Migdal et al, Equation (64)

Average Error: 0.5 → 0.4

Time: 28.8s

Precision: 64

Math

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

↓

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

C

double f(double a1, double a2, double th) {
        double r85631 = th;
        double r85632 = cos(r85631);
        double r85633 = 2.0;
        double r85634 = sqrt(r85633);
        double r85635 = r85632 / r85634;
        double r85636 = a1;
        double r85637 = r85636 * r85636;
        double r85638 = r85635 * r85637;
        double r85639 = a2;
        double r85640 = r85639 * r85639;
        double r85641 = r85635 * r85640;
        double r85642 = r85638 + r85641;
        return r85642;
}

↓

double f(double a1, double a2, double th) {
        double r85643 = a2;
        double r85644 = a1;
        double r85645 = r85644 * r85644;
        double r85646 = fma(r85643, r85643, r85645);
        double r85647 = 2.0;
        double r85648 = sqrt(r85647);
        double r85649 = sqrt(r85648);
        double r85650 = sqrt(r85649);
        double r85651 = r85646 / r85650;
        double r85652 = th;
        double r85653 = cos(r85652);
        double r85654 = r85651 * r85653;
        double r85655 = r85650 * r85649;
        double r85656 = r85654 / r85655;
        return r85656;
}

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-sqr-sqrt 0.5

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

5. Applied sqrt-prod 0.6

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

6. Applied associate-/r* 0.5

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

7. Simplified 0.5

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

9. Applied *-un-lft-identity 0.5

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

10. Applied sqrt-prod 0.5

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

11. Applied sqrt-prod 0.5

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

12. Applied add-sqr-sqrt 0.5

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

13. Applied sqrt-prod 0.5

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

14. Applied sqrt-prod 0.5

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

15. Applied *-un-lft-identity 0.5

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

16. Applied times-frac 0.5

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

17. Applied times-frac 0.5

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

18. Simplified 0.5

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

20. Applied frac-times 0.4

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

21. Applied associate-*r/ 0.4

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

22. Simplified 0.4

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

23. Final simplification 0.4

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

Reproduce

herbie shell --seed 2019196 +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))))