Migdal et al, Equation (64)

Average Error: 0.5 → 0.5

Time: 6.4s

Precision: 64

Math

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

↓

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

C

double f(double a1, double a2, double th) {
        double r76051 = th;
        double r76052 = cos(r76051);
        double r76053 = 2.0;
        double r76054 = sqrt(r76053);
        double r76055 = r76052 / r76054;
        double r76056 = a1;
        double r76057 = r76056 * r76056;
        double r76058 = r76055 * r76057;
        double r76059 = a2;
        double r76060 = r76059 * r76059;
        double r76061 = r76055 * r76060;
        double r76062 = r76058 + r76061;
        return r76062;
}

↓

double f(double a1, double a2, double th) {
        double r76063 = a1;
        double r76064 = a2;
        double r76065 = hypot(r76063, r76064);
        double r76066 = th;
        double r76067 = cos(r76066);
        double r76068 = r76065 * r76067;
        double r76069 = 2.0;
        double r76070 = sqrt(r76069);
        double r76071 = r76065 / r76070;
        double r76072 = r76068 * r76071;
        return r76072;
}

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-sqr-sqrt 0.5

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

5. Applied associate-*r* 0.5

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

6. Simplified 0.5

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

8. Applied *-un-lft-identity 0.5

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

9. Applied sqrt-prod 0.5

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

10. Applied times-frac 0.5

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

11. Simplified 0.5

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

12. Simplified 0.5

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

14. Applied *-commutative 0.5

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

15. Final simplification 0.5

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

Reproduce

herbie shell --seed 2020046 +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))))