Average Error: 0.5 → 0.5
Time: 38.0s
Precision: 64
\[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)\]
↓
\[\frac{\frac{\frac{\cos th \cdot \mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\left|\sqrt[3]{\sqrt{2}}\right|}}{\sqrt{\sqrt[3]{\sqrt{2}}}}}{\sqrt{\sqrt{2}}}\]
\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)
↓
\frac{\frac{\frac{\cos th \cdot \mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\left|\sqrt[3]{\sqrt{2}}\right|}}{\sqrt{\sqrt[3]{\sqrt{2}}}}}{\sqrt{\sqrt{2}}}
double f(double a1, double a2, double th) {
        double r111725 = th;
        double r111726 = cos(r111725);
        double r111727 = 2.0;
        double r111728 = sqrt(r111727);
        double r111729 = r111726 / r111728;
        double r111730 = a1;
        double r111731 = r111730 * r111730;
        double r111732 = r111729 * r111731;
        double r111733 = a2;
        double r111734 = r111733 * r111733;
        double r111735 = r111729 * r111734;
        double r111736 = r111732 + r111735;
        return r111736;
}
↓
double f(double a1, double a2, double th) {
        double r111737 = th;
        double r111738 = cos(r111737);
        double r111739 = a1;
        double r111740 = a2;
        double r111741 = r111740 * r111740;
        double r111742 = fma(r111739, r111739, r111741);
        double r111743 = r111738 * r111742;
        double r111744 = 2.0;
        double r111745 = sqrt(r111744);
        double r111746 = cbrt(r111745);
        double r111747 = fabs(r111746);
        double r111748 = r111743 / r111747;
        double r111749 = sqrt(r111746);
        double r111750 = r111748 / r111749;
        double r111751 = sqrt(r111745);
        double r111752 = r111750 / r111751;
        return r111752;
}

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

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

  9. Applied sqrt-prod0.7

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

  10. Applied associate-/r*0.5

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

  11. Simplified0.5

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

  12. Final simplification0.5

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

Reproduce

herbie shell --seed 2019199 +o rules:numerics
(FPCore (a1 a2 th)
  :name "Migdal et al, Equation (64)"
  (+ (* (/ (cos th) (sqrt 2.0)) (* a1 a1)) (* (/ (cos th) (sqrt 2.0)) (* a2 a2))))