Average Error: 0.5 → 0.5
Time: 42.0s
Precision: 64
\[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)\]
↓
\[\left(\frac{\cos th}{\sqrt{2}} \cdot a2\right) \cdot a2 + \left(\frac{\frac{1}{\sqrt{\sqrt{2}}}}{\sqrt{\sqrt[3]{\sqrt{2}}}} \cdot \frac{\cos th}{\left|\sqrt[3]{\sqrt{2}}\right|}\right) \cdot \left(a1 \cdot a1\right)\]
\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)
↓
\left(\frac{\cos th}{\sqrt{2}} \cdot a2\right) \cdot a2 + \left(\frac{\frac{1}{\sqrt{\sqrt{2}}}}{\sqrt{\sqrt[3]{\sqrt{2}}}} \cdot \frac{\cos th}{\left|\sqrt[3]{\sqrt{2}}\right|}\right) \cdot \left(a1 \cdot a1\right)
double f(double a1, double a2, double th) {
        double r1283734 = th;
        double r1283735 = cos(r1283734);
        double r1283736 = 2.0;
        double r1283737 = sqrt(r1283736);
        double r1283738 = r1283735 / r1283737;
        double r1283739 = a1;
        double r1283740 = r1283739 * r1283739;
        double r1283741 = r1283738 * r1283740;
        double r1283742 = a2;
        double r1283743 = r1283742 * r1283742;
        double r1283744 = r1283738 * r1283743;
        double r1283745 = r1283741 + r1283744;
        return r1283745;
}
↓
double f(double a1, double a2, double th) {
        double r1283746 = th;
        double r1283747 = cos(r1283746);
        double r1283748 = 2.0;
        double r1283749 = sqrt(r1283748);
        double r1283750 = r1283747 / r1283749;
        double r1283751 = a2;
        double r1283752 = r1283750 * r1283751;
        double r1283753 = r1283752 * r1283751;
        double r1283754 = 1.0;
        double r1283755 = sqrt(r1283749);
        double r1283756 = r1283754 / r1283755;
        double r1283757 = cbrt(r1283749);
        double r1283758 = sqrt(r1283757);
        double r1283759 = r1283756 / r1283758;
        double r1283760 = fabs(r1283757);
        double r1283761 = r1283747 / r1283760;
        double r1283762 = r1283759 * r1283761;
        double r1283763 = a1;
        double r1283764 = r1283763 * r1283763;
        double r1283765 = r1283762 * r1283764;
        double r1283766 = r1283753 + r1283765;
        return r1283766;
}

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

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

  4. Applied sqrt-prod0.5

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

  5. Applied associate-/r*0.5

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

  7. Applied add-cube-cbrt0.5

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

  8. Applied sqrt-prod0.6

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

  9. Applied div-inv0.5

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

  10. Applied times-frac0.5

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

  11. Simplified0.5

    \[\leadsto \left(\color{blue}{\frac{\cos th}{\left|\sqrt[3]{\sqrt{2}}\right|}} \cdot \frac{\frac{1}{\sqrt{\sqrt{2}}}}{\sqrt{\sqrt[3]{\sqrt{2}}}}\right) \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)\]
  12. Using strategy rm

  13. Applied associate-*r*0.5

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

  14. Final simplification0.5

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

Reproduce

herbie shell --seed 2019152 
(FPCore (a1 a2 th)
  :name "Migdal et al, Equation (64)"
  (+ (* (/ (cos th) (sqrt 2)) (* a1 a1)) (* (/ (cos th) (sqrt 2)) (* a2 a2))))