Average Error: 0.5 → 0.4
Time: 9.4s
Precision: 64
\[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)\]
↓
\[\frac{\cos th}{\sqrt{\sqrt{\sqrt[3]{2} \cdot \sqrt[3]{2}}}} \cdot \left(\frac{\frac{1}{\sqrt{\sqrt{2}}}}{\sqrt{\sqrt{\sqrt[3]{2}}}} \cdot \left(a1 \cdot a1\right)\right) + \cos th \cdot \frac{{a2}^{2}}{\sqrt{2}}\]
\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}{\sqrt{\sqrt{\sqrt[3]{2} \cdot \sqrt[3]{2}}}} \cdot \left(\frac{\frac{1}{\sqrt{\sqrt{2}}}}{\sqrt{\sqrt{\sqrt[3]{2}}}} \cdot \left(a1 \cdot a1\right)\right) + \cos th \cdot \frac{{a2}^{2}}{\sqrt{2}}
double f(double a1, double a2, double th) {
        double r109904 = th;
        double r109905 = cos(r109904);
        double r109906 = 2.0;
        double r109907 = sqrt(r109906);
        double r109908 = r109905 / r109907;
        double r109909 = a1;
        double r109910 = r109909 * r109909;
        double r109911 = r109908 * r109910;
        double r109912 = a2;
        double r109913 = r109912 * r109912;
        double r109914 = r109908 * r109913;
        double r109915 = r109911 + r109914;
        return r109915;
}
↓
double f(double a1, double a2, double th) {
        double r109916 = th;
        double r109917 = cos(r109916);
        double r109918 = 2.0;
        double r109919 = cbrt(r109918);
        double r109920 = r109919 * r109919;
        double r109921 = sqrt(r109920);
        double r109922 = sqrt(r109921);
        double r109923 = r109917 / r109922;
        double r109924 = 1.0;
        double r109925 = sqrt(r109918);
        double r109926 = sqrt(r109925);
        double r109927 = r109924 / r109926;
        double r109928 = sqrt(r109919);
        double r109929 = sqrt(r109928);
        double r109930 = r109927 / r109929;
        double r109931 = a1;
        double r109932 = r109931 * r109931;
        double r109933 = r109930 * r109932;
        double r109934 = r109923 * r109933;
        double r109935 = a2;
        double r109936 = 2.0;
        double r109937 = pow(r109935, r109936);
        double r109938 = r109937 / r109925;
        double r109939 = r109917 * r109938;
        double r109940 = r109934 + r109939;
        return r109940;
}

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 div-inv0.5

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

  4. Applied associate-*l*0.5

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

  5. Simplified0.5

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

  7. Applied add-sqr-sqrt0.5

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

  8. Applied sqrt-prod0.5

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

  9. Applied associate-/r*0.5

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

  11. Applied add-cube-cbrt0.5

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

  12. Applied sqrt-prod0.5

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

  13. Applied sqrt-prod0.5

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

  14. Applied div-inv0.5

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

  15. Applied times-frac0.4

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

  16. Applied associate-*l*0.4

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

  17. Final simplification0.4

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

Reproduce

herbie shell --seed 2020024 
(FPCore (a1 a2 th)
  :name "Migdal et al, Equation (64)"
  :precision binary64
  (+ (* (/ (cos th) (sqrt 2)) (* a1 a1)) (* (/ (cos th) (sqrt 2)) (* a2 a2))))