Average Error: 0.5 → 0.4
Time: 11.3s
Precision: 64
\[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)\]
↓
\[\frac{\cos th \cdot \mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\sqrt[3]{\sqrt[3]{\sqrt{2}}}} \cdot \left(\frac{\frac{1}{\sqrt{\sqrt[3]{\sqrt[3]{\sqrt{2}}}}}}{\sqrt[3]{\sqrt[3]{\sqrt{2}}}} \cdot \frac{\frac{1}{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}}{\sqrt{\sqrt[3]{\sqrt[3]{\sqrt{2}}}}}\right)\]
\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[3]{\sqrt[3]{\sqrt{2}}}} \cdot \left(\frac{\frac{1}{\sqrt{\sqrt[3]{\sqrt[3]{\sqrt{2}}}}}}{\sqrt[3]{\sqrt[3]{\sqrt{2}}}} \cdot \frac{\frac{1}{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}}{\sqrt{\sqrt[3]{\sqrt[3]{\sqrt{2}}}}}\right)
double f(double a1, double a2, double th) {
        double r124965 = th;
        double r124966 = cos(r124965);
        double r124967 = 2.0;
        double r124968 = sqrt(r124967);
        double r124969 = r124966 / r124968;
        double r124970 = a1;
        double r124971 = r124970 * r124970;
        double r124972 = r124969 * r124971;
        double r124973 = a2;
        double r124974 = r124973 * r124973;
        double r124975 = r124969 * r124974;
        double r124976 = r124972 + r124975;
        return r124976;
}
↓
double f(double a1, double a2, double th) {
        double r124977 = th;
        double r124978 = cos(r124977);
        double r124979 = a1;
        double r124980 = a2;
        double r124981 = r124980 * r124980;
        double r124982 = fma(r124979, r124979, r124981);
        double r124983 = r124978 * r124982;
        double r124984 = 2.0;
        double r124985 = sqrt(r124984);
        double r124986 = cbrt(r124985);
        double r124987 = cbrt(r124986);
        double r124988 = r124983 / r124987;
        double r124989 = 1.0;
        double r124990 = sqrt(r124987);
        double r124991 = r124989 / r124990;
        double r124992 = r124991 / r124987;
        double r124993 = r124986 * r124986;
        double r124994 = r124989 / r124993;
        double r124995 = r124994 / r124990;
        double r124996 = r124992 * r124995;
        double r124997 = r124988 * r124996;
        return r124997;
}

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-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. Using strategy rm

  7. Applied add-cube-cbrt0.5

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

  8. Applied *-un-lft-identity0.5

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

  9. Applied times-frac0.5

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

  11. Applied add-sqr-sqrt0.5

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

  12. Applied div-inv0.5

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

  13. Applied times-frac0.5

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

  14. Applied associate-*r*0.5

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

  15. Simplified0.5

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

  17. Applied div-inv0.5

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

  18. Applied times-frac0.5

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

  19. Applied associate-*l*0.4

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

  20. Final simplification0.4

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

Reproduce

herbie shell --seed 2020062 +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))))