Average Error: 36.6 → 21.5
Time: 7.9s
Precision: binary64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\frac{\cos x \cdot \left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x \cdot \left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}{\cos x \cdot \left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\frac{\cos x \cdot \left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x \cdot \left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}{\cos x \cdot \left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}
(FPCore (x eps) :precision binary64 (- (tan (+ x eps)) (tan x)))
↓
(FPCore (x eps)
 :precision binary64
 (/
  (-
   (* (cos x) (+ (* (sin x) (cos eps)) (* (cos x) (sin eps))))
   (* (sin x) (- (* (cos x) (cos eps)) (* (sin x) (sin eps)))))
  (* (cos x) (- (* (cos x) (cos eps)) (* (sin x) (sin eps))))))
double code(double x, double eps) {
	return tan(x + eps) - tan(x);
}
↓
double code(double x, double eps) {
	return ((cos(x) * ((sin(x) * cos(eps)) + (cos(x) * sin(eps)))) - (sin(x) * ((cos(x) * cos(eps)) - (sin(x) * sin(eps))))) / (cos(x) * ((cos(x) * cos(eps)) - (sin(x) * sin(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

Original36.6
Target15.2
Herbie21.5
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Initial program 36.6

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

  3. Applied tan-quot_binary6436.7

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

  5. Applied sin-sum_binary6435.7

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

  7. Applied cos-sum_binary6421.6

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

  8. Simplified21.6

    \[\leadsto \frac{\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon}{\color{blue}{\cos \varepsilon \cdot \cos x} - \sin x \cdot \sin \varepsilon} - \tan x\]
  9. Using strategy rm

  10. Applied tan-quot_binary6421.5

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

  11. Applied frac-sub_binary6421.5

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

  12. Simplified21.5

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

  13. Simplified21.5

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

  14. Final simplification21.5

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

Reproduce

herbie shell --seed 2020299 
(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)))