Migdal et al, Equation (64)

Average Error: 0.5 → 0.5

Time: 21.9s

Precision: 64

Math

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

↓

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

C

double f(double a1, double a2, double th) {
        double r70567 = th;
        double r70568 = cos(r70567);
        double r70569 = 2.0;
        double r70570 = sqrt(r70569);
        double r70571 = r70568 / r70570;
        double r70572 = a1;
        double r70573 = r70572 * r70572;
        double r70574 = r70571 * r70573;
        double r70575 = a2;
        double r70576 = r70575 * r70575;
        double r70577 = r70571 * r70576;
        double r70578 = r70574 + r70577;
        return r70578;
}

↓

double f(double a1, double a2, double th) {
        double r70579 = th;
        double r70580 = cos(r70579);
        double r70581 = 2.0;
        double r70582 = sqrt(r70581);
        double r70583 = r70580 / r70582;
        double r70584 = a2;
        double r70585 = r70584 * r70584;
        double r70586 = a1;
        double r70587 = 2.0;
        double r70588 = pow(r70586, r70587);
        double r70589 = r70585 + r70588;
        double r70590 = r70583 * r70589;
        return r70590;
}

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

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

4. Applied associate-*l* 0.5

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

5. Simplified 0.5

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

7. Applied add-sqr-sqrt 0.5

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

8. Applied sqrt-prod 0.5

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

9. Applied associate-/r* 0.5

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

11. Applied add-sqr-sqrt 0.5

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

12. Applied sqrt-prod 0.5

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

13. Applied sqrt-prod 0.5

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

14. Applied add-sqr-sqrt 0.5

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

15. Applied sqrt-prod 0.5

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

16. Applied sqrt-prod 0.5

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

17. Applied *-un-lft-identity 0.5

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

18. Applied times-frac 0.5

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

19. Applied times-frac 0.5

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

20. Applied associate-*l* 0.6

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

21. Simplified 0.4

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

23. Applied div-inv 0.4

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

24. Final simplification 0.5

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

Reproduce

herbie shell --seed 2019308 
(FPCore (a1 a2 th)
  :name "Migdal et al, Equation (64)"
  :precision binary64
  (+ (* (/ (cos th) (sqrt 2)) (* a1 a1)) (* (/ (cos th) (sqrt 2)) (* a2 a2))))