Average Error: 37.2 → 0.7
Time: 8.2s
Precision: binary64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\left(1 + \frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2}}\right) \cdot \frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(1 - \frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2}} \cdot \frac{{\left(\sin \varepsilon\right)}^{2}}{{\left(\cos \varepsilon\right)}^{2}}\right)} + \log \left(e^{\left(1 + \frac{{\left(\sin \varepsilon\right)}^{2}}{{\left(\cos \varepsilon\right)}^{2}}\right) \cdot \frac{\sin x}{\cos x \cdot \left(1 - \frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2}} \cdot \frac{{\left(\sin \varepsilon\right)}^{2}}{{\left(\cos \varepsilon\right)}^{2}}\right)} - \frac{\sin x}{\cos x}}\right)\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\left(1 + \frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2}}\right) \cdot \frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(1 - \frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2}} \cdot \frac{{\left(\sin \varepsilon\right)}^{2}}{{\left(\cos \varepsilon\right)}^{2}}\right)} + \log \left(e^{\left(1 + \frac{{\left(\sin \varepsilon\right)}^{2}}{{\left(\cos \varepsilon\right)}^{2}}\right) \cdot \frac{\sin x}{\cos x \cdot \left(1 - \frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2}} \cdot \frac{{\left(\sin \varepsilon\right)}^{2}}{{\left(\cos \varepsilon\right)}^{2}}\right)} - \frac{\sin x}{\cos x}}\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) (1.0 + ((double) (((double) pow(((double) sin(x)), 2.0)) / ((double) pow(((double) cos(x)), 2.0)))))) * ((double) (((double) sin(eps)) / ((double) (((double) cos(eps)) * ((double) (1.0 - ((double) (((double) (((double) pow(((double) sin(x)), 2.0)) / ((double) pow(((double) cos(x)), 2.0)))) * ((double) (((double) pow(((double) sin(eps)), 2.0)) / ((double) pow(((double) cos(eps)), 2.0)))))))))))))) + ((double) log(((double) exp(((double) (((double) (((double) (1.0 + ((double) (((double) pow(((double) sin(eps)), 2.0)) / ((double) pow(((double) cos(eps)), 2.0)))))) * ((double) (((double) sin(x)) / ((double) (((double) cos(x)) * ((double) (1.0 - ((double) (((double) (((double) pow(((double) sin(x)), 2.0)) / ((double) pow(((double) cos(x)), 2.0)))) * ((double) (((double) pow(((double) sin(eps)), 2.0)) / ((double) pow(((double) cos(eps)), 2.0)))))))))))))) - ((double) (((double) sin(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

Original37.2
Target14.9
Herbie0.7
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Initial program 37.2

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

  3. Applied tan-sum22.2

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

  5. Applied flip--22.2

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

  6. Simplified22.2

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

  7. Taylor expanded around inf 22.4

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

  8. Simplified0.6

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

  10. Applied add-log-exp13.0

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

  11. Applied add-log-exp0.7

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

  12. Applied diff-log0.7

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

  13. Simplified0.7

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

  14. Final simplification0.7

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

Reproduce

herbie shell --seed 2020179 
(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)))