Average Error: 0.5 → 0.4
Time: 7.5s
Precision: 64
\[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)\]
↓
\[\frac{\cos th \cdot \mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\sqrt{\sqrt{\sqrt[3]{2} \cdot \sqrt[3]{2}}}} \cdot \frac{\frac{1}{\sqrt{\sqrt{2}}}}{\sqrt{\sqrt{\sqrt[3]{2}}}}\]
\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)
↓
\frac{\cos th \cdot \mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\sqrt{\sqrt{\sqrt[3]{2} \cdot \sqrt[3]{2}}}} \cdot \frac{\frac{1}{\sqrt{\sqrt{2}}}}{\sqrt{\sqrt{\sqrt[3]{2}}}}
double f(double a1, double a2, double th) {
        double r93483 = th;
        double r93484 = cos(r93483);
        double r93485 = 2.0;
        double r93486 = sqrt(r93485);
        double r93487 = r93484 / r93486;
        double r93488 = a1;
        double r93489 = r93488 * r93488;
        double r93490 = r93487 * r93489;
        double r93491 = a2;
        double r93492 = r93491 * r93491;
        double r93493 = r93487 * r93492;
        double r93494 = r93490 + r93493;
        return r93494;
}
↓
double f(double a1, double a2, double th) {
        double r93495 = th;
        double r93496 = cos(r93495);
        double r93497 = a1;
        double r93498 = a2;
        double r93499 = r93498 * r93498;
        double r93500 = fma(r93497, r93497, r93499);
        double r93501 = r93496 * r93500;
        double r93502 = 2.0;
        double r93503 = cbrt(r93502);
        double r93504 = r93503 * r93503;
        double r93505 = sqrt(r93504);
        double r93506 = sqrt(r93505);
        double r93507 = r93501 / r93506;
        double r93508 = 1.0;
        double r93509 = sqrt(r93502);
        double r93510 = sqrt(r93509);
        double r93511 = r93508 / r93510;
        double r93512 = sqrt(r93503);
        double r93513 = sqrt(r93512);
        double r93514 = r93511 / r93513;
        double r93515 = r93507 * r93514;
        return r93515;
}

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-cbrt0.7

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

  9. Applied sqrt-prod0.7

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

  10. Applied sqrt-prod0.7

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

  11. Applied div-inv0.5

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

  12. Applied times-frac0.4

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

  13. Final simplification0.4

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

Reproduce

herbie shell --seed 2020027 +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))))