Average Error: 36.9 → 0.3
Time: 7.6s
Precision: binary64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\frac{\tan \varepsilon \cdot \left(\cos x + \frac{{\sin x}^{2}}{\cos x}\right)}{\cos x \cdot \left(1 - \tan \varepsilon \cdot \tan x\right)}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\frac{\tan \varepsilon \cdot \left(\cos x + \frac{{\sin x}^{2}}{\cos x}\right)}{\cos x \cdot \left(1 - \tan \varepsilon \cdot \tan x\right)}
(FPCore (x eps) :precision binary64 (- (tan (+ x eps)) (tan x)))
↓
(FPCore (x eps)
 :precision binary64
 (/
  (* (tan eps) (+ (cos x) (/ (pow (sin x) 2.0) (cos x))))
  (* (cos x) (- 1.0 (* (tan eps) (tan x))))))
double code(double x, double eps) {
	return tan(x + eps) - tan(x);
}
↓
double code(double x, double eps) {
	return (tan(eps) * (cos(x) + (pow(sin(x), 2.0) / cos(x)))) / (cos(x) * (1.0 - (tan(eps) * tan(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

Original36.9
Target14.9
Herbie0.3
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Initial program 36.9

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

  3. Applied tan-quot_binary6436.9

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

  4. Applied tan-sum_binary6422.1

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

  5. Applied frac-sub_binary6422.1

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

  6. Simplified22.1

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

  7. Simplified22.1

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

  8. Taylor expanded around inf 0.4

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

  9. Simplified0.4

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

  11. Applied quot-tan_binary640.3

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

  12. Final simplification0.3

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

Reproduce

herbie shell --seed 2021173 
(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)))