Average Error: 0.5 → 0.5
Time: 41.1s
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 r2687828 = th;
        double r2687829 = cos(r2687828);
        double r2687830 = 2.0;
        double r2687831 = sqrt(r2687830);
        double r2687832 = r2687829 / r2687831;
        double r2687833 = a1;
        double r2687834 = r2687833 * r2687833;
        double r2687835 = r2687832 * r2687834;
        double r2687836 = a2;
        double r2687837 = r2687836 * r2687836;
        double r2687838 = r2687832 * r2687837;
        double r2687839 = r2687835 + r2687838;
        return r2687839;
}
↓
double f(double a1, double a2, double th) {
        double r2687840 = a2;
        double r2687841 = th;
        double r2687842 = cos(r2687841);
        double r2687843 = r2687840 * r2687842;
        double r2687844 = 2.0;
        double r2687845 = sqrt(r2687844);
        double r2687846 = r2687843 / r2687845;
        double r2687847 = r2687840 * r2687846;
        double r2687848 = a1;
        double r2687849 = r2687848 * r2687848;
        double r2687850 = r2687842 / r2687845;
        double r2687851 = r2687849 * r2687850;
        double r2687852 = r2687847 + r2687851;
        return r2687852;
}

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

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

  3. Simplified0.5

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

  4. 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 +o rules:numerics
(FPCore (a1 a2 th)
  :name "Migdal et al, Equation (64)"
  (+ (* (/ (cos th) (sqrt 2)) (* a1 a1)) (* (/ (cos th) (sqrt 2)) (* a2 a2))))