Average Error: 0.6 → 0.5
Time: 20.4s
Precision: 64
\[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)\]
↓
\[\cos th \cdot \frac{\frac{{a1}^{2}}{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}}{\sqrt[3]{\sqrt{2}}} + \frac{\cos th \cdot {a2}^{2}}{\sqrt{2}}\]
\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)
↓
\cos th \cdot \frac{\frac{{a1}^{2}}{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}}{\sqrt[3]{\sqrt{2}}} + \frac{\cos th \cdot {a2}^{2}}{\sqrt{2}}
double f(double a1, double a2, double th) {
        double r80816 = th;
        double r80817 = cos(r80816);
        double r80818 = 2.0;
        double r80819 = sqrt(r80818);
        double r80820 = r80817 / r80819;
        double r80821 = a1;
        double r80822 = r80821 * r80821;
        double r80823 = r80820 * r80822;
        double r80824 = a2;
        double r80825 = r80824 * r80824;
        double r80826 = r80820 * r80825;
        double r80827 = r80823 + r80826;
        return r80827;
}
↓
double f(double a1, double a2, double th) {
        double r80828 = th;
        double r80829 = cos(r80828);
        double r80830 = a1;
        double r80831 = 2.0;
        double r80832 = pow(r80830, r80831);
        double r80833 = 2.0;
        double r80834 = sqrt(r80833);
        double r80835 = cbrt(r80834);
        double r80836 = r80835 * r80835;
        double r80837 = r80832 / r80836;
        double r80838 = r80837 / r80835;
        double r80839 = r80829 * r80838;
        double r80840 = a2;
        double r80841 = pow(r80840, r80831);
        double r80842 = r80829 * r80841;
        double r80843 = r80842 / r80834;
        double r80844 = r80839 + r80843;
        return r80844;
}

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 0.6

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

  3. Applied div-inv0.6

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

  4. Applied associate-*l*0.6

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

  5. Simplified0.5

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

  6. Taylor expanded around inf 0.5

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

  8. Applied add-cube-cbrt0.5

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

  9. Applied associate-/r*0.5

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

  10. Final simplification0.5

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

Reproduce

herbie shell --seed 2019209 
(FPCore (a1 a2 th)
  :name "Migdal et al, Equation (64)"
  :precision binary64
  (+ (* (/ (cos th) (sqrt 2)) (* a1 a1)) (* (/ (cos th) (sqrt 2)) (* a2 a2))))