Average Error: 0.5 → 0.5
Time: 31.5s
Precision: 64
\[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)\]
↓
\[\frac{\frac{\cos th}{\sqrt{\sqrt{2}}}}{\sqrt{\sqrt{2}}} \cdot \left(a1 \cdot a1\right) + \frac{\frac{\cos th}{\sqrt{\sqrt{2}}}}{\sqrt{\sqrt{2}}} \cdot \left(a2 \cdot a2\right)\]
\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)
↓
\frac{\frac{\cos th}{\sqrt{\sqrt{2}}}}{\sqrt{\sqrt{2}}} \cdot \left(a1 \cdot a1\right) + \frac{\frac{\cos th}{\sqrt{\sqrt{2}}}}{\sqrt{\sqrt{2}}} \cdot \left(a2 \cdot a2\right)
double f(double a1, double a2, double th) {
        double r89732 = th;
        double r89733 = cos(r89732);
        double r89734 = 2.0;
        double r89735 = sqrt(r89734);
        double r89736 = r89733 / r89735;
        double r89737 = a1;
        double r89738 = r89737 * r89737;
        double r89739 = r89736 * r89738;
        double r89740 = a2;
        double r89741 = r89740 * r89740;
        double r89742 = r89736 * r89741;
        double r89743 = r89739 + r89742;
        return r89743;
}
↓
double f(double a1, double a2, double th) {
        double r89744 = th;
        double r89745 = cos(r89744);
        double r89746 = 2.0;
        double r89747 = sqrt(r89746);
        double r89748 = sqrt(r89747);
        double r89749 = r89745 / r89748;
        double r89750 = r89749 / r89748;
        double r89751 = a1;
        double r89752 = r89751 * r89751;
        double r89753 = r89750 * r89752;
        double r89754 = a2;
        double r89755 = r89754 * r89754;
        double r89756 = r89750 * r89755;
        double r89757 = r89753 + r89756;
        return r89757;
}

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

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

  4. Applied sqrt-prod0.5

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

  5. Applied associate-/r*0.5

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

  7. Applied add-sqr-sqrt0.5

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

  8. Applied sqrt-prod0.5

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

  9. Applied associate-/r*0.5

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

  10. Final simplification0.5

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

Reproduce

herbie shell --seed 2019326 
(FPCore (a1 a2 th)
  :name "Migdal et al, Equation (64)"
  :precision binary64
  (+ (* (/ (cos th) (sqrt 2)) (* a1 a1)) (* (/ (cos th) (sqrt 2)) (* a2 a2))))