Migdal et al, Equation (64)

Average Error: 0.5 → 0.4

Time: 26.3s

Precision: 64

Math

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

↓

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

C

double f(double a1, double a2, double th) {
        double r2237254 = th;
        double r2237255 = cos(r2237254);
        double r2237256 = 2.0;
        double r2237257 = sqrt(r2237256);
        double r2237258 = r2237255 / r2237257;
        double r2237259 = a1;
        double r2237260 = r2237259 * r2237259;
        double r2237261 = r2237258 * r2237260;
        double r2237262 = a2;
        double r2237263 = r2237262 * r2237262;
        double r2237264 = r2237258 * r2237263;
        double r2237265 = r2237261 + r2237264;
        return r2237265;
}

↓

double f(double a1, double a2, double th) {
        double r2237266 = a1;
        double r2237267 = 2.0;
        double r2237268 = sqrt(r2237267);
        double r2237269 = sqrt(r2237268);
        double r2237270 = r2237266 / r2237269;
        double r2237271 = r2237270 / r2237269;
        double r2237272 = r2237271 * r2237266;
        double r2237273 = th;
        double r2237274 = cos(r2237273);
        double r2237275 = r2237272 * r2237274;
        double r2237276 = sqrt(r2237269);
        double r2237277 = r2237274 / r2237276;
        double r2237278 = a2;
        double r2237279 = r2237278 * r2237278;
        double r2237280 = r2237269 * r2237276;
        double r2237281 = r2237279 / r2237280;
        double r2237282 = r2237277 * r2237281;
        double r2237283 = r2237275 + r2237282;
        return r2237283;
}

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

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

7. Applied add-sqr-sqrt 0.5

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

8. Applied sqrt-prod 0.5

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

9. Applied associate-/r* 0.5

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

11. Applied add-sqr-sqrt 0.5

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

12. Applied sqrt-prod 0.5

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

13. Applied sqrt-prod 0.5

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

14. Applied div-inv 0.5

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

15. Applied times-frac 0.5

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

16. Applied associate-*l* 0.5

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

17. Simplified 0.4

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

19. Applied add-sqr-sqrt 0.4

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

20. Applied sqrt-prod 0.5

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

21. Applied associate-/r* 0.4

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

22. Final simplification 0.4

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

Reproduce

herbie shell --seed 2019164 
(FPCore (a1 a2 th)
  :name "Migdal et al, Equation (64)"
  (+ (* (/ (cos th) (sqrt 2)) (* a1 a1)) (* (/ (cos th) (sqrt 2)) (* a2 a2))))