Average Error: 0.5 → 0.5
Time: 54.3s
Precision: 64
\[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)\]
↓
\[a2 \cdot \left(a2 \cdot \frac{\cos th}{\sqrt{2}}\right) + \frac{a1 \cdot \cos th}{\frac{\sqrt{2}}{a1}}\]
\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)
↓
a2 \cdot \left(a2 \cdot \frac{\cos th}{\sqrt{2}}\right) + \frac{a1 \cdot \cos th}{\frac{\sqrt{2}}{a1}}
double f(double a1, double a2, double th) {
        double r7429860 = th;
        double r7429861 = cos(r7429860);
        double r7429862 = 2.0;
        double r7429863 = sqrt(r7429862);
        double r7429864 = r7429861 / r7429863;
        double r7429865 = a1;
        double r7429866 = r7429865 * r7429865;
        double r7429867 = r7429864 * r7429866;
        double r7429868 = a2;
        double r7429869 = r7429868 * r7429868;
        double r7429870 = r7429864 * r7429869;
        double r7429871 = r7429867 + r7429870;
        return r7429871;
}
↓
double f(double a1, double a2, double th) {
        double r7429872 = a2;
        double r7429873 = th;
        double r7429874 = cos(r7429873);
        double r7429875 = 2.0;
        double r7429876 = sqrt(r7429875);
        double r7429877 = r7429874 / r7429876;
        double r7429878 = r7429872 * r7429877;
        double r7429879 = r7429872 * r7429878;
        double r7429880 = a1;
        double r7429881 = r7429880 * r7429874;
        double r7429882 = r7429876 / r7429880;
        double r7429883 = r7429881 / r7429882;
        double r7429884 = r7429879 + r7429883;
        return r7429884;
}

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. Taylor expanded around inf 0.5

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

  3. Simplified0.5

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

  5. Applied associate-*r*0.5

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

  6. Final simplification0.5

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

Reproduce

herbie shell --seed 2019104 
(FPCore (a1 a2 th)
  :name "Migdal et al, Equation (64)"
  (+ (* (/ (cos th) (sqrt 2)) (* a1 a1)) (* (/ (cos th) (sqrt 2)) (* a2 a2))))