Average Error: 36.8 → 0.5
Time: 10.3s
Precision: binary64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\frac{\cos x \cdot \left(\sin \varepsilon \cdot \left(\cos x + \frac{{\sin x}^{2}}{\cos x}\right)\right)}{\cos x \cdot \left(\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \left(\cos x \cdot \cos \varepsilon\right)\right)}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\frac{\cos x \cdot \left(\sin \varepsilon \cdot \left(\cos x + \frac{{\sin x}^{2}}{\cos x}\right)\right)}{\cos x \cdot \left(\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \left(\cos x \cdot \cos \varepsilon\right)\right)}
(FPCore (x eps) :precision binary64 (- (tan (+ x eps)) (tan x)))
↓
(FPCore (x eps)
 :precision binary64
 (/
  (* (cos x) (* (sin eps) (+ (cos x) (/ (pow (sin x) 2.0) (cos x)))))
  (* (cos x) (* (- 1.0 (* (tan x) (tan eps))) (* (cos x) (cos eps))))))
double code(double x, double eps) {
	return tan(x + eps) - tan(x);
}
↓
double code(double x, double eps) {
	return (cos(x) * (sin(eps) * (cos(x) + (pow(sin(x), 2.0) / cos(x))))) / (cos(x) * ((1.0 - (tan(x) * tan(eps))) * (cos(x) * cos(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

Original36.8
Target15.1
Herbie0.5
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Initial program 36.8

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

  3. Applied tan-sum_binary64_189421.7

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

  5. Applied div-inv_binary64_175921.7

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

  7. Applied tan-quot_binary64_191821.8

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

  8. Applied tan-quot_binary64_191821.9

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

  9. Applied tan-quot_binary64_191821.8

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

  10. Applied frac-add_binary64_177021.8

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

  11. Applied frac-times_binary64_177221.8

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

  12. Applied frac-sub_binary64_177121.8

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

  13. Simplified21.8

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

  14. Simplified21.8

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

  15. Taylor expanded around inf 0.4

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

  16. Simplified0.5

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

  17. Final simplification0.5

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

Reproduce

herbie shell --seed 2020281 
(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)))