Average Error: 37.2 → 16.0
Time: 10.8s
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -1.4292982892743089 \cdot 10^{-72} \lor \neg \left(\varepsilon \le 7.895037822047127 \cdot 10^{-70}\right):\\ \;\;\;\;\frac{\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \left(1 - \frac{\sin x \cdot \tan \varepsilon}{\cos x}\right) \cdot \sin x}{\left(1 - \frac{\sin x \cdot \tan \varepsilon}{\cos x}\right) \cdot \cos x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({\varepsilon}^{2}, x, \mathsf{fma}\left(\varepsilon, {x}^{2}, \varepsilon\right)\right)\\ \end{array}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -1.4292982892743089 \cdot 10^{-72} \lor \neg \left(\varepsilon \le 7.895037822047127 \cdot 10^{-70}\right):\\
\;\;\;\;\frac{\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \left(1 - \frac{\sin x \cdot \tan \varepsilon}{\cos x}\right) \cdot \sin x}{\left(1 - \frac{\sin x \cdot \tan \varepsilon}{\cos x}\right) \cdot \cos x}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left({\varepsilon}^{2}, x, \mathsf{fma}\left(\varepsilon, {x}^{2}, \varepsilon\right)\right)\\

\end{array}
double f(double x, double eps) {
        double r106536 = x;
        double r106537 = eps;
        double r106538 = r106536 + r106537;
        double r106539 = tan(r106538);
        double r106540 = tan(r106536);
        double r106541 = r106539 - r106540;
        return r106541;
}
↓
double f(double x, double eps) {
        double r106542 = eps;
        double r106543 = -1.429298289274309e-72;
        bool r106544 = r106542 <= r106543;
        double r106545 = 7.895037822047127e-70;
        bool r106546 = r106542 <= r106545;
        double r106547 = !r106546;
        bool r106548 = r106544 || r106547;
        double r106549 = x;
        double r106550 = tan(r106549);
        double r106551 = tan(r106542);
        double r106552 = r106550 + r106551;
        double r106553 = cos(r106549);
        double r106554 = r106552 * r106553;
        double r106555 = 1.0;
        double r106556 = sin(r106549);
        double r106557 = r106556 * r106551;
        double r106558 = r106557 / r106553;
        double r106559 = r106555 - r106558;
        double r106560 = r106559 * r106556;
        double r106561 = r106554 - r106560;
        double r106562 = r106559 * r106553;
        double r106563 = r106561 / r106562;
        double r106564 = 2.0;
        double r106565 = pow(r106542, r106564);
        double r106566 = pow(r106549, r106564);
        double r106567 = fma(r106542, r106566, r106542);
        double r106568 = fma(r106565, r106549, r106567);
        double r106569 = r106548 ? r106563 : r106568;
        return r106569;
}

Error

Bits error versus x

Bits error versus eps

Target

Original37.2
Target15.3
Herbie16.0
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Split input into 2 regimes

  2. if eps < -1.429298289274309e-72 or 7.895037822047127e-70 < eps

    1. Initial program 31.1

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

    3. Applied tan-sum6.5

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

    5. Applied tan-quot6.5

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

    6. Applied associate-*l/6.5

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

    8. Applied tan-quot6.5

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

    9. Applied frac-sub6.6

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

    if -1.429298289274309e-72 < eps < 7.895037822047127e-70

    1. Initial program 47.3

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

    2. Taylor expanded around 0 31.9

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

    3. Simplified31.9

      \[\leadsto \color{blue}{\mathsf{fma}\left({\varepsilon}^{2}, x, \mathsf{fma}\left(\varepsilon, {x}^{2}, \varepsilon\right)\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification16.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1.4292982892743089 \cdot 10^{-72} \lor \neg \left(\varepsilon \le 7.895037822047127 \cdot 10^{-70}\right):\\ \;\;\;\;\frac{\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \left(1 - \frac{\sin x \cdot \tan \varepsilon}{\cos x}\right) \cdot \sin x}{\left(1 - \frac{\sin x \cdot \tan \varepsilon}{\cos x}\right) \cdot \cos x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({\varepsilon}^{2}, x, \mathsf{fma}\left(\varepsilon, {x}^{2}, \varepsilon\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020049 +o rules:numerics
(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)))