Average Error: 36.6 → 0.4
Time: 7.2s
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\frac{\frac{\sin \varepsilon \cdot \cos x}{\cos \varepsilon} + \frac{{\left(\sin x\right)}^{2}}{\frac{\cos x \cdot \cos \varepsilon}{\sin \varepsilon}}}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\frac{\frac{\sin \varepsilon \cdot \cos x}{\cos \varepsilon} + \frac{{\left(\sin x\right)}^{2}}{\frac{\cos x \cdot \cos \varepsilon}{\sin \varepsilon}}}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}
double code(double x, double eps) {
	return (tan((x + eps)) - tan(x));
}
↓
double code(double x, double eps) {
	return ((((sin(eps) * cos(x)) / cos(eps)) + (pow(sin(x), 2.0) / ((cos(x) * cos(eps)) / sin(eps)))) / ((1.0 - (tan(x) * tan(eps))) * cos(x)));
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original36.6
Target15.1
Herbie0.4
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Initial program 36.6

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

  3. Applied tan-quot36.6

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

  4. Applied tan-sum21.6

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

  5. Applied frac-sub21.6

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

  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. Using strategy rm

  8. Applied associate-/l*0.4

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

  9. Final simplification0.4

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

Reproduce

herbie shell --seed 2020106 
(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)))