Average Error: 0.5 → 0.5
Time: 56.4s
Precision: 64
\[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)\]
↓
\[\left(\frac{\frac{1}{\sqrt{\sqrt{2}}}}{\sqrt{\sqrt[3]{\sqrt{2}}}} \cdot \frac{\cos th}{\left|\sqrt[3]{\sqrt{2}}\right|}\right) \cdot \left(a1 \cdot a1\right) + \left(a2 \cdot a2\right) \cdot \left(\frac{\frac{1}{\sqrt{\sqrt{2}}}}{\sqrt{\sqrt{\sqrt{2}}}} \cdot \frac{\cos th}{\sqrt{\sqrt{\sqrt{2}}}}\right)\]
\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{1}{\sqrt{\sqrt{2}}}}{\sqrt{\sqrt[3]{\sqrt{2}}}} \cdot \frac{\cos th}{\left|\sqrt[3]{\sqrt{2}}\right|}\right) \cdot \left(a1 \cdot a1\right) + \left(a2 \cdot a2\right) \cdot \left(\frac{\frac{1}{\sqrt{\sqrt{2}}}}{\sqrt{\sqrt{\sqrt{2}}}} \cdot \frac{\cos th}{\sqrt{\sqrt{\sqrt{2}}}}\right)
double f(double a1, double a2, double th) {
        double r1436342 = th;
        double r1436343 = cos(r1436342);
        double r1436344 = 2.0;
        double r1436345 = sqrt(r1436344);
        double r1436346 = r1436343 / r1436345;
        double r1436347 = a1;
        double r1436348 = r1436347 * r1436347;
        double r1436349 = r1436346 * r1436348;
        double r1436350 = a2;
        double r1436351 = r1436350 * r1436350;
        double r1436352 = r1436346 * r1436351;
        double r1436353 = r1436349 + r1436352;
        return r1436353;
}
↓
double f(double a1, double a2, double th) {
        double r1436354 = 1.0;
        double r1436355 = 2.0;
        double r1436356 = sqrt(r1436355);
        double r1436357 = sqrt(r1436356);
        double r1436358 = r1436354 / r1436357;
        double r1436359 = cbrt(r1436356);
        double r1436360 = sqrt(r1436359);
        double r1436361 = r1436358 / r1436360;
        double r1436362 = th;
        double r1436363 = cos(r1436362);
        double r1436364 = fabs(r1436359);
        double r1436365 = r1436363 / r1436364;
        double r1436366 = r1436361 * r1436365;
        double r1436367 = a1;
        double r1436368 = r1436367 * r1436367;
        double r1436369 = r1436366 * r1436368;
        double r1436370 = a2;
        double r1436371 = r1436370 * r1436370;
        double r1436372 = sqrt(r1436357);
        double r1436373 = r1436358 / r1436372;
        double r1436374 = r1436363 / r1436372;
        double r1436375 = r1436373 * r1436374;
        double r1436376 = r1436371 * r1436375;
        double r1436377 = r1436369 + r1436376;
        return r1436377;
}

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 add-sqr-sqrt0.5

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

  4. Applied associate-/r*0.5

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

  6. Applied add-cube-cbrt0.5

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

  7. Applied sqrt-prod0.6

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

  8. Applied div-inv0.5

    \[\leadsto \frac{\color{blue}{\cos th \cdot \frac{1}{\sqrt{\sqrt{2}}}}}{\sqrt{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}} \cdot \sqrt{\sqrt[3]{\sqrt{2}}}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)\]

  9. Applied times-frac0.5

    \[\leadsto \color{blue}{\left(\frac{\cos th}{\sqrt{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}} \cdot \frac{\frac{1}{\sqrt{\sqrt{2}}}}{\sqrt{\sqrt[3]{\sqrt{2}}}}\right)} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)\]

  10. Simplified0.5

    \[\leadsto \left(\color{blue}{\frac{\cos th}{\left|\sqrt[3]{\sqrt{2}}\right|}} \cdot \frac{\frac{1}{\sqrt{\sqrt{2}}}}{\sqrt{\sqrt[3]{\sqrt{2}}}}\right) \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)\]
  11. Using strategy rm

