Migdal et al, Equation (64)

Average Error: 0.5 → 0.4

Time: 33.9s

Precision: 64

Math

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

↓

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

C

double f(double a1, double a2, double th) {
        double r3103203 = th;
        double r3103204 = cos(r3103203);
        double r3103205 = 2.0;
        double r3103206 = sqrt(r3103205);
        double r3103207 = r3103204 / r3103206;
        double r3103208 = a1;
        double r3103209 = r3103208 * r3103208;
        double r3103210 = r3103207 * r3103209;
        double r3103211 = a2;
        double r3103212 = r3103211 * r3103211;
        double r3103213 = r3103207 * r3103212;
        double r3103214 = r3103210 + r3103213;
        return r3103214;
}

↓

double f(double a1, double a2, double th) {
        double r3103215 = th;
        double r3103216 = cos(r3103215);
        double r3103217 = a2;
        double r3103218 = a1;
        double r3103219 = hypot(r3103217, r3103218);
        double r3103220 = r3103216 * r3103219;
        double r3103221 = 2.0;
        double r3103222 = sqrt(r3103221);
        double r3103223 = r3103219 / r3103222;
        double r3103224 = r3103220 * r3103223;
        return r3103224;
}

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

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

4. Applied associate-/l* 0.5

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

6. Applied *-un-lft-identity 0.5

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

7. Applied *-un-lft-identity 0.5

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

8. Applied sqrt-prod 0.5

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

9. Applied times-frac 0.5

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

10. Applied add-sqr-sqrt 0.5

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

11. Applied times-frac 0.5

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

12. Simplified 0.5

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

13. Simplified 0.5

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

15. Applied associate-*r* 0.4

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

16. Final simplification 0.4

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

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(FPCore (a1 a2 th)
  :name "Migdal et al, Equation (64)"
  (+ (* (/ (cos th) (sqrt 2.0)) (* a1 a1)) (* (/ (cos th) (sqrt 2.0)) (* a2 a2))))