Migdal et al, Equation (64)

Average Error: 0.5 → 0.4

Time: 29.8s

Precision: 64

Math

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

↓

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

C

double f(double a1, double a2, double th) {
        double r2558532 = th;
        double r2558533 = cos(r2558532);
        double r2558534 = 2.0;
        double r2558535 = sqrt(r2558534);
        double r2558536 = r2558533 / r2558535;
        double r2558537 = a1;
        double r2558538 = r2558537 * r2558537;
        double r2558539 = r2558536 * r2558538;
        double r2558540 = a2;
        double r2558541 = r2558540 * r2558540;
        double r2558542 = r2558536 * r2558541;
        double r2558543 = r2558539 + r2558542;
        return r2558543;
}

↓

double f(double a1, double a2, double th) {
        double r2558544 = 1.0;
        double r2558545 = 2.0;
        double r2558546 = sqrt(r2558545);
        double r2558547 = sqrt(r2558546);
        double r2558548 = sqrt(r2558547);
        double r2558549 = r2558544 / r2558548;
        double r2558550 = r2558549 / r2558548;
        double r2558551 = a2;
        double r2558552 = th;
        double r2558553 = cos(r2558552);
        double r2558554 = r2558553 / r2558547;
        double r2558555 = r2558554 * r2558551;
        double r2558556 = r2558551 * r2558555;
        double r2558557 = r2558550 * r2558556;
        double r2558558 = a1;
        double r2558559 = r2558558 * r2558553;
        double r2558560 = r2558558 / r2558546;
        double r2558561 = r2558559 * r2558560;
        double r2558562 = r2558557 + r2558561;
        return r2558562;
}

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. Taylor expanded around inf 0.5

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

3. Simplified 0.5

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

5. Applied add-sqr-sqrt 0.5

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

6. Applied sqrt-prod 0.5

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

7. Applied associate-/r* 0.4

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

9. Applied add-sqr-sqrt 0.4

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

10. Applied sqrt-prod 0.4

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

11. Applied sqrt-prod 0.4

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

12. Applied add-sqr-sqrt 0.4

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

13. Applied sqrt-prod 0.4

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

14. Applied sqrt-prod 0.4

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

15. Applied *-un-lft-identity 0.4

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

16. Applied times-frac 0.5

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

17. Applied times-frac 0.5

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

18. Applied associate-*l* 0.5

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

19. Simplified 0.4

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

21. Applied *-un-lft-identity 0.4

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

22. Applied sqrt-prod 0.4

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

23. Applied times-frac 0.4

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

24. Simplified 0.4

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

25. Final simplification 0.4

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

Reproduce

herbie shell --seed 2019170 
(FPCore (a1 a2 th)
  :name "Migdal et al, Equation (64)"
  (+ (* (/ (cos th) (sqrt 2.0)) (* a1 a1)) (* (/ (cos th) (sqrt 2.0)) (* a2 a2))))