Migdal et al, Equation (64)

Average Error: 0.5 → 0.4

Time: 10.2s

Precision: 64

Math

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

↓

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

C

double f(double a1, double a2, double th) {
        double r104228 = th;
        double r104229 = cos(r104228);
        double r104230 = 2.0;
        double r104231 = sqrt(r104230);
        double r104232 = r104229 / r104231;
        double r104233 = a1;
        double r104234 = r104233 * r104233;
        double r104235 = r104232 * r104234;
        double r104236 = a2;
        double r104237 = r104236 * r104236;
        double r104238 = r104232 * r104237;
        double r104239 = r104235 + r104238;
        return r104239;
}

↓

double f(double a1, double a2, double th) {
        double r104240 = 1.0;
        double r104241 = 2.0;
        double r104242 = sqrt(r104241);
        double r104243 = sqrt(r104242);
        double r104244 = sqrt(r104243);
        double r104245 = r104240 / r104244;
        double r104246 = r104245 / r104244;
        double r104247 = th;
        double r104248 = cos(r104247);
        double r104249 = r104248 / r104243;
        double r104250 = a1;
        double r104251 = r104250 * r104250;
        double r104252 = a2;
        double r104253 = r104252 * r104252;
        double r104254 = r104251 + r104253;
        double r104255 = r104249 * r104254;
        double r104256 = r104246 * r104255;
        return r104256;
}

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. Simplified 0.5

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

4. Applied add-sqr-sqrt 0.5

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

5. Applied sqrt-prod 0.5

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

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 0.5

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

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* 0.6

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

18. Simplified 0.4

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

19. Final simplification 0.4

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

Reproduce

herbie shell --seed 2020047 
(FPCore (a1 a2 th)
  :name "Migdal et al, Equation (64)"
  :precision binary64
  (+ (* (/ (cos th) (sqrt 2)) (* a1 a1)) (* (/ (cos th) (sqrt 2)) (* a2 a2))))