Average Error: 0.5 → 0.4
Time: 16.6s
Precision: 64
\[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)\]
↓
\[\frac{{a1}^{2} \cdot \cos th}{\sqrt{2}} + \frac{1}{\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}} \cdot \frac{\frac{\left(a2 \cdot \cos th\right) \cdot a2}{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}}{\sqrt[3]{\sqrt[3]{\sqrt{2}}}}\]
\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)
↓
\frac{{a1}^{2} \cdot \cos th}{\sqrt{2}} + \frac{1}{\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}} \cdot \frac{\frac{\left(a2 \cdot \cos th\right) \cdot a2}{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}}{\sqrt[3]{\sqrt[3]{\sqrt{2}}}}
double f(double a1, double a2, double th) {
        double r78151 = th;
        double r78152 = cos(r78151);
        double r78153 = 2.0;
        double r78154 = sqrt(r78153);
        double r78155 = r78152 / r78154;
        double r78156 = a1;
        double r78157 = r78156 * r78156;
        double r78158 = r78155 * r78157;
        double r78159 = a2;
        double r78160 = r78159 * r78159;
        double r78161 = r78155 * r78160;
        double r78162 = r78158 + r78161;
        return r78162;
}
↓
double f(double a1, double a2, double th) {
        double r78163 = a1;
        double r78164 = 2.0;
        double r78165 = pow(r78163, r78164);
        double r78166 = th;
        double r78167 = cos(r78166);
        double r78168 = r78165 * r78167;
        double r78169 = 2.0;
        double r78170 = sqrt(r78169);
        double r78171 = r78168 / r78170;
        double r78172 = 1.0;
        double r78173 = cbrt(r78170);
        double r78174 = cbrt(r78173);
        double r78175 = r78174 * r78174;
        double r78176 = r78172 / r78175;
        double r78177 = a2;
        double r78178 = r78177 * r78167;
        double r78179 = r78178 * r78177;
        double r78180 = r78173 * r78173;
        double r78181 = r78179 / r78180;
        double r78182 = r78181 / r78174;
        double r78183 = r78176 * r78182;
        double r78184 = r78171 + r78183;
        return r78184;
}

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 distribute-lft-in0.5

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

  5. Simplified0.5

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

  6. Simplified0.5

    \[\leadsto \frac{{a1}^{2} \cdot \cos th}{\sqrt{2}} + \color{blue}{\frac{\cos th \cdot {a2}^{2}}{\sqrt{2}}}\]
  7. Using strategy rm

  8. Applied unpow20.5

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

  9. Applied associate-*r*0.4

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

  10. Simplified0.4

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

  12. Applied add-cube-cbrt0.4

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

  13. Applied associate-/r*0.5

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

  15. Applied add-cube-cbrt0.5

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

  16. Applied *-un-lft-identity0.5

    \[\leadsto \frac{{a1}^{2} \cdot \cos th}{\sqrt{2}} + \frac{\color{blue}{1 \cdot \frac{\left(a2 \cdot \cos th\right) \cdot a2}{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}}}{\left(\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}\right) \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}}\]

  17. Applied times-frac0.4

    \[\leadsto \frac{{a1}^{2} \cdot \cos th}{\sqrt{2}} + \color{blue}{\frac{1}{\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}} \cdot \frac{\frac{\left(a2 \cdot \cos th\right) \cdot a2}{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}}{\sqrt[3]{\sqrt[3]{\sqrt{2}}}}}\]

  18. Final simplification0.4

    \[\leadsto \frac{{a1}^{2} \cdot \cos th}{\sqrt{2}} + \frac{1}{\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}} \cdot \frac{\frac{\left(a2 \cdot \cos th\right) \cdot a2}{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}}{\sqrt[3]{\sqrt[3]{\sqrt{2}}}}\]

Reproduce

herbie shell --seed 2019347 
(FPCore (a1 a2 th)
  :name "Migdal et al, Equation (64)"
  :precision binary64
  (+ (* (/ (cos th) (sqrt 2)) (* a1 a1)) (* (/ (cos th) (sqrt 2)) (* a2 a2))))