Migdal et al, Equation (64)

Average Error: 0.5 → 0.5

Time: 7.7s

Precision: 64

Math

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

↓

\[\cos th \cdot \left(a1 \cdot \frac{a1}{\sqrt{2}}\right) + \cos th \cdot \frac{\frac{{a2}^{2}}{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}}{\sqrt[3]{\sqrt{2}}}\]

C

double f(double a1, double a2, double th) {
        double r102324 = th;
        double r102325 = cos(r102324);
        double r102326 = 2.0;
        double r102327 = sqrt(r102326);
        double r102328 = r102325 / r102327;
        double r102329 = a1;
        double r102330 = r102329 * r102329;
        double r102331 = r102328 * r102330;
        double r102332 = a2;
        double r102333 = r102332 * r102332;
        double r102334 = r102328 * r102333;
        double r102335 = r102331 + r102334;
        return r102335;
}

↓

double f(double a1, double a2, double th) {
        double r102336 = th;
        double r102337 = cos(r102336);
        double r102338 = a1;
        double r102339 = 2.0;
        double r102340 = sqrt(r102339);
        double r102341 = r102338 / r102340;
        double r102342 = r102338 * r102341;
        double r102343 = r102337 * r102342;
        double r102344 = a2;
        double r102345 = 2.0;
        double r102346 = pow(r102344, r102345);
        double r102347 = cbrt(r102340);
        double r102348 = r102347 * r102347;
        double r102349 = r102346 / r102348;
        double r102350 = r102349 / r102347;
        double r102351 = r102337 * r102350;
        double r102352 = r102343 + r102351;
        return r102352;
}

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

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

4. Applied associate-*l* 0.5

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

5. Simplified 0.5

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

7. Applied div-inv 0.5

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

8. Applied associate-*l* 0.5

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

9. Simplified 0.5

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

11. Applied *-un-lft-identity 0.5

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

12. Applied sqrt-prod 0.5

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

13. Applied add-sqr-sqrt 32.7

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

14. Applied unpow-prod-down 32.7

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

15. Applied times-frac 32.6

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

16. Simplified 32.6

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

17. Simplified 0.5

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

19. Applied add-cube-cbrt 0.5

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

20. Applied associate-/r* 0.5

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

21. Final simplification 0.5

    \[\leadsto \cos th \cdot \left(a1 \cdot \frac{a1}{\sqrt{2}}\right) + \cos th \cdot \frac{\frac{{a2}^{2}}{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}}{\sqrt[3]{\sqrt{2}}}\]

Reproduce

herbie shell --seed 2020001 
(FPCore (a1 a2 th)
  :name "Migdal et al, Equation (64)"
  :precision binary64
  (+ (* (/ (cos th) (sqrt 2)) (* a1 a1)) (* (/ (cos th) (sqrt 2)) (* a2 a2))))