Average Error: 0.5 → 0.5
Time: 26.7s
Precision: 64
\[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)\]
↓
\[\cos th \cdot \left(\frac{\frac{a2 \cdot a2}{\sqrt{\sqrt{\sqrt[3]{2}}} \cdot \sqrt{\sqrt{2}}}}{\sqrt{\sqrt{\sqrt[3]{2} \cdot \sqrt[3]{2}}}} + \frac{a1 \cdot a1}{\sqrt{2}}\right)\]
\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)
↓
\cos th \cdot \left(\frac{\frac{a2 \cdot a2}{\sqrt{\sqrt{\sqrt[3]{2}}} \cdot \sqrt{\sqrt{2}}}}{\sqrt{\sqrt{\sqrt[3]{2} \cdot \sqrt[3]{2}}}} + \frac{a1 \cdot a1}{\sqrt{2}}\right)
double f(double a1, double a2, double th) {
        double r73766 = th;
        double r73767 = cos(r73766);
        double r73768 = 2.0;
        double r73769 = sqrt(r73768);
        double r73770 = r73767 / r73769;
        double r73771 = a1;
        double r73772 = r73771 * r73771;
        double r73773 = r73770 * r73772;
        double r73774 = a2;
        double r73775 = r73774 * r73774;
        double r73776 = r73770 * r73775;
        double r73777 = r73773 + r73776;
        return r73777;
}
↓
double f(double a1, double a2, double th) {
        double r73778 = th;
        double r73779 = cos(r73778);
        double r73780 = a2;
        double r73781 = r73780 * r73780;
        double r73782 = 2.0;
        double r73783 = cbrt(r73782);
        double r73784 = sqrt(r73783);
        double r73785 = sqrt(r73784);
        double r73786 = sqrt(r73782);
        double r73787 = sqrt(r73786);
        double r73788 = r73785 * r73787;
        double r73789 = r73781 / r73788;
        double r73790 = r73783 * r73783;
        double r73791 = sqrt(r73790);
        double r73792 = sqrt(r73791);
        double r73793 = r73789 / r73792;
        double r73794 = a1;
        double r73795 = r73794 * r73794;
        double r73796 = r73795 / r73786;
        double r73797 = r73793 + r73796;
        double r73798 = r73779 * r73797;
        return r73798;
}

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 add-sqr-sqrt0.5

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

  4. Applied sqrt-prod0.5

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

  5. Applied associate-/r*0.5

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

  7. Applied add-cube-cbrt0.6

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

  8. Applied sqrt-prod0.6

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

  9. Applied sqrt-prod0.6

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

  10. Applied div-inv0.5

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

  11. Applied times-frac0.5

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

  12. Applied associate-*l*0.5

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

  13. Final simplification0.5

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

Reproduce

herbie shell --seed 2019297 
(FPCore (a1 a2 th)
  :name "Migdal et al, Equation (64)"
  :precision binary64
  (+ (* (/ (cos th) (sqrt 2)) (* a1 a1)) (* (/ (cos th) (sqrt 2)) (* a2 a2))))