Migdal et al, Equation (64)

Average Error: 0.5 → 0.4

Time: 24.9s

Precision: 64

Math

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

↓

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

C

double f(double a1, double a2, double th) {
        double r1809050 = th;
        double r1809051 = cos(r1809050);
        double r1809052 = 2.0;
        double r1809053 = sqrt(r1809052);
        double r1809054 = r1809051 / r1809053;
        double r1809055 = a1;
        double r1809056 = r1809055 * r1809055;
        double r1809057 = r1809054 * r1809056;
        double r1809058 = a2;
        double r1809059 = r1809058 * r1809058;
        double r1809060 = r1809054 * r1809059;
        double r1809061 = r1809057 + r1809060;
        return r1809061;
}

↓

double f(double a1, double a2, double th) {
        double r1809062 = th;
        double r1809063 = cos(r1809062);
        double r1809064 = a1;
        double r1809065 = a2;
        double r1809066 = hypot(r1809064, r1809065);
        double r1809067 = r1809063 * r1809066;
        double r1809068 = r1809065 * r1809065;
        double r1809069 = fma(r1809064, r1809064, r1809068);
        double r1809070 = 2.0;
        double r1809071 = r1809069 / r1809070;
        double r1809072 = sqrt(r1809071);
        double r1809073 = r1809067 * r1809072;
        return r1809073;
}

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

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

4. Applied *-un-lft-identity 0.5

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

5. Applied sqrt-prod 0.5

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

6. Applied add-sqr-sqrt 0.5

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

7. Applied times-frac 0.5

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

8. Simplified 0.5

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

9. Simplified 0.5

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

11. Applied associate-*r* 0.5

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

13. Applied hypot-udef 0.5

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

14. Applied sqrt-undiv 0.4

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

15. Simplified 0.4

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

16. Final simplification 0.4

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

Reproduce

herbie shell --seed 2019174 +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))))