Average Error: 36.8 → 0.4
Time: 7.9s
Precision: binary64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\frac{\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \left(\cos x + \sin x \cdot \tan x\right)}{\cos x \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\frac{\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \left(\cos x + \sin x \cdot \tan x\right)}{\cos x \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)}
(FPCore (x eps) :precision binary64 (- (tan (+ x eps)) (tan x)))
↓
(FPCore (x eps)
 :precision binary64
 (/
  (* (/ (sin eps) (cos eps)) (+ (cos x) (* (sin x) (tan x))))
  (* (cos x) (- 1.0 (* (tan x) (tan eps))))))
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) + (sin(x) * tan(x)))) / (cos(x) * (1.0 - (tan(x) * tan(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
Target14.8
Herbie0.4
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Initial program 36.8

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

  3. Applied tan-quot_binary6436.8

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

  4. Applied tan-sum_binary6422.0

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

  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 *-un-lft-identity_binary640.4

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

  12. Applied add-sqr-sqrt_binary6433.0

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

  13. Applied unpow-prod-down_binary6433.0

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

  14. Applied times-frac_binary6433.0

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

  15. Simplified33.0

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

  16. Simplified0.4

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

  18. Applied quot-tan_binary640.4

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

  19. Final simplification0.4

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

Reproduce

herbie shell --seed 2021174 
(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)))