Trigonometry A

Average Error: 0.1 → 0.1

Time: 7.4s

Precision: 64

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

\[0.0 \le e \le 1\]

Math

\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]

↓

\[\frac{\sin v}{\mathsf{fma}\left(\cos v, e, 1\right)} \cdot e\]

C

double f(double e, double v) {
        double r13063 = e;
        double r13064 = v;
        double r13065 = sin(r13064);
        double r13066 = r13063 * r13065;
        double r13067 = 1.0;
        double r13068 = cos(r13064);
        double r13069 = r13063 * r13068;
        double r13070 = r13067 + r13069;
        double r13071 = r13066 / r13070;
        return r13071;
}

↓

double f(double e, double v) {
        double r13072 = v;
        double r13073 = sin(r13072);
        double r13074 = cos(r13072);
        double r13075 = e;
        double r13076 = 1.0;
        double r13077 = fma(r13074, r13075, r13076);
        double r13078 = r13073 / r13077;
        double r13079 = r13078 * r13075;
        return r13079;
}

Error

Bits error versus e

Bits error versus v

Derivation

1. Initial program 0.1

   \[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]

2. Simplified 0.1

   \[\leadsto \color{blue}{\frac{e \cdot \sin v}{\mathsf{fma}\left(\cos v, e, 1\right)}}\]
3. Using strategy rm

4. Applied *-un-lft-identity 0.1

   \[\leadsto \frac{e \cdot \sin v}{\color{blue}{1 \cdot \mathsf{fma}\left(\cos v, e, 1\right)}}\]

5. Applied times-frac 0.1

   \[\leadsto \color{blue}{\frac{e}{1} \cdot \frac{\sin v}{\mathsf{fma}\left(\cos v, e, 1\right)}}\]

6. Simplified 0.1

   \[\leadsto \color{blue}{e} \cdot \frac{\sin v}{\mathsf{fma}\left(\cos v, e, 1\right)}\]
7. Using strategy rm

8. Applied add-sqr-sqrt 0.4

   \[\leadsto \color{blue}{\left(\sqrt{e} \cdot \sqrt{e}\right)} \cdot \frac{\sin v}{\mathsf{fma}\left(\cos v, e, 1\right)}\]

9. Applied associate-*l* 0.4

   \[\leadsto \color{blue}{\sqrt{e} \cdot \left(\sqrt{e} \cdot \frac{\sin v}{\mathsf{fma}\left(\cos v, e, 1\right)}\right)}\]
10. Using strategy rm

11. Applied add-sqr-sqrt 0.4

    \[\leadsto \sqrt{e} \cdot \left(\sqrt{\color{blue}{\sqrt{e} \cdot \sqrt{e}}} \cdot \frac{\sin v}{\mathsf{fma}\left(\cos v, e, 1\right)}\right)\]

12. Applied sqrt-prod 0.6

    \[\leadsto \sqrt{e} \cdot \left(\color{blue}{\left(\sqrt{\sqrt{e}} \cdot \sqrt{\sqrt{e}}\right)} \cdot \frac{\sin v}{\mathsf{fma}\left(\cos v, e, 1\right)}\right)\]

13. Applied associate-*l* 0.5

    \[\leadsto \sqrt{e} \cdot \color{blue}{\left(\sqrt{\sqrt{e}} \cdot \left(\sqrt{\sqrt{e}} \cdot \frac{\sin v}{\mathsf{fma}\left(\cos v, e, 1\right)}\right)\right)}\]
14. Using strategy rm

15. Applied pow1 0.5

    \[\leadsto \sqrt{e} \cdot \left(\sqrt{\sqrt{e}} \cdot \left(\sqrt{\sqrt{e}} \cdot \color{blue}{{\left(\frac{\sin v}{\mathsf{fma}\left(\cos v, e, 1\right)}\right)}^{1}}\right)\right)\]

16. Applied pow1 0.5

    \[\leadsto \sqrt{e} \cdot \left(\sqrt{\sqrt{e}} \cdot \left(\color{blue}{{\left(\sqrt{\sqrt{e}}\right)}^{1}} \cdot {\left(\frac{\sin v}{\mathsf{fma}\left(\cos v, e, 1\right)}\right)}^{1}\right)\right)\]

17. Applied pow-prod-down 0.5

    \[\leadsto \sqrt{e} \cdot \left(\sqrt{\sqrt{e}} \cdot \color{blue}{{\left(\sqrt{\sqrt{e}} \cdot \frac{\sin v}{\mathsf{fma}\left(\cos v, e, 1\right)}\right)}^{1}}\right)\]

18. Applied pow1 0.5

    \[\leadsto \sqrt{e} \cdot \left(\color{blue}{{\left(\sqrt{\sqrt{e}}\right)}^{1}} \cdot {\left(\sqrt{\sqrt{e}} \cdot \frac{\sin v}{\mathsf{fma}\left(\cos v, e, 1\right)}\right)}^{1}\right)\]

19. Applied pow-prod-down 0.5

    \[\leadsto \sqrt{e} \cdot \color{blue}{{\left(\sqrt{\sqrt{e}} \cdot \left(\sqrt{\sqrt{e}} \cdot \frac{\sin v}{\mathsf{fma}\left(\cos v, e, 1\right)}\right)\right)}^{1}}\]

20. Applied pow1 0.5

    \[\leadsto \color{blue}{{\left(\sqrt{e}\right)}^{1}} \cdot {\left(\sqrt{\sqrt{e}} \cdot \left(\sqrt{\sqrt{e}} \cdot \frac{\sin v}{\mathsf{fma}\left(\cos v, e, 1\right)}\right)\right)}^{1}\]

21. Applied pow-prod-down 0.5

    \[\leadsto \color{blue}{{\left(\sqrt{e} \cdot \left(\sqrt{\sqrt{e}} \cdot \left(\sqrt{\sqrt{e}} \cdot \frac{\sin v}{\mathsf{fma}\left(\cos v, e, 1\right)}\right)\right)\right)}^{1}}\]

22. Simplified 0.1

    \[\leadsto {\color{blue}{\left(e \cdot \frac{\sin v}{\mathsf{fma}\left(\cos v, e, 1\right)}\right)}}^{1}\]

23. Final simplification 0.1

    \[\leadsto \frac{\sin v}{\mathsf{fma}\left(\cos v, e, 1\right)} \cdot e\]

Reproduce

herbie shell --seed 2020046 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (<= 0.0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))