332553 = r1332551 / r1332552;
        double r1332554 = -2.0;
        double r1332555 = u1;
        double r1332556 = log(r1332555);
        double r1332557 = r1332554 * r1332556;
        double r1332558 = 0.5;
        double r1332559 = pow(r1332557, r1332558);
        double r1332560 = r1332553 * r1332559;
        double r1332561 = 2.0;
        double r1332562 = atan2(1.0, 0.0);
        double r1332563 = r1332561 * r1332562;
        double r1332564 = u2;
        double r1332565 = r1332563 * r1332564;
        double r1332566 = cos(r1332565);
        double r1332567 = r1332560 * r1332566;
        double r1332568 = r1332567 + r1332558;
        return r1332568;
}
↓
double f(double u1, double u2) {
        double r1332569 = atan2(1.0, 0.0);
        double r1332570 = 2.0;
        double r1332571 = r1332569 * r1332570;
        double r1332572 = u2;
        double r1332573 = r1332571 * r1332572;
        double r1332574 = cos(r1332573);
        double r1332575 = 0.16666666666666666;
        double r1332576 = sqrt(r1332575);
        double r1332577 = -2.0;
        double r1332578 = u1;
        double r1332579 = log(r1332578);
        double r1332580 = r1332577 * r1332579;
        double r1332581 = 0.5;
        double r1332582 = pow(r1332580, r1332581);
        double r1332583 = r1332576 * r1332582;
        double r1332584 = r1332583 * r1332576;
        double r1332585 = fma(r1332574, r1332584, r1332581);
        return r1332585;
}

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 2019146 +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))