Average Error: 36.6 → 15.2
Time: 11.7s
Precision: binary64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -8.82532240330736 \cdot 10^{-33} \lor \neg \left(\varepsilon \leq 2.8352874239115425 \cdot 10^{-18}\right):\\ \;\;\;\;\frac{\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \sin x}{\cos x \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \left(\varepsilon + x\right) \cdot \left(\varepsilon \cdot x\right)\\ \end{array}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \leq -8.82532240330736 \cdot 10^{-33} \lor \neg \left(\varepsilon \leq 2.8352874239115425 \cdot 10^{-18}\right):\\
\;\;\;\;\frac{\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \sin x}{\cos x \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)}\\

\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 -8.82532240330736e-33) (not (<= eps 2.8352874239115425e-18)))
   (/
    (-
     (* (+ (tan x) (tan eps)) (cos x))
     (* (- 1.0 (* (tan x) (tan eps))) (sin x)))
    (* (cos x) (- 1.0 (* (tan x) (tan eps)))))
   (+ 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 <= -8.82532240330736e-33) || !(eps <= 2.8352874239115425e-18)) {
		tmp = (((tan(x) + tan(eps)) * cos(x)) - ((1.0 - (tan(x) * tan(eps))) * sin(x))) / (cos(x) * (1.0 - (tan(x) * tan(eps))));
	} 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
Target15.2
Herbie15.2
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Split input into 2 regimes

  2. if eps < -8.82532240330736014e-33 or 2.83528742391154245e-18 < eps

    1. Initial program 29.6

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

    3. Applied tan-quot_binary64_191829.5

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

    4. Applied tan-sum_binary64_18941.7

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

    5. Applied frac-sub_binary64_17711.7

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

    if -8.82532240330736014e-33 < eps < 2.83528742391154245e-18

    1. Initial program 45.0

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

    2. Taylor expanded around 0 31.4

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

    3. Simplified31.2

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

  4. Final simplification15.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -8.82532240330736 \cdot 10^{-33} \lor \neg \left(\varepsilon \leq 2.8352874239115425 \cdot 10^{-18}\right):\\ \;\;\;\;\frac{\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \sin x}{\cos x \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \left(\varepsilon + x\right) \cdot \left(\varepsilon \cdot x\right)\\ \end{array}\]

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)))