Average Error: 0.5 → 0.5
Time: 50.4s
Precision: 64
\[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)\]
↓
\[\frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) + \left(a1 \cdot a1\right) \cdot \sqrt[3]{\frac{\cos th}{\frac{2 \cdot \left(\sqrt{\sqrt{2}} \cdot \sqrt{\sqrt{2}}\right)}{\cos th \cdot \cos th}}}\]
\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)
↓
\frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) + \left(a1 \cdot a1\right) \cdot \sqrt[3]{\frac{\cos th}{\frac{2 \cdot \left(\sqrt{\sqrt{2}} \cdot \sqrt{\sqrt{2}}\right)}{\cos th \cdot \cos th}}}
double f(double a1, double a2, double th) {
        double r1999234 = th;
        double r1999235 = cos(r1999234);
        double r1999236 = 2.0;
        double r1999237 = sqrt(r1999236);
        double r1999238 = r1999235 / r1999237;
        double r1999239 = a1;
        double r1999240 = r1999239 * r1999239;
        double r1999241 = r1999238 * r1999240;
        double r1999242 = a2;
        double r1999243 = r1999242 * r1999242;
        double r1999244 = r1999238 * r1999243;
        double r1999245 = r1999241 + r1999244;
        return r1999245;
}
↓
double f(double a1, double a2, double th) {
        double r1999246 = th;
        double r1999247 = cos(r1999246);
        double r1999248 = 2.0;
        double r1999249 = sqrt(r1999248);
        double r1999250 = r1999247 / r1999249;
        double r1999251 = a2;
        double r1999252 = r1999251 * r1999251;
        double r1999253 = r1999250 * r1999252;
        double r1999254 = a1;
        double r1999255 = r1999254 * r1999254;
        double r1999256 = sqrt(r1999249);
        double r1999257 = r1999256 * r1999256;
        double r1999258 = r1999248 * r1999257;
        double r1999259 = r1999247 * r1999247;
        double r1999260 = r1999258 / r1999259;
        double r1999261 = r1999247 / r1999260;
        double r1999262 = cbrt(r1999261);
        double r1999263 = r1999255 * r1999262;
        double r1999264 = r1999253 + r1999263;
        return r1999264;
}

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 add-sqr-sqrt0.5

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

  4. Applied associate-/r*0.5

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

  6. Applied add-cbrt-cube0.9

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

  7. Applied add-cbrt-cube1.1

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

  8. Applied add-cbrt-cube1.1

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

  9. Applied cbrt-undiv0.8

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

  10. Applied cbrt-undiv0.5

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

  11. Simplified0.5

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

  12. Final simplification0.5

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

Reproduce

herbie shell --seed 2019130 
(FPCore (a1 a2 th)
  :name "Migdal et al, Equation (64)"
  (+ (* (/ (cos th) (sqrt 2)) (* a1 a1)) (* (/ (cos th) (sqrt 2)) (* a2 a2))))