2tan (problem 3.3.2)

Average Error: 37.0 → 0.4

Time: 8.4min

Precision: binary64

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

Math

\[\tan \left(x + \varepsilon\right) - \tan x\]

↓

\[\frac{\cos x \cdot \left(\left(\cos \varepsilon \cdot \left(\tan x \cdot \tan \varepsilon\right)\right) \cdot \sin x + \sin \varepsilon \cdot \cos x\right)}{\left(\left(\cos x \cdot \cos \varepsilon\right) \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)\right) \cdot \cos x}\]

C

double code(double x, double eps) {
	return ((double) (((double) tan(((double) (x + eps)))) - ((double) tan(x))));
}

↓

double code(double x, double eps) {
	return (((double) (((double) cos(x)) * ((double) (((double) (((double) (((double) cos(eps)) * ((double) (((double) tan(x)) * ((double) tan(eps)))))) * ((double) sin(x)))) + ((double) (((double) sin(eps)) * ((double) cos(x)))))))) / ((double) (((double) (((double) (((double) cos(x)) * ((double) cos(eps)))) * ((double) (1.0 - ((double) (((double) tan(x)) * ((double) tan(eps)))))))) * ((double) cos(x)))));
}

Error

Bits error versus x

Bits error versus eps

Target

Original	37.0
Target	14.9
Herbie	0.4

\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

1. Initial program 37.0

   \[\tan \left(x + \varepsilon\right) - \tan x\]
2. Using strategy rm

3. Applied tan-sum 22.1

   \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x\]
4. Using strategy rm

5. Applied div-inv 22.1

   \[\leadsto \color{blue}{\left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon}} - \tan x\]
6. Using strategy rm

7. Applied tan-quot 22.2

   \[\leadsto \left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon} - \color{blue}{\frac{\sin x}{\cos x}}\]

8. Applied tan-quot 22.3

   \[\leadsto \left(\tan x + \color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon}}\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon} - \frac{\sin x}{\cos x}\]

9. Applied tan-quot 22.2

   \[\leadsto \left(\color{blue}{\frac{\sin x}{\cos x}} + \frac{\sin \varepsilon}{\cos \varepsilon}\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon} - \frac{\sin x}{\cos x}\]

10. Applied frac-add 22.2

    \[\leadsto \color{blue}{\frac{\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon}} \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon} - \frac{\sin x}{\cos x}\]

11. Applied frac-times 22.2

    \[\leadsto \color{blue}{\frac{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) \cdot 1}{\left(\cos x \cdot \cos \varepsilon\right) \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)}} - \frac{\sin x}{\cos x}\]

12. Applied frac-sub 22.2

    \[\leadsto \color{blue}{\frac{\left(\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) \cdot 1\right) \cdot \cos x - \left(\left(\cos x \cdot \cos \varepsilon\right) \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)\right) \cdot \sin x}{\left(\left(\cos x \cdot \cos \varepsilon\right) \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)\right) \cdot \cos x}}\]

13. Simplified 0.4

    \[\leadsto \frac{\color{blue}{\cos x \cdot \left(\sin x \cdot \left(\left(\tan x \cdot \tan \varepsilon + 0\right) \cdot \cos \varepsilon\right) + \sin \varepsilon \cdot \cos x\right)}}{\left(\left(\cos x \cdot \cos \varepsilon\right) \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)\right) \cdot \cos x}\]

14. Final simplification 0.4

    \[\leadsto \frac{\cos x \cdot \left(\left(\cos \varepsilon \cdot \left(\tan x \cdot \tan \varepsilon\right)\right) \cdot \sin x + \sin \varepsilon \cdot \cos x\right)}{\left(\left(\cos x \cdot \cos \varepsilon\right) \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)\right) \cdot \cos x}\]

Reproduce

herbie shell --seed 2020181 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"
  :precision binary64

  :herbie-target
  (/ (sin eps) (* (cos x) (cos (+ x eps))))

  (- (tan (+ x eps)) (tan x)))