Migdal et al, Equation (64)

Average Error: 0.6 → 0.5

Time: 13.2s

Precision: 64

Math

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

↓

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

C

double f(double a1, double a2, double th) {
        double r140156 = th;
        double r140157 = cos(r140156);
        double r140158 = 2.0;
        double r140159 = sqrt(r140158);
        double r140160 = r140157 / r140159;
        double r140161 = a1;
        double r140162 = r140161 * r140161;
        double r140163 = r140160 * r140162;
        double r140164 = a2;
        double r140165 = r140164 * r140164;
        double r140166 = r140160 * r140165;
        double r140167 = r140163 + r140166;
        return r140167;
}

↓

double f(double a1, double a2, double th) {
        double r140168 = 1.0;
        double r140169 = 2.0;
        double r140170 = sqrt(r140169);
        double r140171 = sqrt(r140170);
        double r140172 = sqrt(r140171);
        double r140173 = r140168 / r140172;
        double r140174 = r140173 / r140172;
        double r140175 = th;
        double r140176 = cos(r140175);
        double r140177 = r140168 / r140171;
        double r140178 = r140176 * r140177;
        double r140179 = a1;
        double r140180 = r140179 * r140179;
        double r140181 = a2;
        double r140182 = r140181 * r140181;
        double r140183 = r140180 + r140182;
        double r140184 = r140178 * r140183;
        double r140185 = r140174 * r140184;
        return r140185;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus th

Derivation

1. Initial program 0.6

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

2. Simplified 0.6

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

4. Applied add-sqr-sqrt 0.6

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

5. Applied sqrt-prod 0.6

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

6. Applied associate-/r* 0.5

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

8. Applied add-sqr-sqrt 0.5

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

9. Applied sqrt-prod 0.5

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

10. Applied sqrt-prod 0.5

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

11. Applied add-sqr-sqrt 0.5

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

12. Applied sqrt-prod 0.5

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

13. Applied sqrt-prod 0.5

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

14. Applied *-un-lft-identity 0.5

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

15. Applied times-frac 0.6

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

16. Applied times-frac 0.6

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

17. Applied associate-*l* 0.7

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

18. Simplified 0.5

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

20. Applied div-inv 0.5

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

21. Final simplification 0.5

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

Reproduce

herbie shell --seed 2020083 
(FPCore (a1 a2 th)
  :name "Migdal et al, Equation (64)"
  :precision binary64
  (+ (* (/ (cos th) (sqrt 2)) (* a1 a1)) (* (/ (cos th) (sqrt 2)) (* a2 a2))))