Migdal et al, Equation (64)

Average Error: 0.5 → 0.5

Time: 5.1m

Precision: 64

Math

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

↓

\[a2 \cdot \left(a2 \cdot \frac{\cos th}{\sqrt{2}}\right) + \left(\left(\frac{\cos th}{\sqrt{\sqrt[3]{\sqrt{2}}}} \cdot \frac{1}{\sqrt{\sqrt{2}}}\right) \cdot \frac{1}{\left|\sqrt[3]{\sqrt{2}}\right|}\right) \cdot \left(a1 \cdot a1\right)\]

C

double f(double a1, double a2, double th) {
        double r1884031 = th;
        double r1884032 = cos(r1884031);
        double r1884033 = 2.0;
        double r1884034 = sqrt(r1884033);
        double r1884035 = r1884032 / r1884034;
        double r1884036 = a1;
        double r1884037 = r1884036 * r1884036;
        double r1884038 = r1884035 * r1884037;
        double r1884039 = a2;
        double r1884040 = r1884039 * r1884039;
        double r1884041 = r1884035 * r1884040;
        double r1884042 = r1884038 + r1884041;
        return r1884042;
}

↓

double f(double a1, double a2, double th) {
        double r1884043 = a2;
        double r1884044 = th;
        double r1884045 = cos(r1884044);
        double r1884046 = 2.0;
        double r1884047 = sqrt(r1884046);
        double r1884048 = r1884045 / r1884047;
        double r1884049 = r1884043 * r1884048;
        double r1884050 = r1884043 * r1884049;
        double r1884051 = cbrt(r1884047);
        double r1884052 = sqrt(r1884051);
        double r1884053 = r1884045 / r1884052;
        double r1884054 = 1.0;
        double r1884055 = sqrt(r1884047);
        double r1884056 = r1884054 / r1884055;
        double r1884057 = r1884053 * r1884056;
        double r1884058 = fabs(r1884051);
        double r1884059 = r1884054 / r1884058;
        double r1884060 = r1884057 * r1884059;
        double r1884061 = a1;
        double r1884062 = r1884061 * r1884061;
        double r1884063 = r1884060 * r1884062;
        double r1884064 = r1884050 + r1884063;
        return r1884064;
}

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 add-sqr-sqrt 0.5

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

4. Applied sqrt-prod 0.5

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

5. Applied associate-/r* 0.5

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

7. Applied *-un-lft-identity 0.5

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

8. Applied associate-/l* 0.5

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

10. Applied associate-*r* 0.5

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

12. Applied add-cube-cbrt 0.5

    \[\leadsto \frac{1}{\frac{\sqrt{\sqrt{2}}}{\frac{\cos th}{\sqrt{\color{blue}{\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \sqrt[3]{\sqrt{2}}}}}}} \cdot \left(a1 \cdot a1\right) + \left(\frac{\cos th}{\sqrt{2}} \cdot a2\right) \cdot a2\]

13. Applied sqrt-prod 0.7

    \[\leadsto \frac{1}{\frac{\sqrt{\sqrt{2}}}{\frac{\cos th}{\color{blue}{\sqrt{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}} \cdot \sqrt{\sqrt[3]{\sqrt{2}}}}}}} \cdot \left(a1 \cdot a1\right) + \left(\frac{\cos th}{\sqrt{2}} \cdot a2\right) \cdot a2\]

14. Applied *-un-lft-identity 0.7

    \[\leadsto \frac{1}{\frac{\sqrt{\sqrt{2}}}{\frac{\color{blue}{1 \cdot \cos th}}{\sqrt{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}} \cdot \sqrt{\sqrt[3]{\sqrt{2}}}}}} \cdot \left(a1 \cdot a1\right) + \left(\frac{\cos th}{\sqrt{2}} \cdot a2\right) \cdot a2\]

15. Applied times-frac 0.7

    \[\leadsto \frac{1}{\frac{\sqrt{\sqrt{2}}}{\color{blue}{\frac{1}{\sqrt{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}} \cdot \frac{\cos th}{\sqrt{\sqrt[3]{\sqrt{2}}}}}}} \cdot \left(a1 \cdot a1\right) + \left(\frac{\cos th}{\sqrt{2}} \cdot a2\right) \cdot a2\]

