Average Error: 0.5 → 0.5
Time: 46.5s
Precision: 64
\[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)\]
↓
\[a2 \cdot \frac{a2 \cdot \cos th}{\sqrt{2}} + \left(a1 \cdot a1\right) \cdot \frac{\cos th}{\sqrt{2}}\]
\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)
↓
a2 \cdot \frac{a2 \cdot \cos th}{\sqrt{2}} + \left(a1 \cdot a1\right) \cdot \frac{\cos th}{\sqrt{2}}
double f(double a1, double a2, double th) {
        double r3079843 = th;
        double r3079844 = cos(r3079843);
        double r3079845 = 2.0;
        double r3079846 = sqrt(r3079845);
        double r3079847 = r3079844 / r3079846;
        double r3079848 = a1;
        double r3079849 = r3079848 * r3079848;
        double r3079850 = r3079847 * r3079849;
        double r3079851 = a2;
        double r3079852 = r3079851 * r3079851;
        double r3079853 = r3079847 * r3079852;
        double r3079854 = r3079850 + r3079853;
        return r3079854;
}
↓
double f(double a1, double a2, double th) {
        double r3079855 = a2;
        double r3079856 = th;
        double r3079857 = cos(r3079856);
        double r3079858 = r3079855 * r3079857;
        double r3079859 = 2.0;
        double r3079860 = sqrt(r3079859);
        double r3079861 = r3079858 / r3079860;
        double r3079862 = r3079855 * r3079861;
        double r3079863 = a1;
        double r3079864 = r3079863 * r3079863;
        double r3079865 = r3079857 / r3079860;
        double r3079866 = r3079864 * r3079865;
        double r3079867 = r3079862 + r3079866;
        return r3079867;
}

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 associate-*r*0.5

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

  4. Taylor expanded around inf 0.5

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

  5. Final simplification0.5

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

Reproduce

herbie shell --seed 2019143 
(FPCore (a1 a2 th)
  :name "Migdal et al, Equation (64)"
  (+ (* (/ (cos th) (sqrt 2)) (* a1 a1)) (* (/ (cos th) (sqrt 2)) (* a2 a2))))