Average Error: 0.5 → 0.5
Time: 15.0s
Precision: 64
\[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)\]
↓
\[\frac{\frac{\frac{\cos th \cdot \mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\left|\sqrt[3]{\sqrt{2}}\right| \cdot \sqrt{\sqrt{2}}}}{\sqrt{\sqrt[3]{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}}}}{\sqrt{\sqrt[3]{\sqrt[3]{\sqrt{2}}}}}\]
\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)
↓
\frac{\frac{\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[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}}}}{\sqrt{\sqrt[3]{\sqrt[3]{\sqrt{2}}}}}
double f(double a1, double a2, double th) {
        double r141036 = th;
        double r141037 = cos(r141036);
        double r141038 = 2.0;
        double r141039 = sqrt(r141038);
        double r141040 = r141037 / r141039;
        double r141041 = a1;
        double r141042 = r141041 * r141041;
        double r141043 = r141040 * r141042;
        double r141044 = a2;
        double r141045 = r141044 * r141044;
        double r141046 = r141040 * r141045;
        double r141047 = r141043 + r141046;
        return r141047;
}
↓
double f(double a1, double a2, double th) {
        double r141048 = th;
        double r141049 = cos(r141048);
        double r141050 = a1;
        double r141051 = a2;
        double r141052 = r141051 * r141051;
        double r141053 = fma(r141050, r141050, r141052);
        double r141054 = r141049 * r141053;
        double r141055 = 2.0;
        double r141056 = sqrt(r141055);
        double r141057 = cbrt(r141056);
        double r141058 = fabs(r141057);
        double r141059 = sqrt(r141056);
        double r141060 = r141058 * r141059;
        double r141061 = r141054 / r141060;
        double r141062 = r141057 * r141057;
        double r141063 = cbrt(r141062);
        double r141064 = sqrt(r141063);
        double r141065 = r141061 / r141064;
        double r141066 = cbrt(r141057);
        double r141067 = sqrt(r141066);
        double r141068 = r141065 / r141067;
        return r141068;
}

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. Simplified0.4

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

  13. Applied add-cube-cbrt0.4

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

  14. Applied cbrt-prod0.4

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

  15. Applied sqrt-prod0.4

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

  16. Applied associate-/r*0.5

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

  17. Final simplification0.5

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

Reproduce

herbie shell --seed 2020036 +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))))