  12. Applied add-sqr-sqrt0.5

    \[\leadsto \left(\frac{\cos th}{\left|\sqrt[3]{\sqrt{2}}\right|} \cdot \frac{\frac{1}{\sqrt{\sqrt{2}}}}{\sqrt{\sqrt[3]{\sqrt{2}}}}\right) \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{\color{blue}{\sqrt{2} \cdot \sqrt{2}}}} \cdot \left(a2 \cdot a2\right)\]

  13. Applied sqrt-prod0.5

    \[\leadsto \left(\frac{\cos th}{\left|\sqrt[3]{\sqrt{2}}\right|} \cdot \frac{\frac{1}{\sqrt{\sqrt{2}}}}{\sqrt{\sqrt[3]{\sqrt{2}}}}\right) \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\color{blue}{\sqrt{\sqrt{2}} \cdot \sqrt{\sqrt{2}}}} \cdot \left(a2 \cdot a2\right)\]

  14. Applied associate-/r*0.5

    \[\leadsto \left(\frac{\cos th}{\left|\sqrt[3]{\sqrt{2}}\right|} \cdot \frac{\frac{1}{\sqrt{\sqrt{2}}}}{\sqrt{\sqrt[3]{\sqrt{2}}}}\right) \cdot \left(a1 \cdot a1\right) + \color{blue}{\frac{\frac{\cos th}{\sqrt{\sqrt{2}}}}{\sqrt{\sqrt{2}}}} \cdot \left(a2 \cdot a2\right)\]
  15. Using strategy rm

  16. Applied add-sqr-sqrt0.5

    \[\leadsto \left(\frac{\cos th}{\left|\sqrt[3]{\sqrt{2}}\right|} \cdot \frac{\frac{1}{\sqrt{\sqrt{2}}}}{\sqrt{\sqrt[3]{\sqrt{2}}}}\right) \cdot \left(a1 \cdot a1\right) + \frac{\frac{\cos th}{\sqrt{\sqrt{2}}}}{\color{blue}{\sqrt{\sqrt{\sqrt{2}}} \cdot \sqrt{\sqrt{\sqrt{2}}}}} \cdot \left(a2 \cdot a2\right)\]

  17. Applied div-inv0.5

    \[\leadsto \left(\frac{\cos th}{\left|\sqrt[3]{\sqrt{2}}\right|} \cdot \frac{\frac{1}{\sqrt{\sqrt{2}}}}{\sqrt{\sqrt[3]{\sqrt{2}}}}\right) \cdot \left(a1 \cdot a1\right) + \frac{\color{blue}{\cos th \cdot \frac{1}{\sqrt{\sqrt{2}}}}}{\sqrt{\sqrt{\sqrt{2}}} \cdot \sqrt{\sqrt{\sqrt{2}}}} \cdot \left(a2 \cdot a2\right)\]

  18. Applied times-frac0.5

    \[\leadsto \left(\frac{\cos th}{\left|\sqrt[3]{\sqrt{2}}\right|} \cdot \frac{\frac{1}{\sqrt{\sqrt{2}}}}{\sqrt{\sqrt[3]{\sqrt{2}}}}\right) \cdot \left(a1 \cdot a1\right) + \color{blue}{\left(\frac{\cos th}{\sqrt{\sqrt{\sqrt{2}}}} \cdot \frac{\frac{1}{\sqrt{\sqrt{2}}}}{\sqrt{\sqrt{\sqrt{2}}}}\right)} \cdot \left(a2 \cdot a2\right)\]

  19. Final simplification0.5

    \[\leadsto \left(\frac{\frac{1}{\sqrt{\sqrt{2}}}}{\sqrt{\sqrt[3]{\sqrt{2}}}} \cdot \frac{\cos th}{\left|\sqrt[3]{\sqrt{2}}\right|}\right) \cdot \left(a1 \cdot a1\right) + \left(a2 \cdot a2\right) \cdot \left(\frac{\frac{1}{\sqrt{\sqrt{2}}}}{\sqrt{\sqrt{\sqrt{2}}}} \cdot \frac{\cos th}{\sqrt{\sqrt{\sqrt{2}}}}\right)\]

Reproduce

herbie shell --seed 2019132 
(FPCore (a1 a2 th)
  :name "Migdal et al, Equation (64)"
  (+ (* (/ (cos th) (sqrt 2)) (* a1 a1)) (* (/ (cos th) (sqrt 2)) (* a2 a2))))