Migdal et al, Equation (64)

Average Error: 0.5 → 0.4

Time: 28.6s

Precision: 64

Math

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

↓

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

C

double f(double a1, double a2, double th) {
        double r3491237 = th;
        double r3491238 = cos(r3491237);
        double r3491239 = 2.0;
        double r3491240 = sqrt(r3491239);
        double r3491241 = r3491238 / r3491240;
        double r3491242 = a1;
        double r3491243 = r3491242 * r3491242;
        double r3491244 = r3491241 * r3491243;
        double r3491245 = a2;
        double r3491246 = r3491245 * r3491245;
        double r3491247 = r3491241 * r3491246;
        double r3491248 = r3491244 + r3491247;
        return r3491248;
}

↓

double f(double a1, double a2, double th) {
        double r3491249 = th;
        double r3491250 = cos(r3491249);
        double r3491251 = 2.0;
        double r3491252 = sqrt(r3491251);
        double r3491253 = sqrt(r3491252);
        double r3491254 = sqrt(r3491253);
        double r3491255 = r3491250 / r3491254;
        double r3491256 = a1;
        double r3491257 = a2;
        double r3491258 = r3491257 * r3491257;
        double r3491259 = fma(r3491256, r3491256, r3491258);
        double r3491260 = r3491254 * r3491253;
        double r3491261 = r3491259 / r3491260;
        double r3491262 = r3491255 * r3491261;
        return r3491262;
}

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

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

4. Applied add-sqr-sqrt 0.5

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

5. Applied sqrt-prod 0.5

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

6. Applied associate-/r* 0.5

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

8. Applied add-sqr-sqrt 0.5

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

9. Applied sqrt-prod 0.5

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

10. Applied sqrt-prod 0.5

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

11. Applied div-inv 0.6

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

12. Applied times-frac 0.5

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

13. Applied associate-*l* 0.5

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

14. Simplified 0.4

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

15. Final simplification 0.4

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

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))))