Average Error: 0.5 → 0.5
Time: 7.8s
Precision: 64
\[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)\]
↓
\[\frac{\frac{\cos th \cdot \mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\left|\sqrt[3]{\sqrt{2}}\right| \cdot \sqrt{\sqrt{2}}}}{\sqrt{\sqrt[3]{\sqrt{2}}}}\]
\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)}{\left|\sqrt[3]{\sqrt{2}}\right| \cdot \sqrt{\sqrt{2}}}}{\sqrt{\sqrt[3]{\sqrt{2}}}}
double f(double a1, double a2, double th) {
        double r102492 = th;
        double r102493 = cos(r102492);
        double r102494 = 2.0;
        double r102495 = sqrt(r102494);
        double r102496 = r102493 / r102495;
        double r102497 = a1;
        double r102498 = r102497 * r102497;
        double r102499 = r102496 * r102498;
        double r102500 = a2;
        double r102501 = r102500 * r102500;
        double r102502 = r102496 * r102501;
        double r102503 = r102499 + r102502;
        return r102503;
}
↓
double f(double a1, double a2, double th) {
        double r102504 = th;
        double r102505 = cos(r102504);
        double r102506 = a1;
        double r102507 = a2;
        double r102508 = r102507 * r102507;
        double r102509 = fma(r102506, r102506, r102508);
        double r102510 = r102505 * r102509;
        double r102511 = 2.0;
        double r102512 = sqrt(r102511);
        double r102513 = cbrt(r102512);
        double r102514 = fabs(r102513);
        double r102515 = sqrt(r102512);
        double r102516 = r102514 * r102515;
        double r102517 = r102510 / r102516;
        double r102518 = sqrt(r102513);
        double r102519 = r102517 / r102518;
        return r102519;
}

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. 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-sqr-sqrt0.5

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

  5. Applied sqrt-prod0.6

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

  6. Applied associate-/r*0.5

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

  8. Applied add-cube-cbrt0.5

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

  9. Applied sqrt-prod0.7

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

  10. Applied associate-/r*0.5

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

  11. Simplified0.5

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

  12. Final simplification0.5

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

Reproduce

herbie shell --seed 2019356 +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))))