Migdal et al, Equation (64)

Average Error: 0.5 → 0.5

Time: 32.6s

Precision: 64

Math

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

↓

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

C

double f(double a1, double a2, double th) {
        double r63766 = th;
        double r63767 = cos(r63766);
        double r63768 = 2.0;
        double r63769 = sqrt(r63768);
        double r63770 = r63767 / r63769;
        double r63771 = a1;
        double r63772 = r63771 * r63771;
        double r63773 = r63770 * r63772;
        double r63774 = a2;
        double r63775 = r63774 * r63774;
        double r63776 = r63770 * r63775;
        double r63777 = r63773 + r63776;
        return r63777;
}

↓

double f(double a1, double a2, double th) {
        double r63778 = th;
        double r63779 = cos(r63778);
        double r63780 = a2;
        double r63781 = 2.0;
        double r63782 = pow(r63780, r63781);
        double r63783 = r63779 * r63782;
        double r63784 = 2.0;
        double r63785 = sqrt(r63784);
        double r63786 = r63783 / r63785;
        double r63787 = a1;
        double r63788 = sqrt(r63785);
        double r63789 = r63787 / r63788;
        double r63790 = r63779 * r63789;
        double r63791 = r63790 * r63789;
        double r63792 = r63786 + r63791;
        return r63792;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus th

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 *-un-lft-identity 0.5

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

4. Applied *-un-lft-identity 0.5

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

5. Applied times-frac 0.5

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

6. Applied associate-*l* 0.5

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

7. Simplified 0.5

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

9. Applied div-inv 0.5

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

10. Applied associate-*l* 0.5

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

11. Simplified 0.5

    \[\leadsto \cos th \cdot \color{blue}{\frac{{a1}^{2}}{\sqrt{2}}} + \frac{1}{1} \cdot \frac{\cos th \cdot {a2}^{2}}{\sqrt{2}}\]
12. Using strategy rm

13. Applied add-sqr-sqrt 0.5

    \[\leadsto \cos th \cdot \frac{{a1}^{2}}{\sqrt{\color{blue}{\sqrt{2} \cdot \sqrt{2}}}} + \frac{1}{1} \cdot \frac{\cos th \cdot {a2}^{2}}{\sqrt{2}}\]

14. Applied sqrt-prod 0.5

    \[\leadsto \cos th \cdot \frac{{a1}^{2}}{\color{blue}{\sqrt{\sqrt{2}} \cdot \sqrt{\sqrt{2}}}} + \frac{1}{1} \cdot \frac{\cos th \cdot {a2}^{2}}{\sqrt{2}}\]

15. Applied unpow2 0.5

    \[\leadsto \cos th \cdot \frac{\color{blue}{a1 \cdot a1}}{\sqrt{\sqrt{2}} \cdot \sqrt{\sqrt{2}}} + \frac{1}{1} \cdot \frac{\cos th \cdot {a2}^{2}}{\sqrt{2}}\]

16. Applied times-frac 0.5

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

17. Applied associate-*r* 0.5

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

18. Final simplification 0.5

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

Reproduce

herbie shell --seed 2019199 
(FPCore (a1 a2 th)
  :name "Migdal et al, Equation (64)"
  (+ (* (/ (cos th) (sqrt 2.0)) (* a1 a1)) (* (/ (cos th) (sqrt 2.0)) (* a2 a2))))