Migdal et al, Equation (64)

Average Error: 0.5 → 0.5

Time: 15.8s

Precision: 64

Math

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

↓

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

C

double f(double a1, double a2, double th) {
        double r268 = th;
        double r269 = cos(r268);
        double r270 = 2.0;
        double r271 = sqrt(r270);
        double r272 = r269 / r271;
        double r273 = a1;
        double r274 = r273 * r273;
        double r275 = r272 * r274;
        double r276 = a2;
        double r277 = r276 * r276;
        double r278 = r272 * r277;
        double r279 = r275 + r278;
        return r279;
}

↓

double f(double a1, double a2, double th) {
        double r280 = th;
        double r281 = cos(r280);
        double r282 = 2.0;
        double r283 = sqrt(r282);
        double r284 = sqrt(r283);
        double r285 = sqrt(r284);
        double r286 = r281 / r285;
        double r287 = a1;
        double r288 = a2;
        double r289 = r288 * r288;
        double r290 = fma(r287, r287, r289);
        double r291 = r290 / r284;
        double r292 = r291 / r285;
        double r293 = r286 * r292;
        return r293;
}

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

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

5. Applied sqrt-prod 0.5

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

6. Applied associate-/r* 0.5

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

8. Applied *-un-lft-identity 0.5

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

9. Applied sqrt-prod 0.5

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

10. Applied sqrt-prod 0.5

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

11. Applied add-sqr-sqrt 0.5

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

12. Applied sqrt-prod 0.5

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

13. Applied sqrt-prod 0.5

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

14. Applied times-frac 0.4

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

15. Applied times-frac 0.4

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

16. Simplified 0.4

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

17. Simplified 0.5

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

18. Final simplification 0.5

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

Reproduce

herbie shell --seed 2020025 +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))))