Migdal et al, Equation (64)

Average Error: 0.5 → 0.5

Time: 7.8s

Precision: 64

Math

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

↓

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

C

double f(double a1, double a2, double th) {
        double r99336 = th;
        double r99337 = cos(r99336);
        double r99338 = 2.0;
        double r99339 = sqrt(r99338);
        double r99340 = r99337 / r99339;
        double r99341 = a1;
        double r99342 = r99341 * r99341;
        double r99343 = r99340 * r99342;
        double r99344 = a2;
        double r99345 = r99344 * r99344;
        double r99346 = r99340 * r99345;
        double r99347 = r99343 + r99346;
        return r99347;
}

↓

double f(double a1, double a2, double th) {
        double r99348 = th;
        double r99349 = cos(r99348);
        double r99350 = a2;
        double r99351 = 2.0;
        double r99352 = sqrt(r99351);
        double r99353 = r99350 / r99352;
        double r99354 = r99349 * r99353;
        double r99355 = r99354 * r99350;
        double r99356 = a1;
        double r99357 = 2.0;
        double r99358 = pow(r99356, r99357);
        double r99359 = r99352 / r99349;
        double r99360 = r99358 / r99359;
        double r99361 = r99355 + r99360;
        return r99361;
}

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 clear-num 0.5

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

7. Applied *-un-lft-identity 0.5

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

8. Applied *-un-lft-identity 0.5

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

9. Applied times-frac 0.5

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

10. Applied *-un-lft-identity 0.5

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

11. Applied times-frac 0.5

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

12. Applied associate-*l* 0.5

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

13. Simplified 0.5

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

15. Applied div-inv 0.5

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

16. Applied associate-*l* 0.5

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

17. Simplified 0.5

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

18. Final simplification 0.5

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

Reproduce

herbie shell --seed 2019354 
(FPCore (a1 a2 th)
  :name "Migdal et al, Equation (64)"
  :precision binary64
  (+ (* (/ (cos th) (sqrt 2)) (* a1 a1)) (* (/ (cos th) (sqrt 2)) (* a2 a2))))