Migdal et al, Equation (64)

Average Error: 0.5 → 0.5

Time: 28.3s

Precision: 64

Math

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

↓

\[\frac{\cos th \cdot \left(a1 \cdot a1\right)}{\sqrt{2}} + \left(\frac{\frac{\cos th}{\sqrt[3]{\sqrt{\sqrt{\sqrt{2}}}} \cdot \sqrt[3]{\sqrt{\sqrt{\sqrt{2}}}}}}{\sqrt{\sqrt{\sqrt{2}}}} \cdot \frac{\frac{1}{\sqrt{\sqrt{2}}}}{\sqrt[3]{\sqrt{\sqrt{\sqrt{2}}}}}\right) \cdot \left(a2 \cdot a2\right)\]

C

double f(double a1, double a2, double th) {
        double r83447 = th;
        double r83448 = cos(r83447);
        double r83449 = 2.0;
        double r83450 = sqrt(r83449);
        double r83451 = r83448 / r83450;
        double r83452 = a1;
        double r83453 = r83452 * r83452;
        double r83454 = r83451 * r83453;
        double r83455 = a2;
        double r83456 = r83455 * r83455;
        double r83457 = r83451 * r83456;
        double r83458 = r83454 + r83457;
        return r83458;
}

↓

double f(double a1, double a2, double th) {
        double r83459 = th;
        double r83460 = cos(r83459);
        double r83461 = a1;
        double r83462 = r83461 * r83461;
        double r83463 = r83460 * r83462;
        double r83464 = 2.0;
        double r83465 = sqrt(r83464);
        double r83466 = r83463 / r83465;
        double r83467 = sqrt(r83465);
        double r83468 = sqrt(r83467);
        double r83469 = cbrt(r83468);
        double r83470 = r83469 * r83469;
        double r83471 = r83460 / r83470;
        double r83472 = r83471 / r83468;
        double r83473 = 1.0;
        double r83474 = r83473 / r83467;
        double r83475 = r83474 / r83469;
        double r83476 = r83472 * r83475;
        double r83477 = a2;
        double r83478 = r83477 * r83477;
        double r83479 = r83476 * r83478;
        double r83480 = r83466 + r83479;
        return r83480;
}

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 add-sqr-sqrt 0.5

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

4. Applied sqrt-prod 0.5

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

5. Applied associate-/r* 0.5

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

7. Applied associate-*l/ 0.5

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

9. Applied add-sqr-sqrt 0.5

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

10. Applied sqrt-prod 0.5

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

11. Applied sqrt-prod 0.5

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

12. Applied *-un-lft-identity 0.5

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

13. Applied sqrt-prod 0.5

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

14. Applied sqrt-prod 0.5

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

15. Applied *-un-lft-identity 0.5

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

16. Applied times-frac 0.5

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

17. Applied times-frac 0.5

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

18. Simplified 0.5

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

20. Applied add-cube-cbrt 0.5

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

21. Applied div-inv 0.5

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

22. Applied times-frac 0.5

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

23. Applied associate-*r* 0.5

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

24. Simplified 0.5

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

25. Final simplification 0.5

    \[\leadsto \frac{\cos th \cdot \left(a1 \cdot a1\right)}{\sqrt{2}} + \left(\frac{\frac{\cos th}{\sqrt[3]{\sqrt{\sqrt{\sqrt{2}}}} \cdot \sqrt[3]{\sqrt{\sqrt{\sqrt{2}}}}}}{\sqrt{\sqrt{\sqrt{2}}}} \cdot \frac{\frac{1}{\sqrt{\sqrt{2}}}}{\sqrt[3]{\sqrt{\sqrt{\sqrt{2}}}}}\right) \cdot \left(a2 \cdot a2\right)\]

Reproduce

herbie shell --seed 2019195 +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))))