Average Error: 0.5 → 0.5
Time: 1.2m
Precision: 64
\[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)\]
↓
\[a1 \cdot \left(\frac{\cos th}{\sqrt{2}} \cdot a1\right) + \left(\frac{\cos th}{\left|\sqrt[3]{\sqrt{2}}\right|} \cdot \frac{\frac{1}{\sqrt{\sqrt{2}}}}{\sqrt{\sqrt[3]{\sqrt{2}}}}\right) \cdot \left(a2 \cdot a2\right)\]
double f(double a1, double a2, double th) {
        double r9153359 = th;
        double r9153360 = cos(r9153359);
        double r9153361 = 2.0;
        double r9153362 = sqrt(r9153361);
        double r9153363 = r9153360 / r9153362;
        double r9153364 = a1;
        double r9153365 = r9153364 * r9153364;
        double r9153366 = r9153363 * r9153365;
        double r9153367 = a2;
        double r9153368 = r9153367 * r9153367;
        double r9153369 = r9153363 * r9153368;
        double r9153370 = r9153366 + r9153369;
        return r9153370;
}
↓
double f(double a1, double a2, double th) {
        double r9153371 = a1;
        double r9153372 = th;
        double r9153373 = cos(r9153372);
        double r9153374 = 2.0;
        double r9153375 = sqrt(r9153374);
        double r9153376 = r9153373 / r9153375;
        double r9153377 = r9153376 * r9153371;
        double r9153378 = r9153371 * r9153377;
        double r9153379 = cbrt(r9153375);
        double r9153380 = fabs(r9153379);
        double r9153381 = r9153373 / r9153380;
        double r9153382 = 1.0;
        double r9153383 = sqrt(r9153375);
        double r9153384 = r9153382 / r9153383;
        double r9153385 = sqrt(r9153379);
        double r9153386 = r9153384 / r9153385;
        double r9153387 = r9153381 * r9153386;
        double r9153388 = a2;
        double r9153389 = r9153388 * r9153388;
        double r9153390 = r9153387 * r9153389;
        double r9153391 = r9153378 + r9153390;
        return r9153391;
}


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

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

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

  5. Applied add-sqr-sqrt0.5

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

  6. Applied sqrt-prod0.5

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

  7. Applied associate-/r*0.5

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

  9. Applied add-cube-cbrt0.5

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

  10. Applied sqrt-prod0.6

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

  11. Applied div-inv0.5

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

  12. Applied times-frac0.5

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

  13. Simplified0.5

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

  14. Final simplification0.5

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

Reproduce

herbie shell --seed 2019101 
(FPCore (a1 a2 th)
  :name "Migdal et al, Equation (64)"
  (+ (* (/ (cos th) (sqrt 2)) (* a1 a1)) (* (/ (cos th) (sqrt 2)) (* a2 a2))))