VandenBroeck and Keller, Equation (6)

Average Error: 16.6 → 12.6

Time: 10.2s

Precision: binary64

• could not determine a ground truth for program body

  1. F = -1.678408177611356e-233
  2. l = 2340279284988967.5

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

Math

\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]

↓

\[\pi \cdot \ell - \frac{1}{F} \cdot \left(1 \cdot \frac{\tan \left(\pi \cdot \ell\right)}{F}\right)\]

C

double code(double F, double l) {
	return ((double) (((double) (((double) M_PI) * l)) - ((double) ((1.0 / ((double) (F * F))) * ((double) tan(((double) (((double) M_PI) * l))))))));
}

↓

double code(double F, double l) {
	return ((double) (((double) (((double) M_PI) * l)) - ((double) ((1.0 / F) * ((double) (1.0 * (((double) tan(((double) (((double) M_PI) * l)))) / F)))))));
}

Error

Bits error versus F

Bits error versus l

Derivation

1. Initial program 16.6

   \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
2. Using strategy rm

3. Applied *-un-lft-identity 16.6

   \[\leadsto \pi \cdot \ell - \frac{\color{blue}{1 \cdot 1}}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]

4. Applied times-frac 16.6

   \[\leadsto \pi \cdot \ell - \color{blue}{\left(\frac{1}{F} \cdot \frac{1}{F}\right)} \cdot \tan \left(\pi \cdot \ell\right)\]

5. Applied associate-*l* 12.6

   \[\leadsto \pi \cdot \ell - \color{blue}{\frac{1}{F} \cdot \left(\frac{1}{F} \cdot \tan \left(\pi \cdot \ell\right)\right)}\]
6. Using strategy rm

7. Applied div-inv 12.6

   \[\leadsto \pi \cdot \ell - \frac{1}{F} \cdot \left(\color{blue}{\left(1 \cdot \frac{1}{F}\right)} \cdot \tan \left(\pi \cdot \ell\right)\right)\]

8. Applied associate-*l* 12.6

   \[\leadsto \pi \cdot \ell - \frac{1}{F} \cdot \color{blue}{\left(1 \cdot \left(\frac{1}{F} \cdot \tan \left(\pi \cdot \ell\right)\right)\right)}\]

9. Simplified 12.6

   \[\leadsto \pi \cdot \ell - \frac{1}{F} \cdot \left(1 \cdot \color{blue}{\frac{\tan \left(\pi \cdot \ell\right)}{F}}\right)\]

10. Final simplification 12.6

    \[\leadsto \pi \cdot \ell - \frac{1}{F} \cdot \left(1 \cdot \frac{\tan \left(\pi \cdot \ell\right)}{F}\right)\]

Reproduce

herbie shell --seed 2020182 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))