Average Error: 0.6 → 0.5
Time: 17.8s
Precision: 64
\[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)\]
↓
\[\frac{\cos th \cdot \frac{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\sqrt[3]{\sqrt{2}}}}{{\left(\sqrt[3]{\sqrt[3]{\sqrt{2}}}\right)}^{3} \cdot \sqrt[3]{\sqrt{2}}}\]
\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)
↓
\frac{\cos th \cdot \frac{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\sqrt[3]{\sqrt{2}}}}{{\left(\sqrt[3]{\sqrt[3]{\sqrt{2}}}\right)}^{3} \cdot \sqrt[3]{\sqrt{2}}}
double f(double a1, double a2, double th) {
        double r178059 = th;
        double r178060 = cos(r178059);
        double r178061 = 2.0;
        double r178062 = sqrt(r178061);
        double r178063 = r178060 / r178062;
        double r178064 = a1;
        double r178065 = r178064 * r178064;
        double r178066 = r178063 * r178065;
        double r178067 = a2;
        double r178068 = r178067 * r178067;
        double r178069 = r178063 * r178068;
        double r178070 = r178066 + r178069;
        return r178070;
}
↓
double f(double a1, double a2, double th) {
        double r178071 = th;
        double r178072 = cos(r178071);
        double r178073 = a1;
        double r178074 = a2;
        double r178075 = r178074 * r178074;
        double r178076 = fma(r178073, r178073, r178075);
        double r178077 = 2.0;
        double r178078 = sqrt(r178077);
        double r178079 = cbrt(r178078);
        double r178080 = r178076 / r178079;
        double r178081 = r178072 * r178080;
        double r178082 = cbrt(r178079);
        double r178083 = 3.0;
        double r178084 = pow(r178082, r178083);
        double r178085 = r178084 * r178079;
        double r178086 = r178081 / r178085;
        return r178086;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus th

Derivation

  1. Initial program 0.6

    \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)\]

  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 times-frac0.7

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

  9. Applied times-frac0.5

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

  11. Applied associate-*l/0.5

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

  13. Applied frac-times0.6

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

  14. Applied associate-/l/0.7

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

  15. Simplified0.5

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

  16. Final simplification0.5

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

Reproduce

herbie shell --seed 2020018 +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))))