Average Error: 37.3 → 15.8
Time: 16.5s
Precision: binary64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.5441009276949707 \cdot 10^{-53}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{{\left(\sin x \cdot \sin \varepsilon\right)}^{3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^{3}}} \cdot \left(1 + \left(\tan x \cdot \tan \varepsilon + \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)\right) - \tan x\\ \mathbf{elif}\;\varepsilon \leq 1.914951132073428 \cdot 10^{-114}:\\ \;\;\;\;\varepsilon + \left(\varepsilon + x\right) \cdot \left(\varepsilon \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \sin x \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)}{\cos x \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)}\\ \end{array}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \leq -2.5441009276949707 \cdot 10^{-53}:\\
\;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{{\left(\sin x \cdot \sin \varepsilon\right)}^{3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^{3}}} \cdot \left(1 + \left(\tan x \cdot \tan \varepsilon + \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)\right) - \tan x\\

\mathbf{elif}\;\varepsilon \leq 1.914951132073428 \cdot 10^{-114}:\\
\;\;\;\;\varepsilon + \left(\varepsilon + x\right) \cdot \left(\varepsilon \cdot x\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \sin x \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)}{\cos x \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)}\\

\end{array}
(FPCore (x eps) :precision binary64 (- (tan (+ x eps)) (tan x)))
↓
(FPCore (x eps)
 :precision binary64
 (if (<= eps -2.5441009276949707e-53)
   (-
    (*
     (/
      (+ (tan x) (tan eps))
      (-
       1.0
       (/ (pow (* (sin x) (sin eps)) 3.0) (pow (* (cos x) (cos eps)) 3.0))))
     (+
      1.0
      (+
       (* (tan x) (tan eps))
       (* (* (tan x) (tan eps)) (* (tan x) (tan eps))))))
    (tan x))
   (if (<= eps 1.914951132073428e-114)
     (+ eps (* (+ eps x) (* eps x)))
     (/
      (-
       (* (+ (tan x) (tan eps)) (cos x))
       (* (sin x) (- 1.0 (* (tan x) (tan eps)))))
      (* (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) {
	double tmp;
	if (eps <= -2.5441009276949707e-53) {
		tmp = (((tan(x) + tan(eps)) / (1.0 - (pow((sin(x) * sin(eps)), 3.0) / pow((cos(x) * cos(eps)), 3.0)))) * (1.0 + ((tan(x) * tan(eps)) + ((tan(x) * tan(eps)) * (tan(x) * tan(eps)))))) - tan(x);
	} else if (eps <= 1.914951132073428e-114) {
		tmp = eps + ((eps + x) * (eps * x));
	} else {
		tmp = (((tan(x) + tan(eps)) * cos(x)) - (sin(x) * (1.0 - (tan(x) * tan(eps))))) / (cos(x) * (1.0 - (tan(x) * tan(eps))));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.3
Target15.1
Herbie15.8
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Split input into 3 regimes

  2. if eps < -2.5441009276949707e-53

    1. Initial program 29.1

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

    3. Applied tan-sum_binary64_18944.0

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

    5. Applied flip3--_binary64_17664.0

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

    6. Applied associate-/r/_binary64_17104.0

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

    7. Simplified4.0

      \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}}} \cdot \left(1 \cdot 1 + \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right) + 1 \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)\right) - \tan x\]
    8. Using strategy rm

    9. Applied tan-quot_binary64_19184.1

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

    10. Applied tan-quot_binary64_19184.1

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

    11. Applied frac-times_binary64_17724.1

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

    12. Applied cube-div_binary64_17884.1

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

    if -2.5441009276949707e-53 < eps < 1.9149511320734281e-114

    1. Initial program 48.5

      \[\tan \left(x + \varepsilon\right) - \tan x\]

    2. Taylor expanded around 0 31.5

      \[\leadsto \color{blue}{x \cdot {\varepsilon}^{2} + \left(\varepsilon + {x}^{2} \cdot \varepsilon\right)}\]

    3. Simplified31.3

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

    if 1.9149511320734281e-114 < eps

    1. Initial program 32.3

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

    3. Applied tan-quot_binary64_191832.2

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

    4. Applied tan-sum_binary64_18949.2

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

    5. Applied frac-sub_binary64_17719.2

      \[\leadsto \color{blue}{\frac{\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \sin x}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification15.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.5441009276949707 \cdot 10^{-53}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{{\left(\sin x \cdot \sin \varepsilon\right)}^{3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^{3}}} \cdot \left(1 + \left(\tan x \cdot \tan \varepsilon + \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)\right) - \tan x\\ \mathbf{elif}\;\varepsilon \leq 1.914951132073428 \cdot 10^{-114}:\\ \;\;\;\;\varepsilon + \left(\varepsilon + x\right) \cdot \left(\varepsilon \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \sin x \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)}{\cos x \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020280 
(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)))