Migdal et al, Equation (64)

Average Error: 0.5 → 0.4

Time: 26.4s

Precision: 64

Math

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

↓

\[\frac{\mathsf{hypot}\left(a1, a2\right) \cdot \cos th}{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}} \cdot \frac{\mathsf{hypot}\left(a1, a2\right)}{\sqrt[3]{\sqrt{2}}}\]

C

double f(double a1, double a2, double th) {
        double r93458 = th;
        double r93459 = cos(r93458);
        double r93460 = 2.0;
        double r93461 = sqrt(r93460);
        double r93462 = r93459 / r93461;
        double r93463 = a1;
        double r93464 = r93463 * r93463;
        double r93465 = r93462 * r93464;
        double r93466 = a2;
        double r93467 = r93466 * r93466;
        double r93468 = r93462 * r93467;
        double r93469 = r93465 + r93468;
        return r93469;
}

↓

double f(double a1, double a2, double th) {
        double r93470 = a1;
        double r93471 = a2;
        double r93472 = hypot(r93470, r93471);
        double r93473 = th;
        double r93474 = cos(r93473);
        double r93475 = r93472 * r93474;
        double r93476 = 2.0;
        double r93477 = sqrt(r93476);
        double r93478 = cbrt(r93477);
        double r93479 = r93478 * r93478;
        double r93480 = r93475 / r93479;
        double r93481 = r93472 / r93478;
        double r93482 = r93480 * r93481;
        return r93482;
}

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

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

4. Applied add-cube-cbrt 0.5

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

5. Applied add-sqr-sqrt 0.5

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

6. Applied times-frac 0.5

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

7. Applied associate-*r* 0.5

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

8. Simplified 0.5

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

10. Applied *-un-lft-identity 0.5

    \[\leadsto \frac{\cos th}{\color{blue}{1 \cdot \frac{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}{\mathsf{hypot}\left(a1, a2\right)}}} \cdot \frac{\sqrt{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}}{\sqrt[3]{\sqrt{2}}}\]

11. Applied *-un-lft-identity 0.5

    \[\leadsto \frac{\color{blue}{1 \cdot \cos th}}{1 \cdot \frac{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}{\mathsf{hypot}\left(a1, a2\right)}} \cdot \frac{\sqrt{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}}{\sqrt[3]{\sqrt{2}}}\]

12. Applied times-frac 0.5

    \[\leadsto \color{blue}{\left(\frac{1}{1} \cdot \frac{\cos th}{\frac{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}{\mathsf{hypot}\left(a1, a2\right)}}\right)} \cdot \frac{\sqrt{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}}{\sqrt[3]{\sqrt{2}}}\]

13. Applied associate-*l* 0.5

    \[\leadsto \color{blue}{\frac{1}{1} \cdot \left(\frac{\cos th}{\frac{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}{\mathsf{hypot}\left(a1, a2\right)}} \cdot \frac{\sqrt{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}}{\sqrt[3]{\sqrt{2}}}\right)}\]

14. Simplified 0.4

    \[\leadsto \frac{1}{1} \cdot \color{blue}{\left(\frac{\mathsf{hypot}\left(a1, a2\right)}{\sqrt[3]{\sqrt{2}}} \cdot \frac{\cos th \cdot \mathsf{hypot}\left(a1, a2\right)}{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}\right)}\]

15. Final simplification 0.4

    \[\leadsto \frac{\mathsf{hypot}\left(a1, a2\right) \cdot \cos th}{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}} \cdot \frac{\mathsf{hypot}\left(a1, a2\right)}{\sqrt[3]{\sqrt{2}}}\]

Reproduce

herbie shell --seed 2019194 +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))))