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 r102030 = th;
        double r102031 = cos(r102030);
        double r102032 = 2.0;
        double r102033 = sqrt(r102032);
        double r102034 = r102031 / r102033;
        double r102035 = a1;
        double r102036 = r102035 * r102035;
        double r102037 = r102034 * r102036;
        double r102038 = a2;
        double r102039 = r102038 * r102038;
        double r102040 = r102034 * r102039;
        double r102041 = r102037 + r102040;
        return r102041;
}

↓

double f(double a1, double a2, double th) {
        double r102042 = th;
        double r102043 = cos(r102042);
        double r102044 = a1;
        double r102045 = fabs(r102044);
        double r102046 = 2.0;
        double r102047 = sqrt(r102046);
        double r102048 = r102045 / r102047;
        double r102049 = r102045 * r102048;
        double r102050 = r102043 * r102049;
        double r102051 = a2;
        double r102052 = 2.0;
        double r102053 = pow(r102051, r102052);
        double r102054 = cbrt(r102047);
        double r102055 = r102054 * r102054;
        double r102056 = r102053 / r102055;
        double r102057 = r102056 / r102054;
        double r102058 = r102043 * r102057;
        double r102059 = r102050 + r102058;
        return r102059;
}

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))))