Average Error: 36.8 → 0.4
Time: 8.8s
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\frac{-\left(\frac{\sin \varepsilon \cdot \cos x}{\cos \varepsilon} + \frac{\sin \varepsilon}{\frac{\cos x \cdot \cos \varepsilon}{{\left(\sin x\right)}^{2}}}\right)}{\left(-\left(1 - \tan \varepsilon \cdot \tan x\right)\right) \cdot \cos x}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\frac{-\left(\frac{\sin \varepsilon \cdot \cos x}{\cos \varepsilon} + \frac{\sin \varepsilon}{\frac{\cos x \cdot \cos \varepsilon}{{\left(\sin x\right)}^{2}}}\right)}{\left(-\left(1 - \tan \varepsilon \cdot \tan x\right)\right) \cdot \cos x}
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) (((double) sin(eps)) * ((double) cos(x)))) / ((double) cos(eps)))) + ((double) (((double) sin(eps)) / ((double) (((double) (((double) cos(x)) * ((double) cos(eps)))) / ((double) pow(((double) sin(x)), 2.0)))))))))) / ((double) (((double) -(((double) (1.0 - ((double) (((double) tan(eps)) * ((double) tan(x)))))))) * ((double) 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

Original36.8
Target14.9
Herbie0.4
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Initial program 36.8

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

  3. Applied tan-sum21.8

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

  4. Simplified21.8

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

  6. Applied tan-quot21.9

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

  7. Applied frac-2neg21.9

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

  8. Applied frac-sub21.9

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

  9. Simplified21.9

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

  10. Taylor expanded around inf 0.4

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

  12. Applied *-commutative0.4

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

  13. Applied associate-/l*0.4

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

  14. Final simplification0.4

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

Reproduce

herbie shell --seed 2020114 
(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)))