Migdal et al, Equation (64)

Average Error: 0.5 → 0.4

Time: 33.7s

Precision: 64

Math

\[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)\]

↓

\[\left(\cos th \cdot \mathsf{hypot}\left(a2, a1\right)\right) \cdot \frac{\mathsf{hypot}\left(a2, a1\right)}{\sqrt{2}}\]

C

double f(double a1, double a2, double th) {
        double r3103801 = th;
        double r3103802 = cos(r3103801);
        double r3103803 = 2.0;
        double r3103804 = sqrt(r3103803);
        double r3103805 = r3103802 / r3103804;
        double r3103806 = a1;
        double r3103807 = r3103806 * r3103806;
        double r3103808 = r3103805 * r3103807;
        double r3103809 = a2;
        double r3103810 = r3103809 * r3103809;
        double r3103811 = r3103805 * r3103810;
        double r3103812 = r3103808 + r3103811;
        return r3103812;
}

↓

double f(double a1, double a2, double th) {
        double r3103813 = th;
        double r3103814 = cos(r3103813);
        double r3103815 = a2;
        double r3103816 = a1;
        double r3103817 = hypot(r3103815, r3103816);
        double r3103818 = r3103814 * r3103817;
        double r3103819 = 2.0;
        double r3103820 = sqrt(r3103819);
        double r3103821 = r3103817 / r3103820;
        double r3103822 = r3103818 * r3103821;
        return r3103822;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus th

Derivation

1. Initial program 0.5

   \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)\]

2. Simplified 0.5

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \cdot \cos th}{\sqrt{2}}}\]
3. Using strategy rm

4. Applied associate-/l* 0.5

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\frac{\sqrt{2}}{\cos th}}}\]
5. Using strategy rm

6. Applied *-un-lft-identity 0.5

   \[\leadsto \frac{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\frac{\sqrt{2}}{\color{blue}{1 \cdot \cos th}}}\]

7. Applied *-un-lft-identity 0.5

   \[\leadsto \frac{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\frac{\sqrt{\color{blue}{1 \cdot 2}}}{1 \cdot \cos th}}\]

8. Applied sqrt-prod 0.5

   \[\leadsto \frac{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\frac{\color{blue}{\sqrt{1} \cdot \sqrt{2}}}{1 \cdot \cos th}}\]

9. Applied times-frac 0.5

   \[\leadsto \frac{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\color{blue}{\frac{\sqrt{1}}{1} \cdot \frac{\sqrt{2}}{\cos th}}}\]

10. Applied add-sqr-sqrt 0.5

    \[\leadsto \frac{\color{blue}{\sqrt{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)} \cdot \sqrt{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}}}{\frac{\sqrt{1}}{1} \cdot \frac{\sqrt{2}}{\cos th}}\]

11. Applied times-frac 0.5

    \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}}{\frac{\sqrt{1}}{1}} \cdot \frac{\sqrt{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}}{\frac{\sqrt{2}}{\cos th}}}\]

12. Simplified 0.5

    \[\leadsto \color{blue}{\mathsf{hypot}\left(a2, a1\right)} \cdot \frac{\sqrt{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}}{\frac{\sqrt{2}}{\cos th}}\]

13. Simplified 0.5

    \[\leadsto \mathsf{hypot}\left(a2, a1\right) \cdot \color{blue}{\left(\cos th \cdot \frac{\mathsf{hypot}\left(a2, a1\right)}{\sqrt{2}}\right)}\]
14. Using strategy rm

15. Applied associate-*r* 0.4

    \[\leadsto \color{blue}{\left(\mathsf{hypot}\left(a2, a1\right) \cdot \cos th\right) \cdot \frac{\mathsf{hypot}\left(a2, a1\right)}{\sqrt{2}}}\]

16. Final simplification 0.4

    \[\leadsto \left(\cos th \cdot \mathsf{hypot}\left(a2, a1\right)\right) \cdot \frac{\mathsf{hypot}\left(a2, a1\right)}{\sqrt{2}}\]

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(FPCore (a1 a2 th)
  :name "Migdal et al, Equation (64)"
  (+ (* (/ (cos th) (sqrt 2.0)) (* a1 a1)) (* (/ (cos th) (sqrt 2.0)) (* a2 a2))))