Average Error: 37.0 → 15.8
Time: 13.0s
Precision: binary64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.960934173966239 \cdot 10^{-23}:\\ \;\;\;\;\left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 8.297387541937686 \cdot 10^{-47}:\\ \;\;\;\;\varepsilon + \left(\varepsilon + x\right) \cdot \left(\varepsilon \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos x - \frac{1 - \tan x \cdot \tan \varepsilon}{\tan x + \tan \varepsilon} \cdot \sin x}{\cos x \cdot \frac{1 - \tan x \cdot \tan \varepsilon}{\tan x + \tan \varepsilon}}\\ \end{array}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \leq -4.960934173966239 \cdot 10^{-23}:\\
\;\;\;\;\left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\

\mathbf{elif}\;\varepsilon \leq 8.297387541937686 \cdot 10^{-47}:\\
\;\;\;\;\varepsilon + \left(\varepsilon + x\right) \cdot \left(\varepsilon \cdot x\right)\\

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

\end{array}
(FPCore (x eps) :precision binary64 (- (tan (+ x eps)) (tan x)))
↓
(FPCore (x eps)
 :precision binary64
 (if (<= eps -4.960934173966239e-23)
   (- (* (+ (tan x) (tan eps)) (/ 1.0 (- 1.0 (* (tan x) (tan eps))))) (tan x))
   (if (<= eps 8.297387541937686e-47)
     (+ eps (* (+ eps x) (* eps x)))
     (/
      (-
       (cos x)
       (* (/ (- 1.0 (* (tan x) (tan eps))) (+ (tan x) (tan eps))) (sin x)))
      (* (cos x) (/ (- 1.0 (* (tan x) (tan eps))) (+ (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 <= -4.960934173966239e-23) {
		tmp = ((tan(x) + tan(eps)) * (1.0 / (1.0 - (tan(x) * tan(eps))))) - tan(x);
	} else if (eps <= 8.297387541937686e-47) {
		tmp = eps + ((eps + x) * (eps * x));
	} else {
		tmp = (cos(x) - (((1.0 - (tan(x) * tan(eps))) / (tan(x) + tan(eps))) * sin(x))) / (cos(x) * ((1.0 - (tan(x) * tan(eps))) / (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

Original37.0
Target15.0
Herbie15.8
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Split input into 3 regimes

  2. if eps < -4.96093417396623927e-23

    1. Initial program 29.6

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

    3. Applied tan-sum_binary64_19341.5

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

    5. Applied div-inv_binary64_17961.6

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

    if -4.96093417396623927e-23 < eps < 8.2973875419376863e-47

    1. Initial program 45.7

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

    2. Taylor expanded around 0 32.0

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

    3. Simplified31.8

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

    if 8.2973875419376863e-47 < eps

    1. Initial program 30.1

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

    3. Applied tan-sum_binary64_19343.2

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

    5. Applied clear-num_binary64_17983.3

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

    7. Applied tan-quot_binary64_19583.3

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

    8. Applied frac-sub_binary64_18083.4

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

    9. Simplified3.4

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

  4. Final simplification15.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.960934173966239 \cdot 10^{-23}:\\ \;\;\;\;\left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 8.297387541937686 \cdot 10^{-47}:\\ \;\;\;\;\varepsilon + \left(\varepsilon + x\right) \cdot \left(\varepsilon \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos x - \frac{1 - \tan x \cdot \tan \varepsilon}{\tan x + \tan \varepsilon} \cdot \sin x}{\cos x \cdot \frac{1 - \tan x \cdot \tan \varepsilon}{\tan x + \tan \varepsilon}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020292 
(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)))