Migdal et al, Equation (64)

Average Error: 0.5 → 0.5

Time: 17.2s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

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

C

double f(double a1, double a2, double th) {
        double r95700 = th;
        double r95701 = cos(r95700);
        double r95702 = 2.0;
        double r95703 = sqrt(r95702);
        double r95704 = r95701 / r95703;
        double r95705 = a1;
        double r95706 = r95705 * r95705;
        double r95707 = r95704 * r95706;
        double r95708 = a2;
        double r95709 = r95708 * r95708;
        double r95710 = r95704 * r95709;
        double r95711 = r95707 + r95710;
        return r95711;
}

↓

double f(double a1, double a2, double th) {
        double r95712 = a1;
        double r95713 = a2;
        double r95714 = r95713 * r95713;
        double r95715 = fma(r95712, r95712, r95714);
        double r95716 = th;
        double r95717 = cos(r95716);
        double r95718 = r95715 * r95717;
        double r95719 = 1.0;
        double r95720 = 2.0;
        double r95721 = sqrt(r95720);
        double r95722 = sqrt(r95721);
        double r95723 = r95719 / r95722;
        double r95724 = r95723 / r95722;
        double r95725 = r95718 * r95724;
        return r95725;
}

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

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

4. Applied add-sqr-sqrt 0.5

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

5. Applied sqrt-prod 0.5

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

6. Applied associate-/r* 0.5

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

8. Applied *-un-lft-identity 0.5

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

9. Applied sqrt-prod 0.5

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

10. Applied sqrt-prod 0.5

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

11. Applied div-inv 0.6

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

12. Applied times-frac 0.5

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

13. Applied associate-*r* 0.5

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

14. Simplified 0.5

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

15. Final simplification 0.5

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

Reproduce

herbie shell --seed 2020047 +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))))