Average Error: 36.7 → 0.5
Time: 5.7s
Precision: binary64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\frac{\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \left(\cos x + \frac{{\sin x}^{2}}{\cos x}\right)}{\cos x \cdot \left(1 - \sqrt[3]{{\left(\tan x \cdot \tan \varepsilon\right)}^{3}}\right)}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\frac{\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \left(\cos x + \frac{{\sin x}^{2}}{\cos x}\right)}{\cos x \cdot \left(1 - \sqrt[3]{{\left(\tan x \cdot \tan \varepsilon\right)}^{3}}\right)}
(FPCore (x eps) :precision binary64 (- (tan (+ x eps)) (tan x)))
↓
(FPCore (x eps)
 :precision binary64
 (/
  (* (/ (sin eps) (cos eps)) (+ (cos x) (/ (pow (sin x) 2.0) (cos x))))
  (* (cos x) (- 1.0 (cbrt (pow (* (tan x) (tan eps)) 3.0))))))
double code(double x, double eps) {
	return tan(x + eps) - tan(x);
}
↓
double code(double x, double eps) {
	return ((sin(eps) / cos(eps)) * (cos(x) + (pow(sin(x), 2.0) / cos(x)))) / (cos(x) * (1.0 - cbrt(pow((tan(x) * tan(eps)), 3.0))));
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original36.7
Target14.8
Herbie0.5
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Initial program 36.7

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

  3. Applied tan-quot_binary6436.7

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

  4. Applied tan-sum_binary6421.9

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

  5. Applied frac-sub_binary6421.9

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

  6. Taylor expanded around inf 0.4

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

  7. Simplified0.4

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

  9. Applied add-cbrt-cube_binary640.5

    \[\leadsto \frac{\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \left(\cos x + \frac{{\sin x}^{2}}{\cos x}\right)}{\left(1 - \tan x \cdot \color{blue}{\sqrt[3]{\left(\tan \varepsilon \cdot \tan \varepsilon\right) \cdot \tan \varepsilon}}\right) \cdot \cos x}\]

  10. Applied add-cbrt-cube_binary640.5

    \[\leadsto \frac{\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \left(\cos x + \frac{{\sin x}^{2}}{\cos x}\right)}{\left(1 - \color{blue}{\sqrt[3]{\left(\tan x \cdot \tan x\right) \cdot \tan x}} \cdot \sqrt[3]{\left(\tan \varepsilon \cdot \tan \varepsilon\right) \cdot \tan \varepsilon}\right) \cdot \cos x}\]

  11. Applied cbrt-unprod_binary640.5

    \[\leadsto \frac{\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \left(\cos x + \frac{{\sin x}^{2}}{\cos x}\right)}{\left(1 - \color{blue}{\sqrt[3]{\left(\left(\tan x \cdot \tan x\right) \cdot \tan x\right) \cdot \left(\left(\tan \varepsilon \cdot \tan \varepsilon\right) \cdot \tan \varepsilon\right)}}\right) \cdot \cos x}\]

  12. Simplified0.5

    \[\leadsto \frac{\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \left(\cos x + \frac{{\sin x}^{2}}{\cos x}\right)}{\left(1 - \sqrt[3]{\color{blue}{{\left(\tan x \cdot \tan \varepsilon\right)}^{3}}}\right) \cdot \cos x}\]

  13. Final simplification0.5

    \[\leadsto \frac{\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \left(\cos x + \frac{{\sin x}^{2}}{\cos x}\right)}{\cos x \cdot \left(1 - \sqrt[3]{{\left(\tan x \cdot \tan \varepsilon\right)}^{3}}\right)}\]

Reproduce

herbie shell --seed 2020231 
(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)))