Migdal et al, Equation (64)

Average Error: 0.5 → 0.5

Time: 7.3s

Precision: 64

Math

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

↓

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

C

double f(double a1, double a2, double th) {
        double r102042 = th;
        double r102043 = cos(r102042);
        double r102044 = 2.0;
        double r102045 = sqrt(r102044);
        double r102046 = r102043 / r102045;
        double r102047 = a1;
        double r102048 = r102047 * r102047;
        double r102049 = r102046 * r102048;
        double r102050 = a2;
        double r102051 = r102050 * r102050;
        double r102052 = r102046 * r102051;
        double r102053 = r102049 + r102052;
        return r102053;
}

↓

double f(double a1, double a2, double th) {
        double r102054 = th;
        double r102055 = cos(r102054);
        double r102056 = a1;
        double r102057 = fabs(r102056);
        double r102058 = 2.0;
        double r102059 = sqrt(r102058);
        double r102060 = r102057 / r102059;
        double r102061 = r102057 * r102060;
        double r102062 = r102055 * r102061;
        double r102063 = a2;
        double r102064 = 2.0;
        double r102065 = pow(r102063, r102064);
        double r102066 = cbrt(r102059);
        double r102067 = r102066 * r102066;
        double r102068 = r102065 / r102067;
        double r102069 = r102068 / r102066;
        double r102070 = r102055 * r102069;
        double r102071 = r102062 + r102070;
        return r102071;
}

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 div-inv 0.5

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

4. Applied associate-*l* 0.5

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

5. Simplified 0.5

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

7. Applied div-inv 0.5

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

8. Applied associate-*l* 0.5

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

9. Simplified 0.5

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

11. Applied *-un-lft-identity 0.5

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

12. Applied sqrt-prod 0.5

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

13. Applied add-sqr-sqrt 0.5

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

14. Applied times-frac 0.5

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

15. Simplified 0.5

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

16. Simplified 0.5

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

18. Applied add-cube-cbrt 0.5

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

19. Applied associate-/r* 0.5

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

20. Final simplification 0.5

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

Reproduce

herbie shell --seed 2020001 
(FPCore (a1 a2 th)
  :name "Migdal et al, Equation (64)"
  :precision binary64
  (+ (* (/ (cos th) (sqrt 2)) (* a1 a1)) (* (/ (cos th) (sqrt 2)) (* a2 a2))))