16. Applied *-un-lft-identity 0.7

    \[\leadsto \frac{1}{\frac{\sqrt{\color{blue}{1 \cdot \sqrt{2}}}}{\frac{1}{\sqrt{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}} \cdot \frac{\cos th}{\sqrt{\sqrt[3]{\sqrt{2}}}}}} \cdot \left(a1 \cdot a1\right) + \left(\frac{\cos th}{\sqrt{2}} \cdot a2\right) \cdot a2\]

17. Applied sqrt-prod 0.7

    \[\leadsto \frac{1}{\frac{\color{blue}{\sqrt{1} \cdot \sqrt{\sqrt{2}}}}{\frac{1}{\sqrt{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}} \cdot \frac{\cos th}{\sqrt{\sqrt[3]{\sqrt{2}}}}}} \cdot \left(a1 \cdot a1\right) + \left(\frac{\cos th}{\sqrt{2}} \cdot a2\right) \cdot a2\]

18. Applied times-frac 0.5

    \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{1}}{\frac{1}{\sqrt{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}}} \cdot \frac{\sqrt{\sqrt{2}}}{\frac{\cos th}{\sqrt{\sqrt[3]{\sqrt{2}}}}}}} \cdot \left(a1 \cdot a1\right) + \left(\frac{\cos th}{\sqrt{2}} \cdot a2\right) \cdot a2\]

19. Applied add-cube-cbrt 0.5

    \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\frac{\sqrt{1}}{\frac{1}{\sqrt{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}}} \cdot \frac{\sqrt{\sqrt{2}}}{\frac{\cos th}{\sqrt{\sqrt[3]{\sqrt{2}}}}}} \cdot \left(a1 \cdot a1\right) + \left(\frac{\cos th}{\sqrt{2}} \cdot a2\right) \cdot a2\]

20. Applied times-frac 0.5

    \[\leadsto \color{blue}{\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\frac{\sqrt{1}}{\frac{1}{\sqrt{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}}}} \cdot \frac{\sqrt[3]{1}}{\frac{\sqrt{\sqrt{2}}}{\frac{\cos th}{\sqrt{\sqrt[3]{\sqrt{2}}}}}}\right)} \cdot \left(a1 \cdot a1\right) + \left(\frac{\cos th}{\sqrt{2}} \cdot a2\right) \cdot a2\]

21. Simplified 0.5

    \[\leadsto \left(\color{blue}{\frac{1}{\left|\sqrt[3]{\sqrt{2}}\right|}} \cdot \frac{\sqrt[3]{1}}{\frac{\sqrt{\sqrt{2}}}{\frac{\cos th}{\sqrt{\sqrt[3]{\sqrt{2}}}}}}\right) \cdot \left(a1 \cdot a1\right) + \left(\frac{\cos th}{\sqrt{2}} \cdot a2\right) \cdot a2\]

22. Simplified 0.5

    \[\leadsto \left(\frac{1}{\left|\sqrt[3]{\sqrt{2}}\right|} \cdot \color{blue}{\left(\frac{1}{\sqrt{\sqrt{2}}} \cdot \frac{\cos th}{\sqrt{\sqrt[3]{\sqrt{2}}}}\right)}\right) \cdot \left(a1 \cdot a1\right) + \left(\frac{\cos th}{\sqrt{2}} \cdot a2\right) \cdot a2\]

23. Final simplification 0.5

    \[\leadsto a2 \cdot \left(a2 \cdot \frac{\cos th}{\sqrt{2}}\right) + \left(\left(\frac{\cos th}{\sqrt{\sqrt[3]{\sqrt{2}}}} \cdot \frac{1}{\sqrt{\sqrt{2}}}\right) \cdot \frac{1}{\left|\sqrt[3]{\sqrt{2}}\right|}\right) \cdot \left(a1 \cdot a1\right)\]

Reproduce

herbie shell --seed 2019151 +o rules:numerics
(FPCore (a1 a2 th)
  :name "Migdal et al, Equation (64)"
  (+ (* (/ (cos th) (sqrt 2)) (* a1 a1)) (* (/ (cos th) (sqrt 2)) (* a2 a2))))