Migdal et al, Equation (64)

Average Error: 0.5 → 0.5

Time: 11.2s

Precision: 64

Math

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

↓

\[\frac{\frac{\cos th \cdot \mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\sqrt{\sqrt[3]{\sqrt[3]{\sqrt{2}}}}}}{\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}} \cdot \frac{\frac{1}{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}}{\sqrt{\sqrt[3]{\sqrt[3]{\sqrt{2}}}}}\]

C

double f(double a1, double a2, double th) {
        double r124894 = th;
        double r124895 = cos(r124894);
        double r124896 = 2.0;
        double r124897 = sqrt(r124896);
        double r124898 = r124895 / r124897;
        double r124899 = a1;
        double r124900 = r124899 * r124899;
        double r124901 = r124898 * r124900;
        double r124902 = a2;
        double r124903 = r124902 * r124902;
        double r124904 = r124898 * r124903;
        double r124905 = r124901 + r124904;
        return r124905;
}

↓

double f(double a1, double a2, double th) {
        double r124906 = th;
        double r124907 = cos(r124906);
        double r124908 = a1;
        double r124909 = a2;
        double r124910 = r124909 * r124909;
        double r124911 = fma(r124908, r124908, r124910);
        double r124912 = r124907 * r124911;
        double r124913 = 2.0;
        double r124914 = sqrt(r124913);
        double r124915 = cbrt(r124914);
        double r124916 = cbrt(r124915);
        double r124917 = sqrt(r124916);
        double r124918 = r124912 / r124917;
        double r124919 = r124916 * r124916;
        double r124920 = r124918 / r124919;
        double r124921 = 1.0;
        double r124922 = r124915 * r124915;
        double r124923 = r124921 / r124922;
        double r124924 = r124923 / r124917;
        double r124925 = r124920 * r124924;
        return r124925;
}

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

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

4. Applied add-cube-cbrt 0.5

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

5. Applied associate-/r* 0.5

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

7. Applied add-cube-cbrt 0.5

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

8. Applied *-un-lft-identity 0.5

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

9. Applied times-frac 0.5

   \[\leadsto \color{blue}{\frac{1}{\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}} \cdot \frac{\frac{\cos th \cdot \mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}}{\sqrt[3]{\sqrt[3]{\sqrt{2}}}}}\]
10. Using strategy rm

11. Applied add-sqr-sqrt 0.5

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

12. Applied div-inv 0.5

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

13. Applied times-frac 0.5

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

14. Applied associate-*r* 0.5

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

15. Simplified 0.5

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

16. Final simplification 0.5

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

Reproduce

herbie shell --seed 2020062 +o rules:numerics
(FPCore (a1 a2 th)
  :name "Migdal et al, Equation (64)"
  :precision binary64
  (+ (* (/ (cos th) (sqrt 2)) (* a1 a1)) (* (/ (cos th) (sqrt 2)) (* a2 a2))))