Average Error: 0.5 → 0.5
Time: 14.6s
Precision: 64
\[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)\]
↓
\[\cos th \cdot \frac{{a2}^{2} + {a1}^{2}}{\sqrt{2}}\]
\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{{a2}^{2} + {a1}^{2}}{\sqrt{2}}
double f(double a1, double a2, double th) {
        double r244 = th;
        double r245 = cos(r244);
        double r246 = 2.0;
        double r247 = sqrt(r246);
        double r248 = r245 / r247;
        double r249 = a1;
        double r250 = r249 * r249;
        double r251 = r248 * r250;
        double r252 = a2;
        double r253 = r252 * r252;
        double r254 = r248 * r253;
        double r255 = r251 + r254;
        return r255;
}
↓
double f(double a1, double a2, double th) {
        double r256 = th;
        double r257 = cos(r256);
        double r258 = a2;
        double r259 = 2.0;
        double r260 = pow(r258, r259);
        double r261 = a1;
        double r262 = pow(r261, r259);
        double r263 = r260 + r262;
        double r264 = 2.0;
        double r265 = sqrt(r264);
        double r266 = r263 / r265;
        double r267 = r257 * r266;
        return r267;
}

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. Simplified0.5

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

  4. Applied div-inv0.5

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

  5. Applied associate-*l*0.5

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

  6. Simplified0.5

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

  7. Final simplification0.5

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

Reproduce

herbie shell --seed 2020025 
(FPCore (a1 a2 th)
  :name "Migdal et al, Equation (64)"
  :precision binary64
  (+ (* (/ (cos th) (sqrt 2)) (* a1 a1)) (* (/ (cos th) (sqrt 2)) (* a2 a2))))