Average Error: 0.6 → 0.5
Time: 19.9s
Precision: 64
\[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)\]
↓
\[\frac{\frac{\cos th \cdot \mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}}{\sqrt[3]{\sqrt{2}}}\]
\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)
↓
\frac{\frac{\cos th \cdot \mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}}{\sqrt[3]{\sqrt{2}}}
double f(double a1, double a2, double th) {
        double r85353 = th;
        double r85354 = cos(r85353);
        double r85355 = 2.0;
        double r85356 = sqrt(r85355);
        double r85357 = r85354 / r85356;
        double r85358 = a1;
        double r85359 = r85358 * r85358;
        double r85360 = r85357 * r85359;
        double r85361 = a2;
        double r85362 = r85361 * r85361;
        double r85363 = r85357 * r85362;
        double r85364 = r85360 + r85363;
        return r85364;
}
↓
double f(double a1, double a2, double th) {
        double r85365 = th;
        double r85366 = cos(r85365);
        double r85367 = a1;
        double r85368 = a2;
        double r85369 = r85368 * r85368;
        double r85370 = fma(r85367, r85367, r85369);
        double r85371 = r85366 * r85370;
        double r85372 = 2.0;
        double r85373 = sqrt(r85372);
        double r85374 = cbrt(r85373);
        double r85375 = r85374 * r85374;
        double r85376 = r85371 / r85375;
        double r85377 = r85376 / r85374;
        return r85377;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus th

Derivation

  1. Initial program 0.6

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

  2. Simplified0.5

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

  4. Applied add-cube-cbrt0.5

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

  5. Applied associate-/r*0.5

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

  6. Final simplification0.5

    \[\leadsto \frac{\frac{\cos th \cdot \mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}}{\sqrt[3]{\sqrt{2}}}\]

Reproduce

herbie shell --seed 2019209 +o rules:numerics
(FPCore (a1 a2 th)
  :name "Migdal et al, Equation (64)"
  :precision binary64
  (+ (* (/ (cos th) (sqrt 2)) (* a1 a1)) (* (/ (cos th) (sqrt 2)) (* a2 a2))))