Average Error: 0.4 → 0.4
Time: 11.0s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\mathsf{fma}\left(0.166666666666666657 \cdot {\left({-1}^{1} \cdot \left({-2}^{1} \cdot {\left(\log \left(\frac{1}{u1}\right)\right)}^{1}\right)\right)}^{0.5}, \cos \left(\left(\left(2 \cdot \pi\right) \cdot \sqrt{u2}\right) \cdot \sqrt{u2}\right), 0.5\right)\]
\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(0.166666666666666657 \cdot {\left({-1}^{1} \cdot \left({-2}^{1} \cdot {\left(\log \left(\frac{1}{u1}\right)\right)}^{1}\right)\right)}^{0.5}, \cos \left(\left(\left(2 \cdot \pi\right) \cdot \sqrt{u2}\right) \cdot \sqrt{u2}\right), 0.5\right)
double f(double u1, double u2) {
        double r64 = 1.0;
        double r65 = 6.0;
        double r66 = r64 / r65;
        double r67 = -2.0;
        double r68 = u1;
        double r69 = log(r68);
        double r70 = r67 * r69;
        double r71 = 0.5;
        double r72 = pow(r70, r71);
        double r73 = r66 * r72;
        double r74 = 2.0;
        double r75 = atan2(1.0, 0.0);
        double r76 = r74 * r75;
        double r77 = u2;
        double r78 = r76 * r77;
        double r79 = cos(r78);
        double r80 = r73 * r79;
        double r81 = r80 + r71;
        return r81;
}
↓
double f(double u1, double u2) {
        double r82 = 0.16666666666666666;
        double r83 = -1.0;
        double r84 = 1.0;
        double r85 = pow(r83, r84);
        double r86 = -2.0;
        double r87 = pow(r86, r84);
        double r88 = 1.0;
        double r89 = u1;
        double r90 = r88 / r89;
        double r91 = log(r90);
        double r92 = pow(r91, r84);
        double r93 = r87 * r92;
        double r94 = r85 * r93;
        double r95 = 0.5;
        double r96 = pow(r94, r95);
        double r97 = r82 * r96;
        double r98 = 2.0;
        double r99 = atan2(1.0, 0.0);
        double r100 = r98 * r99;
        double r101 = u2;
        double r102 = sqrt(r101);
        double r103 = r100 * r102;
        double r104 = r103 * r102;
        double r105 = cos(r104);
        double r106 = fma(r97, r105, r95);
        return r106;
}

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 associate-*l/0.3

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

  6. Applied add-sqr-sqrt0.3

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

  7. Applied associate-*r*0.3

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

  9. Applied clear-num0.3

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

  10. Taylor expanded around inf 0.4

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

  11. Final simplification0.4

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

Reproduce

herbie shell --seed 2020025 +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))