Average Error: 37.0 → 12.9
Time: 6.1s
Precision: binary64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(1 - \frac{\sin \varepsilon \cdot \sin x}{\cos \varepsilon \cdot \cos x}\right)} + \frac{\sin x}{\cos x} \cdot \log \left(\frac{e^{\frac{1}{1 - \frac{\sin \varepsilon \cdot \sin x}{\cos \varepsilon \cdot \cos x}}}}{e}\right)\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(1 - \frac{\sin \varepsilon \cdot \sin x}{\cos \varepsilon \cdot \cos x}\right)} + \frac{\sin x}{\cos x} \cdot \log \left(\frac{e^{\frac{1}{1 - \frac{\sin \varepsilon \cdot \sin x}{\cos \varepsilon \cdot \cos x}}}}{e}\right)
(FPCore (x eps) :precision binary64 (- (tan (+ x eps)) (tan x)))
↓
(FPCore (x eps)
 :precision binary64
 (+
  (/
   (sin eps)
   (* (cos eps) (- 1.0 (/ (* (sin eps) (sin x)) (* (cos eps) (cos x))))))
  (*
   (/ (sin x) (cos x))
   (log
    (/
     (exp (/ 1.0 (- 1.0 (/ (* (sin eps) (sin x)) (* (cos eps) (cos x))))))
     E)))))
double code(double x, double eps) {
	return tan(x + eps) - tan(x);
}
↓
double code(double x, double eps) {
	return (sin(eps) / (cos(eps) * (1.0 - ((sin(eps) * sin(x)) / (cos(eps) * cos(x)))))) + ((sin(x) / cos(x)) * log(exp(1.0 / (1.0 - ((sin(eps) * sin(x)) / (cos(eps) * cos(x))))) / ((double) M_E)));
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.0
Target15.3
Herbie12.9
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Initial program 37.0

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

  3. Applied tan-sum_binary6421.6

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

  5. Applied tan-quot_binary6421.6

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

  6. Applied tan-quot_binary6421.6

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

  7. Applied frac-times_binary6421.6

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

  8. Taylor expanded around inf 21.7

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

  9. Simplified12.8

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

  11. Applied add-log-exp_binary6412.8

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

  12. Applied add-log-exp_binary6412.9

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

  13. Applied sum-log_binary6412.9

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

  14. Simplified12.9

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

  15. Final simplification12.9

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

Reproduce

herbie shell --seed 2020219 
(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)))