Average Error: 36.6 → 15.2
Time: 10.1s
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -1.770078066117039689688378079023999204188 \cdot 10^{-29}:\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}}, -\tan x\right)\\ \mathbf{elif}\;\varepsilon \le 1.567290905506321661293182837865132571461 \cdot 10^{-52}:\\ \;\;\;\;\mathsf{fma}\left({\varepsilon}^{2}, x, \mathsf{fma}\left(\varepsilon, {x}^{2}, \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} \cdot \frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x \cdot \tan x}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} + \tan x}\\ \end{array}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -1.770078066117039689688378079023999204188 \cdot 10^{-29}:\\
\;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}}, -\tan x\right)\\

\mathbf{elif}\;\varepsilon \le 1.567290905506321661293182837865132571461 \cdot 10^{-52}:\\
\;\;\;\;\mathsf{fma}\left({\varepsilon}^{2}, x, \mathsf{fma}\left(\varepsilon, {x}^{2}, \varepsilon\right)\right)\\

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

\end{array}
double f(double x, double eps) {
        double r135368 = x;
        double r135369 = eps;
        double r135370 = r135368 + r135369;
        double r135371 = tan(r135370);
        double r135372 = tan(r135368);
        double r135373 = r135371 - r135372;
        return r135373;
}
↓
double f(double x, double eps) {
        double r135374 = eps;
        double r135375 = -1.7700780661170397e-29;
        bool r135376 = r135374 <= r135375;
        double r135377 = x;
        double r135378 = tan(r135377);
        double r135379 = tan(r135374);
        double r135380 = r135378 + r135379;
        double r135381 = 1.0;
        double r135382 = sin(r135374);
        double r135383 = r135378 * r135382;
        double r135384 = cos(r135374);
        double r135385 = r135383 / r135384;
        double r135386 = r135381 - r135385;
        double r135387 = r135381 / r135386;
        double r135388 = -r135378;
        double r135389 = fma(r135380, r135387, r135388);
        double r135390 = 1.5672909055063217e-52;
        bool r135391 = r135374 <= r135390;
        double r135392 = 2.0;
        double r135393 = pow(r135374, r135392);
        double r135394 = pow(r135377, r135392);
        double r135395 = fma(r135374, r135394, r135374);
        double r135396 = fma(r135393, r135377, r135395);
        double r135397 = r135378 * r135379;
        double r135398 = r135381 - r135397;
        double r135399 = r135380 / r135398;
        double r135400 = r135399 * r135399;
        double r135401 = r135378 * r135378;
        double r135402 = r135400 - r135401;
        double r135403 = r135399 + r135378;
        double r135404 = r135402 / r135403;
        double r135405 = r135391 ? r135396 : r135404;
        double r135406 = r135376 ? r135389 : r135405;
        return r135406;
}

Error

Bits error versus x

Bits error versus eps

Target

Original36.6
Target14.9
Herbie15.2
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Split input into 3 regimes

  2. if eps < -1.7700780661170397e-29

    1. Initial program 29.1

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

    3. Applied tan-sum1.9

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

    5. Applied div-inv1.9

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

    6. Applied fma-neg1.9

      \[\leadsto \color{blue}{\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right)}\]
    7. Using strategy rm

    8. Applied tan-quot1.9

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

    9. Applied associate-*r/1.9

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

    if -1.7700780661170397e-29 < eps < 1.5672909055063217e-52

    1. Initial program 46.3

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

    2. Taylor expanded around 0 31.1

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

    3. Simplified31.1

      \[\leadsto \color{blue}{\mathsf{fma}\left({\varepsilon}^{2}, x, \mathsf{fma}\left(\varepsilon, {x}^{2}, \varepsilon\right)\right)}\]

    if 1.5672909055063217e-52 < eps

    1. Initial program 29.3

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

    3. Applied tan-sum3.8

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

    5. Applied flip--3.9

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

  4. Final simplification15.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1.770078066117039689688378079023999204188 \cdot 10^{-29}:\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}}, -\tan x\right)\\ \mathbf{elif}\;\varepsilon \le 1.567290905506321661293182837865132571461 \cdot 10^{-52}:\\ \;\;\;\;\mathsf{fma}\left({\varepsilon}^{2}, x, \mathsf{fma}\left(\varepsilon, {x}^{2}, \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} \cdot \frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x \cdot \tan x}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} + \tan x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019356 +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)))