Average Error: 0.5 → 0.5
Time: 24.3s
Precision: 64
\[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)\]
↓
\[\frac{\frac{\cos th \cdot \sqrt{a2 \cdot a2 + a1 \cdot a1}}{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}}{\sqrt[3]{\sqrt{2}}} \cdot \sqrt{a2 \cdot a2 + a1 \cdot a1}\]
\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)
↓
\frac{\frac{\cos th \cdot \sqrt{a2 \cdot a2 + a1 \cdot a1}}{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}}{\sqrt[3]{\sqrt{2}}} \cdot \sqrt{a2 \cdot a2 + a1 \cdot a1}
double f(double a1, double a2, double th) {
        double r2306714 = th;
        double r2306715 = cos(r2306714);
        double r2306716 = 2.0;
        double r2306717 = sqrt(r2306716);
        double r2306718 = r2306715 / r2306717;
        double r2306719 = a1;
        double r2306720 = r2306719 * r2306719;
        double r2306721 = r2306718 * r2306720;
        double r2306722 = a2;
        double r2306723 = r2306722 * r2306722;
        double r2306724 = r2306718 * r2306723;
        double r2306725 = r2306721 + r2306724;
        return r2306725;
}
↓
double f(double a1, double a2, double th) {
        double r2306726 = th;
        double r2306727 = cos(r2306726);
        double r2306728 = a2;
        double r2306729 = r2306728 * r2306728;
        double r2306730 = a1;
        double r2306731 = r2306730 * r2306730;
        double r2306732 = r2306729 + r2306731;
        double r2306733 = sqrt(r2306732);
        double r2306734 = r2306727 * r2306733;
        double r2306735 = 2.0;
        double r2306736 = sqrt(r2306735);
        double r2306737 = cbrt(r2306736);
        double r2306738 = r2306737 * r2306737;
        double r2306739 = r2306734 / r2306738;
        double r2306740 = r2306739 / r2306737;
        double r2306741 = r2306740 * r2306733;
        return r2306741;
}

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. Simplified0.8

    \[\leadsto \color{blue}{\frac{\cos th}{\frac{\sqrt{2}}{a1 \cdot a1 + a2 \cdot a2}}}\]
  3. Using strategy rm

  4. Applied add-sqr-sqrt0.8

    \[\leadsto \frac{\cos th}{\frac{\sqrt{2}}{\color{blue}{\sqrt{a1 \cdot a1 + a2 \cdot a2} \cdot \sqrt{a1 \cdot a1 + a2 \cdot a2}}}}\]

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

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

  6. Applied sqrt-prod0.8

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

  7. Applied times-frac0.8

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

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

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

  9. Applied times-frac0.5

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

  10. Simplified0.5

    \[\leadsto \color{blue}{\sqrt{a2 \cdot a2 + a1 \cdot a1}} \cdot \frac{\cos th}{\frac{\sqrt{2}}{\sqrt{a1 \cdot a1 + a2 \cdot a2}}}\]

  11. Simplified0.5

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

  13. Applied add-cube-cbrt0.5

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

  14. Applied associate-/r*0.5

    \[\leadsto \sqrt{a2 \cdot a2 + a1 \cdot a1} \cdot \color{blue}{\frac{\frac{\cos th \cdot \sqrt{a2 \cdot a2 + a1 \cdot a1}}{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}}{\sqrt[3]{\sqrt{2}}}}\]

  15. Final simplification0.5

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

Reproduce

herbie shell --seed 2019162 
(FPCore (a1 a2 th)
  :name "Migdal et al, Equation (64)"
  (+ (* (/ (cos th) (sqrt 2)) (* a1 a1)) (* (/ (cos th) (sqrt 2)) (* a2 a2))))