Average Error: 37.1 → 0.4
Time: 7.8s
Precision: binary64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\frac{\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \cos x + \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{{\left(\sin x\right)}^{2}}{\cos x}}{\cos x \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\frac{\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \cos x + \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{{\left(\sin x\right)}^{2}}{\cos x}}{\cos x \cdot \left(1 - \tan x \cdot \tan \varepsilon\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) (((double) (((double) (((double) sin(eps)) / ((double) cos(eps)))) * ((double) cos(x)))) + ((double) (((double) (((double) sin(eps)) / ((double) cos(eps)))) * ((double) (((double) pow(((double) sin(x)), 2.0)) / ((double) cos(x)))))))) / ((double) (((double) cos(x)) * ((double) (1.0 - ((double) (((double) tan(x)) * ((double) tan(eps))))))))));
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.1
Target14.9
Herbie0.4
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Initial program 37.1

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

  3. Applied tan-quot37.1

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

  4. Applied tan-sum22.2

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

  5. Applied frac-sub22.2

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

  6. Taylor expanded around inf 0.4

    \[\leadsto \frac{\color{blue}{\frac{\sin \varepsilon \cdot \cos x}{\cos \varepsilon} + \frac{{\left(\sin x\right)}^{2} \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon}}}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}\]

  7. Simplified0.4

    \[\leadsto \frac{\color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \left(\cos x + \frac{{\left(\sin x\right)}^{2}}{\cos x}\right)}}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}\]
  8. Using strategy rm

  9. Applied distribute-lft-in0.4

    \[\leadsto \frac{\color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \cos x + \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{{\left(\sin x\right)}^{2}}{\cos x}}}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}\]

  10. Final simplification0.4

    \[\leadsto \frac{\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \cos x + \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{{\left(\sin x\right)}^{2}}{\cos x}}{\cos x \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)}\]

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)))