normal distribution

Average Error: 0.4 → 0.3

Time: 18.7s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]

Math

\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]

↓

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

C

double f(double u1, double u2) {
        double r64802 = 1.0;
        double r64803 = 6.0;
        double r64804 = r64802 / r64803;
        double r64805 = -2.0;
        double r64806 = u1;
        double r64807 = log(r64806);
        double r64808 = r64805 * r64807;
        double r64809 = 0.5;
        double r64810 = pow(r64808, r64809);
        double r64811 = r64804 * r64810;
        double r64812 = 2.0;
        double r64813 = atan2(1.0, 0.0);
        double r64814 = r64812 * r64813;
        double r64815 = u2;
        double r64816 = r64814 * r64815;
        double r64817 = cos(r64816);
        double r64818 = r64811 * r64817;
        double r64819 = r64818 + r64809;
        return r64819;
}

↓

double f(double u1, double u2) {
        double r64820 = -2.0;
        double r64821 = u1;
        double r64822 = log(r64821);
        double r64823 = r64820 * r64822;
        double r64824 = 0.5;
        double r64825 = pow(r64823, r64824);
        double r64826 = 1.0;
        double r64827 = r64825 * r64826;
        double r64828 = 6.0;
        double r64829 = r64827 / r64828;
        double r64830 = 2.0;
        double r64831 = atan2(1.0, 0.0);
        double r64832 = r64830 * r64831;
        double r64833 = u2;
        double r64834 = r64832 * r64833;
        double r64835 = cos(r64834);
        double r64836 = fma(r64829, r64835, r64824);
        return r64836;
}

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. Simplified 0.4

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

4. Applied add-sqr-sqrt 0.4

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

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. Using strategy rm

7. Applied sqrt-div 0.8

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

8. Applied associate-*l/ 0.5

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

9. Applied sqrt-div 1.1

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

10. Applied frac-times 0.8

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

11. Simplified 0.8

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

12. Simplified 0.3

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

13. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019350 +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))