Migdal et al, Equation (64)

Average Error: 0.5 → 0.4

Time: 12.9s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

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

C

double f(double a1, double a2, double th) {
        double r104400 = th;
        double r104401 = cos(r104400);
        double r104402 = 2.0;
        double r104403 = sqrt(r104402);
        double r104404 = r104401 / r104403;
        double r104405 = a1;
        double r104406 = r104405 * r104405;
        double r104407 = r104404 * r104406;
        double r104408 = a2;
        double r104409 = r104408 * r104408;
        double r104410 = r104404 * r104409;
        double r104411 = r104407 + r104410;
        return r104411;
}

↓

double f(double a1, double a2, double th) {
        double r104412 = 1.0;
        double r104413 = 2.0;
        double r104414 = sqrt(r104413);
        double r104415 = sqrt(r104414);
        double r104416 = sqrt(r104415);
        double r104417 = r104412 / r104416;
        double r104418 = r104417 / r104416;
        double r104419 = a1;
        double r104420 = th;
        double r104421 = cos(r104420);
        double r104422 = r104421 / r104415;
        double r104423 = r104422 * r104419;
        double r104424 = r104419 * r104423;
        double r104425 = r104418 * r104424;
        double r104426 = a2;
        double r104427 = r104421 * r104426;
        double r104428 = r104427 / r104414;
        double r104429 = r104428 * r104426;
        double r104430 = r104425 + r104429;
        return r104430;
}

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 associate-*r* 0.5

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

5. Applied add-sqr-sqrt 0.5

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

6. Applied sqrt-prod 0.5

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

7. Applied associate-/r* 0.5

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

9. Applied associate-*l/ 0.5

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

11. Applied add-sqr-sqrt 0.5

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

12. Applied sqrt-prod 0.5

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

13. Applied sqrt-prod 0.5

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

14. Applied add-sqr-sqrt 0.5

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

15. Applied sqrt-prod 0.5

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

16. Applied sqrt-prod 0.5

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

17. Applied *-un-lft-identity 0.5

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

18. Applied times-frac 0.5

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

19. Applied times-frac 0.5

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

20. Applied associate-*l* 0.5

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

21. Simplified 0.4

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

22. Final simplification 0.4

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

Reproduce

herbie shell --seed 2020047 
(FPCore (a1 a2 th)
  :name "Migdal et al, Equation (64)"
  :precision binary64
  (+ (* (/ (cos th) (sqrt 2)) (* a1 a1)) (* (/ (cos th) (sqrt 2)) (* a2 a2))))