Migdal et al, Equation (64)

Average Error: 0.5 → 0.4

Time: 9.3s

Precision: 64

Math

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

↓

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

C

double f(double a1, double a2, double th) {
        double r116160 = th;
        double r116161 = cos(r116160);
        double r116162 = 2.0;
        double r116163 = sqrt(r116162);
        double r116164 = r116161 / r116163;
        double r116165 = a1;
        double r116166 = r116165 * r116165;
        double r116167 = r116164 * r116166;
        double r116168 = a2;
        double r116169 = r116168 * r116168;
        double r116170 = r116164 * r116169;
        double r116171 = r116167 + r116170;
        return r116171;
}

↓

double f(double a1, double a2, double th) {
        double r116172 = th;
        double r116173 = cos(r116172);
        double r116174 = 2.0;
        double r116175 = sqrt(r116174);
        double r116176 = cbrt(r116175);
        double r116177 = cbrt(r116176);
        double r116178 = r116177 * r116177;
        double r116179 = r116173 / r116178;
        double r116180 = a1;
        double r116181 = a2;
        double r116182 = hypot(r116180, r116181);
        double r116183 = r116182 / r116177;
        double r116184 = r116176 * r116176;
        double r116185 = r116182 / r116184;
        double r116186 = r116183 * r116185;
        double r116187 = r116179 * r116186;
        return r116187;
}

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-sqr-sqrt 0.5

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

8. Applied associate-*r* 0.5

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

9. Simplified 0.5

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

11. Applied *-un-lft-identity 0.5

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

12. Applied times-frac 0.7

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

13. Applied times-frac 0.7

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

14. Simplified 0.7

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

15. Simplified 0.4

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

17. Applied add-cube-cbrt 0.4

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

18. Applied times-frac 0.4

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

19. Applied associate-*l* 0.4

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

20. Final simplification 0.4

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

Reproduce

herbie shell --seed 2020059 +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))))