Average Error: 36.6 → 15.3
Time: 15.3s
Precision: binary64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.5520522219363244 \cdot 10^{-74} \lor \neg \left(\varepsilon \leq 2.207117436105011 \cdot 10^{-19}\right):\\ \;\;\;\;\left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \left(\varepsilon + x\right) \cdot \left(\varepsilon \cdot x\right)\\ \end{array}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \leq -2.5520522219363244 \cdot 10^{-74} \lor \neg \left(\varepsilon \leq 2.207117436105011 \cdot 10^{-19}\right):\\
\;\;\;\;\left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\

\mathbf{else}:\\
\;\;\;\;\varepsilon + \left(\varepsilon + x\right) \cdot \left(\varepsilon \cdot x\right)\\

\end{array}
(FPCore (x eps) :precision binary64 (- (tan (+ x eps)) (tan x)))
↓
(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -2.5520522219363244e-74) (not (<= eps 2.207117436105011e-19)))
   (- (* (+ (tan x) (tan eps)) (/ 1.0 (- 1.0 (* (tan x) (tan eps))))) (tan x))
   (+ eps (* (+ eps x) (* eps x)))))
double code(double x, double eps) {
	return tan(x + eps) - tan(x);
}
↓
double code(double x, double eps) {
	double tmp;
	if ((eps <= -2.5520522219363244e-74) || !(eps <= 2.207117436105011e-19)) {
		tmp = ((tan(x) + tan(eps)) * (1.0 / (1.0 - (tan(x) * tan(eps))))) - tan(x);
	} else {
		tmp = eps + ((eps + x) * (eps * x));
	}
	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

Original36.6
Target15.3
Herbie15.3
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Split input into 2 regimes

  2. if eps < -2.55205222193632443e-74 or 2.20711743610501114e-19 < eps

    1. Initial program 30.4

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

    3. Applied tan-sum_binary64_15903.8

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

    5. Applied div-inv_binary64_14523.9

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

    if -2.55205222193632443e-74 < eps < 2.20711743610501114e-19

    1. Initial program 45.1

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

    2. Taylor expanded around 0 31.0

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

    3. Simplified30.8

      \[\leadsto \color{blue}{\varepsilon + \left(x + \varepsilon\right) \cdot \left(x \cdot \varepsilon\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification15.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.5520522219363244 \cdot 10^{-74} \lor \neg \left(\varepsilon \leq 2.207117436105011 \cdot 10^{-19}\right):\\ \;\;\;\;\left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \left(\varepsilon + x\right) \cdot \left(\varepsilon \cdot x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020288 
(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)))