Average Error: 0.5 → 0.5
Time: 21.0s
Precision: 64
\[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)\]
↓
\[\frac{\frac{\cos th}{\sqrt{\sqrt{\sqrt{2}}}} \cdot \frac{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\sqrt{\sqrt{\sqrt{2}}}}}{\sqrt{\sqrt{2}}}\]
\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)
↓
\frac{\frac{\cos th}{\sqrt{\sqrt{\sqrt{2}}}} \cdot \frac{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\sqrt{\sqrt{\sqrt{2}}}}}{\sqrt{\sqrt{2}}}
double f(double a1, double a2, double th) {
        double r69822 = th;
        double r69823 = cos(r69822);
        double r69824 = 2.0;
        double r69825 = sqrt(r69824);
        double r69826 = r69823 / r69825;
        double r69827 = a1;
        double r69828 = r69827 * r69827;
        double r69829 = r69826 * r69828;
        double r69830 = a2;
        double r69831 = r69830 * r69830;
        double r69832 = r69826 * r69831;
        double r69833 = r69829 + r69832;
        return r69833;
}
↓
double f(double a1, double a2, double th) {
        double r69834 = th;
        double r69835 = cos(r69834);
        double r69836 = 2.0;
        double r69837 = sqrt(r69836);
        double r69838 = sqrt(r69837);
        double r69839 = sqrt(r69838);
        double r69840 = r69835 / r69839;
        double r69841 = a1;
        double r69842 = a2;
        double r69843 = r69842 * r69842;
        double r69844 = fma(r69841, r69841, r69843);
        double r69845 = r69844 / r69839;
        double r69846 = r69840 * r69845;
        double r69847 = r69846 / r69838;
        return r69847;
}

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-sqr-sqrt0.5

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

  9. Applied sqrt-prod0.5

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

  10. Applied sqrt-prod0.5

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

  11. Applied times-frac0.5

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

  12. Final simplification0.5

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

Reproduce

herbie shell --seed 2019305 +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))))