Average Error: 37.1 → 16.0
Time: 17.8s
Precision: binary64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -6.18785078965368 \cdot 10^{-114}:\\ \;\;\;\;\frac{\frac{\sin x}{\cos x} + \frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 5.140335607377363 \cdot 10^{-104}:\\ \;\;\;\;\varepsilon + \left(\varepsilon + x\right) \cdot \left(\varepsilon \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos x \cdot \left(\tan x + \tan \varepsilon\right) - \sin x \cdot \frac{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}}{1 + \tan x \cdot \left(\tan \varepsilon \cdot \left(1 + \tan x \cdot \tan \varepsilon\right)\right)}}{\cos x \cdot \frac{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}}{1 + \tan x \cdot \left(\tan \varepsilon \cdot \left(1 + \tan x \cdot \tan \varepsilon\right)\right)}}\\ \end{array}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \leq -6.18785078965368 \cdot 10^{-114}:\\
\;\;\;\;\frac{\frac{\sin x}{\cos x} + \frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\

\mathbf{elif}\;\varepsilon \leq 5.140335607377363 \cdot 10^{-104}:\\
\;\;\;\;\varepsilon + \left(\varepsilon + x\right) \cdot \left(\varepsilon \cdot x\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{\cos x \cdot \left(\tan x + \tan \varepsilon\right) - \sin x \cdot \frac{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}}{1 + \tan x \cdot \left(\tan \varepsilon \cdot \left(1 + \tan x \cdot \tan \varepsilon\right)\right)}}{\cos x \cdot \frac{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}}{1 + \tan x \cdot \left(\tan \varepsilon \cdot \left(1 + \tan x \cdot \tan \varepsilon\right)\right)}}\\

\end{array}
(FPCore (x eps) :precision binary64 (- (tan (+ x eps)) (tan x)))
↓
(FPCore (x eps)
 :precision binary64
 (if (<= eps -6.18785078965368e-114)
   (-
    (/
     (+ (/ (sin x) (cos x)) (/ (sin eps) (cos eps)))
     (- 1.0 (* (tan x) (tan eps))))
    (tan x))
   (if (<= eps 5.140335607377363e-104)
     (+ eps (* (+ eps x) (* eps x)))
     (/
      (-
       (* (cos x) (+ (tan x) (tan eps)))
       (*
        (sin x)
        (/
         (- 1.0 (pow (* (tan x) (tan eps)) 3.0))
         (+ 1.0 (* (tan x) (* (tan eps) (+ 1.0 (* (tan x) (tan eps)))))))))
      (*
       (cos x)
       (/
        (- 1.0 (pow (* (tan x) (tan eps)) 3.0))
        (+ 1.0 (* (tan x) (* (tan eps) (+ 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 <= -6.18785078965368e-114) {
		tmp = (((sin(x) / cos(x)) + (sin(eps) / cos(eps))) / (1.0 - (tan(x) * tan(eps)))) - tan(x);
	} else if (eps <= 5.140335607377363e-104) {
		tmp = eps + ((eps + x) * (eps * x));
	} else {
		tmp = ((cos(x) * (tan(x) + tan(eps))) - (sin(x) * ((1.0 - pow((tan(x) * tan(eps)), 3.0)) / (1.0 + (tan(x) * (tan(eps) * (1.0 + (tan(x) * tan(eps))))))))) / (cos(x) * ((1.0 - pow((tan(x) * tan(eps)), 3.0)) / (1.0 + (tan(x) * (tan(eps) * (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

Original37.1
Target15.3
Herbie16.0
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Split input into 3 regimes

  2. if eps < -6.1878507896536801e-114

    1. Initial program 31.5

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

    3. Applied tan-sum_binary649.3

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

    4. Taylor expanded around inf 9.4

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

    if -6.1878507896536801e-114 < eps < 5.1403356073773628e-104

    1. Initial program 49.5

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

    2. Taylor expanded around 0 31.8

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

    3. Simplified31.5

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

    if 5.1403356073773628e-104 < eps

    1. Initial program 30.9

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

    3. Applied tan-sum_binary648.0

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

    5. Applied flip3--_binary648.0

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

    6. Simplified8.0

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

    7. Simplified8.0

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

    9. Applied tan-quot_binary648.1

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

    10. Applied frac-sub_binary648.1

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

  4. Final simplification16.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -6.18785078965368 \cdot 10^{-114}:\\ \;\;\;\;\frac{\frac{\sin x}{\cos x} + \frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 5.140335607377363 \cdot 10^{-104}:\\ \;\;\;\;\varepsilon + \left(\varepsilon + x\right) \cdot \left(\varepsilon \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos x \cdot \left(\tan x + \tan \varepsilon\right) - \sin x \cdot \frac{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}}{1 + \tan x \cdot \left(\tan \varepsilon \cdot \left(1 + \tan x \cdot \tan \varepsilon\right)\right)}}{\cos x \cdot \frac{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}}{1 + \tan x \cdot \left(\tan \varepsilon \cdot \left(1 + \tan x \cdot \tan \varepsilon\right)\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020253 
(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)))