Migdal et al, Equation (64)

Average Error: 0.5 → 0.4

Time: 27.3s

Precision: 64

Math

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

↓

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

C

double f(double a1, double a2, double th) {
        double r101360 = th;
        double r101361 = cos(r101360);
        double r101362 = 2.0;
        double r101363 = sqrt(r101362);
        double r101364 = r101361 / r101363;
        double r101365 = a1;
        double r101366 = r101365 * r101365;
        double r101367 = r101364 * r101366;
        double r101368 = a2;
        double r101369 = r101368 * r101368;
        double r101370 = r101364 * r101369;
        double r101371 = r101367 + r101370;
        return r101371;
}

↓

double f(double a1, double a2, double th) {
        double r101372 = 1.0;
        double r101373 = 2.0;
        double r101374 = sqrt(r101373);
        double r101375 = sqrt(r101374);
        double r101376 = sqrt(r101375);
        double r101377 = r101372 / r101376;
        double r101378 = r101377 / r101376;
        double r101379 = th;
        double r101380 = cos(r101379);
        double r101381 = a1;
        double r101382 = r101381 * r101381;
        double r101383 = a2;
        double r101384 = r101383 * r101383;
        double r101385 = r101382 + r101384;
        double r101386 = r101380 * r101385;
        double r101387 = r101386 / r101375;
        double r101388 = r101378 * r101387;
        return r101388;
}

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

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

4. Applied add-sqr-sqrt 0.5

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

5. Applied sqrt-prod 0.5

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

6. Applied associate-/r* 0.5

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

8. Applied add-sqr-sqrt 0.5

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

9. Applied sqrt-prod 0.5

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

10. Applied sqrt-prod 0.5

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

11. Applied add-sqr-sqrt 0.5

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

12. Applied sqrt-prod 0.5

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

13. Applied sqrt-prod 0.5

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

14. Applied *-un-lft-identity 0.5

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

15. Applied times-frac 0.5

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

16. Applied times-frac 0.6

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

17. Applied associate-*l* 0.6

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

18. Simplified 0.4

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

19. Final simplification 0.4

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

Reproduce

herbie shell --seed 2019303 
(FPCore (a1 a2 th)
  :name "Migdal et al, Equation (64)"
  :precision binary64
  (+ (* (/ (cos th) (sqrt 2)) (* a1 a1)) (* (/ (cos th) (sqrt 2)) (* a2 a2))))