Migdal et al, Equation (64)

Average Error: 0.5 → 0.5

Time: 7.9s

Precision: 64

Math

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

↓

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

C

double f(double a1, double a2, double th) {
        double r97327 = th;
        double r97328 = cos(r97327);
        double r97329 = 2.0;
        double r97330 = sqrt(r97329);
        double r97331 = r97328 / r97330;
        double r97332 = a1;
        double r97333 = r97332 * r97332;
        double r97334 = r97331 * r97333;
        double r97335 = a2;
        double r97336 = r97335 * r97335;
        double r97337 = r97331 * r97336;
        double r97338 = r97334 + r97337;
        return r97338;
}

↓

double f(double a1, double a2, double th) {
        double r97339 = a1;
        double r97340 = fabs(r97339);
        double r97341 = th;
        double r97342 = cos(r97341);
        double r97343 = r97340 * r97342;
        double r97344 = 2.0;
        double r97345 = pow(r97339, r97344);
        double r97346 = sqrt(r97345);
        double r97347 = 2.0;
        double r97348 = sqrt(r97347);
        double r97349 = r97346 / r97348;
        double r97350 = r97343 * r97349;
        double r97351 = a2;
        double r97352 = fabs(r97351);
        double r97353 = r97352 / r97348;
        double r97354 = r97352 * r97353;
        double r97355 = r97342 * r97354;
        double r97356 = r97350 + r97355;
        return r97356;
}

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. Using strategy rm

3. Applied div-inv 0.5

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

4. Applied associate-*l* 0.5

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

5. Simplified 0.5

   \[\leadsto \frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \cos th \cdot \color{blue}{\frac{{a2}^{2}}{\sqrt{2}}}\]
6. Using strategy rm

7. Applied div-inv 0.5

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

8. Applied associate-*l* 0.5

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

9. Simplified 0.5

   \[\leadsto \cos th \cdot \color{blue}{\frac{{a1}^{2}}{\sqrt{2}}} + \cos th \cdot \frac{{a2}^{2}}{\sqrt{2}}\]
10. Using strategy rm

11. Applied *-un-lft-identity 0.5

    \[\leadsto \cos th \cdot \frac{{a1}^{2}}{\sqrt{\color{blue}{1 \cdot 2}}} + \cos th \cdot \frac{{a2}^{2}}{\sqrt{2}}\]

12. Applied sqrt-prod 0.5

    \[\leadsto \cos th \cdot \frac{{a1}^{2}}{\color{blue}{\sqrt{1} \cdot \sqrt{2}}} + \cos th \cdot \frac{{a2}^{2}}{\sqrt{2}}\]

13. Applied add-sqr-sqrt 0.5

    \[\leadsto \cos th \cdot \frac{\color{blue}{\sqrt{{a1}^{2}} \cdot \sqrt{{a1}^{2}}}}{\sqrt{1} \cdot \sqrt{2}} + \cos th \cdot \frac{{a2}^{2}}{\sqrt{2}}\]

14. Applied times-frac 0.5

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

15. Applied associate-*r* 0.5

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

16. Simplified 0.5

    \[\leadsto \color{blue}{\left(\left|a1\right| \cdot \cos th\right)} \cdot \frac{\sqrt{{a1}^{2}}}{\sqrt{2}} + \cos th \cdot \frac{{a2}^{2}}{\sqrt{2}}\]
17. Using strategy rm

18. Applied *-un-lft-identity 0.5

    \[\leadsto \left(\left|a1\right| \cdot \cos th\right) \cdot \frac{\sqrt{{a1}^{2}}}{\sqrt{2}} + \cos th \cdot \frac{{a2}^{2}}{\sqrt{\color{blue}{1 \cdot 2}}}\]

19. Applied sqrt-prod 0.5

    \[\leadsto \left(\left|a1\right| \cdot \cos th\right) \cdot \frac{\sqrt{{a1}^{2}}}{\sqrt{2}} + \cos th \cdot \frac{{a2}^{2}}{\color{blue}{\sqrt{1} \cdot \sqrt{2}}}\]

20. Applied add-sqr-sqrt 0.5

    \[\leadsto \left(\left|a1\right| \cdot \cos th\right) \cdot \frac{\sqrt{{a1}^{2}}}{\sqrt{2}} + \cos th \cdot \frac{\color{blue}{\sqrt{{a2}^{2}} \cdot \sqrt{{a2}^{2}}}}{\sqrt{1} \cdot \sqrt{2}}\]

21. Applied times-frac 0.5

    \[\leadsto \left(\left|a1\right| \cdot \cos th\right) \cdot \frac{\sqrt{{a1}^{2}}}{\sqrt{2}} + \cos th \cdot \color{blue}{\left(\frac{\sqrt{{a2}^{2}}}{\sqrt{1}} \cdot \frac{\sqrt{{a2}^{2}}}{\sqrt{2}}\right)}\]

22. Simplified 0.5

    \[\leadsto \left(\left|a1\right| \cdot \cos th\right) \cdot \frac{\sqrt{{a1}^{2}}}{\sqrt{2}} + \cos th \cdot \left(\color{blue}{\left|a2\right|} \cdot \frac{\sqrt{{a2}^{2}}}{\sqrt{2}}\right)\]

23. Simplified 0.5

    \[\leadsto \left(\left|a1\right| \cdot \cos th\right) \cdot \frac{\sqrt{{a1}^{2}}}{\sqrt{2}} + \cos th \cdot \left(\left|a2\right| \cdot \color{blue}{\frac{\left|a2\right|}{\sqrt{2}}}\right)\]

24. Final simplification 0.5

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

Reproduce

herbie shell --seed 2020033 
(FPCore (a1 a2 th)
  :name "Migdal et al, Equation (64)"
  :precision binary64
  (+ (* (/ (cos th) (sqrt 2)) (* a1 a1)) (* (/ (cos th) (sqrt 2)) (* a2 a2))))