Average Error: 0.5 → 0.5
Time: 12.9s
Precision: 64
\[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)\]
↓
\[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\frac{\cos th}{\sqrt{\sqrt{\sqrt[3]{2} \cdot \sqrt[3]{2}}}}}{\frac{\sqrt{\sqrt{2}}}{\frac{a2}{\sqrt{\sqrt{\sqrt[3]{2}}}}}} \cdot a2\]
\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}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\frac{\cos th}{\sqrt{\sqrt{\sqrt[3]{2} \cdot \sqrt[3]{2}}}}}{\frac{\sqrt{\sqrt{2}}}{\frac{a2}{\sqrt{\sqrt{\sqrt[3]{2}}}}}} \cdot a2
double f(double a1, double a2, double th) {
        double r96909 = th;
        double r96910 = cos(r96909);
        double r96911 = 2.0;
        double r96912 = sqrt(r96911);
        double r96913 = r96910 / r96912;
        double r96914 = a1;
        double r96915 = r96914 * r96914;
        double r96916 = r96913 * r96915;
        double r96917 = a2;
        double r96918 = r96917 * r96917;
        double r96919 = r96913 * r96918;
        double r96920 = r96916 + r96919;
        return r96920;
}
↓
double f(double a1, double a2, double th) {
        double r96921 = th;
        double r96922 = cos(r96921);
        double r96923 = 2.0;
        double r96924 = sqrt(r96923);
        double r96925 = r96922 / r96924;
        double r96926 = a1;
        double r96927 = r96926 * r96926;
        double r96928 = r96925 * r96927;
        double r96929 = cbrt(r96923);
        double r96930 = r96929 * r96929;
        double r96931 = sqrt(r96930);
        double r96932 = sqrt(r96931);
        double r96933 = r96922 / r96932;
        double r96934 = sqrt(r96924);
        double r96935 = a2;
        double r96936 = sqrt(r96929);
        double r96937 = sqrt(r96936);
        double r96938 = r96935 / r96937;
        double r96939 = r96934 / r96938;
        double r96940 = r96933 / r96939;
        double r96941 = r96940 * r96935;
        double r96942 = r96928 + r96941;
        return r96942;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus th

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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. Using strategy rm

  3. Applied associate-*r*0.5

    \[\leadsto \frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \color{blue}{\left(\frac{\cos th}{\sqrt{2}} \cdot a2\right) \cdot a2}\]
  4. Using strategy rm

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

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

  6. Applied associate-*l*0.5

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

  7. Simplified0.5

    \[\leadsto \frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \left(1 \cdot \color{blue}{\frac{\cos th \cdot a2}{\sqrt{2}}}\right) \cdot a2\]
  8. Using strategy rm

  9. Applied add-sqr-sqrt0.5

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

  10. Applied sqrt-prod0.5

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

  11. Applied associate-/r*0.5

    \[\leadsto \frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \left(1 \cdot \color{blue}{\frac{\frac{\cos th \cdot a2}{\sqrt{\sqrt{2}}}}{\sqrt{\sqrt{2}}}}\right) \cdot a2\]
  12. Using strategy rm

  13. Applied add-cube-cbrt0.6

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

  14. Applied sqrt-prod0.6

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

  15. Applied sqrt-prod0.6

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

  16. Applied times-frac0.5

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

  17. Applied associate-/l*0.5

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

  18. Final simplification0.5

    \[\leadsto \frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\frac{\cos th}{\sqrt{\sqrt{\sqrt[3]{2} \cdot \sqrt[3]{2}}}}}{\frac{\sqrt{\sqrt{2}}}{\frac{a2}{\sqrt{\sqrt{\sqrt[3]{2}}}}}} \cdot a2\]

Reproduce

herbie shell --seed 2019350 
(FPCore (a1 a2 th)
  :name "Migdal et al, Equation (64)"
  :precision binary64
  (+ (* (/ (cos th) (sqrt 2)) (* a1 a1)) (* (/ (cos th) (sqrt 2)) (* a2 a2))))