Average Error: 0.4 → 0.4
Time: 38.2s
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 r1609684 = 1.0;
        double r1609685 = 6.0;
        double r1609686 = r1609684 / r1609685;
        double r1609687 = -2.0;
        double r1609688 = u1;
        double r1609689 = log(r1609688);
        double r1609690 = r1609687 * r1609689;
        double r1609691 = 0.5;
        double r1609692 = pow(r1609690, r1609691);
        double r1609693 = r1609686 * r1609692;
        double r1609694 = 2.0;
        double r1609695 = atan2(1.0, 0.0);
        double r1609696 = r1609694 * r1609695;
        double r1609697 = u2;
        double r1609698 = r1609696 * r1609697;
        double r1609699 = cos(r1609698);
        double r1609700 = r1609693 * r1609699;
        double r1609701 = r1609700 + r1609691;
        return r1609701;
}
↓
double f(double u1, double u2) {
        double r1609702 = atan2(1.0, 0.0);
        double r1609703 = 2.0;
        double r1609704 = r1609702 * r1609703;
        double r1609705 = u2;
        double r1609706 = r1609704 * r1609705;
        double r1609707 = cos(r1609706);
        double r1609708 = 0.16666666666666666;
        double r1609709 = sqrt(r1609708);
        double r1609710 = -2.0;
        double r1609711 = u1;
        double r1609712 = log(r1609711);
        double r1609713 = r1609710 * r1609712;
        double r1609714 = 0.5;
        double r1609715 = pow(r1609713, r1609714);
        double r1609716 = r1609709 * r1609715;
        double r1609717 = r1609716 * r1609709;
        double r1609718 = fma(r1609707, r1609717, r1609714);
        return r1609718;
}

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 2019139 +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))