Average Error: 0.5 → 0.5
Time: 9.1s
Precision: 64
\[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)\]
↓
\[\left(\left(\frac{1}{\sqrt{\sqrt{\sqrt{2}}}} \cdot \frac{\frac{\cos th}{\sqrt{\sqrt{2}}}}{\sqrt{\sqrt{\sqrt{2}}}}\right) \cdot a1\right) \cdot a1 + \frac{\frac{\cos th}{\sqrt{\sqrt{2}}}}{\sqrt{\sqrt{2}}} \cdot \left(a2 \cdot a2\right)\]
\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)
↓
\left(\left(\frac{1}{\sqrt{\sqrt{\sqrt{2}}}} \cdot \frac{\frac{\cos th}{\sqrt{\sqrt{2}}}}{\sqrt{\sqrt{\sqrt{2}}}}\right) \cdot a1\right) \cdot a1 + \frac{\frac{\cos th}{\sqrt{\sqrt{2}}}}{\sqrt{\sqrt{2}}} \cdot \left(a2 \cdot a2\right)
double f(double a1, double a2, double th) {
        double r97347 = th;
        double r97348 = cos(r97347);
        double r97349 = 2.0;
        double r97350 = sqrt(r97349);
        double r97351 = r97348 / r97350;
        double r97352 = a1;
        double r97353 = r97352 * r97352;
        double r97354 = r97351 * r97353;
        double r97355 = a2;
        double r97356 = r97355 * r97355;
        double r97357 = r97351 * r97356;
        double r97358 = r97354 + r97357;
        return r97358;
}
↓
double f(double a1, double a2, double th) {
        double r97359 = 1.0;
        double r97360 = 2.0;
        double r97361 = sqrt(r97360);
        double r97362 = sqrt(r97361);
        double r97363 = sqrt(r97362);
        double r97364 = r97359 / r97363;
        double r97365 = th;
        double r97366 = cos(r97365);
        double r97367 = r97366 / r97362;
        double r97368 = r97367 / r97363;
        double r97369 = r97364 * r97368;
        double r97370 = a1;
        double r97371 = r97369 * r97370;
        double r97372 = r97371 * r97370;
        double r97373 = r97367 / r97362;
        double r97374 = a2;
        double r97375 = r97374 * r97374;
        double r97376 = r97373 * r97375;
        double r97377 = r97372 + r97376;
        return r97377;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus th

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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

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

  5. Applied add-sqr-sqrt0.5

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

  6. Applied sqrt-prod0.5

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

  7. Applied associate-/r*0.5

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

  9. Applied *-un-lft-identity0.5

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

  10. Applied sqrt-prod0.5

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

  11. Applied sqrt-prod0.5

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

  12. Applied add-sqr-sqrt0.5

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

  13. Applied sqrt-prod0.5

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

  14. Applied sqrt-prod0.5

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

  15. Applied *-un-lft-identity0.5

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

  16. Applied times-frac0.5

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

  17. Applied times-frac0.5

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

  18. Simplified0.5

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

  19. Simplified0.5

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

  21. Applied add-sqr-sqrt0.5

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

  22. Applied sqrt-prod0.5

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

  23. Applied associate-/r*0.5

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

  24. Final simplification0.5

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

Reproduce

herbie shell --seed 2020062 
(FPCore (a1 a2 th)
  :name "Migdal et al, Equation (64)"
  :precision binary64
  (+ (* (/ (cos th) (sqrt 2)) (* a1 a1)) (* (/ (cos th) (sqrt 2)) (* a2 a2))))