Migdal et al, Equation (64)

Average Error: 0.5 → 0.4

Time: 26.8s

Precision: 64

Math

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

↓

\[\frac{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \cdot \cos th}{\left|\sqrt[3]{\sqrt{2}}\right|} \cdot \frac{\frac{1}{\sqrt{\sqrt{2}}}}{\sqrt{\sqrt[3]{\sqrt{2}}}}\]

C

double f(double a1, double a2, double th) {
        double r2036273 = th;
        double r2036274 = cos(r2036273);
        double r2036275 = 2.0;
        double r2036276 = sqrt(r2036275);
        double r2036277 = r2036274 / r2036276;
        double r2036278 = a1;
        double r2036279 = r2036278 * r2036278;
        double r2036280 = r2036277 * r2036279;
        double r2036281 = a2;
        double r2036282 = r2036281 * r2036281;
        double r2036283 = r2036277 * r2036282;
        double r2036284 = r2036280 + r2036283;
        return r2036284;
}

↓

double f(double a1, double a2, double th) {
        double r2036285 = a1;
        double r2036286 = a2;
        double r2036287 = r2036286 * r2036286;
        double r2036288 = fma(r2036285, r2036285, r2036287);
        double r2036289 = th;
        double r2036290 = cos(r2036289);
        double r2036291 = r2036288 * r2036290;
        double r2036292 = 2.0;
        double r2036293 = sqrt(r2036292);
        double r2036294 = cbrt(r2036293);
        double r2036295 = fabs(r2036294);
        double r2036296 = r2036291 / r2036295;
        double r2036297 = 1.0;
        double r2036298 = sqrt(r2036293);
        double r2036299 = r2036297 / r2036298;
        double r2036300 = sqrt(r2036294);
        double r2036301 = r2036299 / r2036300;
        double r2036302 = r2036296 * r2036301;
        return r2036302;
}

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 0.8

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

4. Applied div-inv 0.8

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

5. Simplified 0.5

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

7. Applied add-sqr-sqrt 0.5

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

8. Applied sqrt-prod 0.6

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

9. Applied associate-/r* 0.5

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

11. Applied add-cube-cbrt 0.5

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

12. Applied sqrt-prod 0.7

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

13. Applied div-inv 0.5

    \[\leadsto \cos th \cdot \frac{\color{blue}{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \cdot \frac{1}{\sqrt{\sqrt{2}}}}}{\sqrt{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}} \cdot \sqrt{\sqrt[3]{\sqrt{2}}}}\]

14. Applied times-frac 0.4

    \[\leadsto \cos th \cdot \color{blue}{\left(\frac{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\sqrt{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}} \cdot \frac{\frac{1}{\sqrt{\sqrt{2}}}}{\sqrt{\sqrt[3]{\sqrt{2}}}}\right)}\]

15. Applied associate-*r* 0.4

    \[\leadsto \color{blue}{\left(\cos th \cdot \frac{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\sqrt{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}}\right) \cdot \frac{\frac{1}{\sqrt{\sqrt{2}}}}{\sqrt{\sqrt[3]{\sqrt{2}}}}}\]

16. Simplified 0.4

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \cdot \cos th}{\left|\sqrt[3]{\sqrt{2}}\right|}} \cdot \frac{\frac{1}{\sqrt{\sqrt{2}}}}{\sqrt{\sqrt[3]{\sqrt{2}}}}\]

17. Final simplification 0.4

    \[\leadsto \frac{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \cdot \cos th}{\left|\sqrt[3]{\sqrt{2}}\right|} \cdot \frac{\frac{1}{\sqrt{\sqrt{2}}}}{\sqrt{\sqrt[3]{\sqrt{2}}}}\]

Reproduce

herbie shell --seed 2019162 +o rules:numerics
(FPCore (a1 a2 th)
  :name "Migdal et al, Equation (64)"
  (+ (* (/ (cos th) (sqrt 2)) (* a1 a1)) (* (/ (cos th) (sqrt 2)) (* a2 a2))))