Average Error: 36.8 → 12.9
Time: 6.5s
Precision: binary64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(1 - \frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)} + \frac{\sin x}{\cos x} \cdot \left(\frac{1}{1 - \frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}} + -1\right)\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(1 - \frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)} + \frac{\sin x}{\cos x} \cdot \left(\frac{1}{1 - \frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}} + -1\right)
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) sin(eps)) / ((double) (((double) cos(eps)) * ((double) (1.0 - ((double) ((((double) sin(x)) / ((double) cos(x))) * (((double) sin(eps)) / ((double) cos(eps)))))))))) + ((double) ((((double) sin(x)) / ((double) cos(x))) * ((double) ((1.0 / ((double) (1.0 - ((double) ((((double) sin(x)) / ((double) cos(x))) * (((double) sin(eps)) / ((double) cos(eps)))))))) + -1.0))))));
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original36.8
Target14.6
Herbie12.9
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Initial program 36.8

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

  3. Applied tan-sum22.1

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

  5. Applied tan-quot22.1

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

  6. Applied tan-quot22.1

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

  7. Applied frac-times22.1

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

  8. Taylor expanded around inf 22.2

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

  9. Simplified12.9

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

  10. Final simplification12.9

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

Reproduce

herbie shell --seed 2020199 
(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)))