Average Error: 36.8 → 15.0
Time: 10.1s
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -5.97198873028037458 \cdot 10^{-27} \lor \neg \left(\varepsilon \le 2.0534977977042638 \cdot 10^{-22}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{\frac{1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)}{1 + \tan x \cdot \tan \varepsilon}} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot x\right) \cdot \left(x + \varepsilon\right) + \varepsilon\\ \end{array}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -5.97198873028037458 \cdot 10^{-27} \lor \neg \left(\varepsilon \le 2.0534977977042638 \cdot 10^{-22}\right):\\
\;\;\;\;\frac{\tan x + \tan \varepsilon}{\frac{1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)}{1 + \tan x \cdot \tan \varepsilon}} - \tan x\\

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

\end{array}
double f(double x, double eps) {
        double r146494 = x;
        double r146495 = eps;
        double r146496 = r146494 + r146495;
        double r146497 = tan(r146496);
        double r146498 = tan(r146494);
        double r146499 = r146497 - r146498;
        return r146499;
}
↓
double f(double x, double eps) {
        double r146500 = eps;
        double r146501 = -5.9719887302803746e-27;
        bool r146502 = r146500 <= r146501;
        double r146503 = 2.0534977977042638e-22;
        bool r146504 = r146500 <= r146503;
        double r146505 = !r146504;
        bool r146506 = r146502 || r146505;
        double r146507 = x;
        double r146508 = tan(r146507);
        double r146509 = tan(r146500);
        double r146510 = r146508 + r146509;
        double r146511 = 1.0;
        double r146512 = r146508 * r146509;
        double r146513 = r146512 * r146512;
        double r146514 = r146511 - r146513;
        double r146515 = r146511 + r146512;
        double r146516 = r146514 / r146515;
        double r146517 = r146510 / r146516;
        double r146518 = r146517 - r146508;
        double r146519 = r146500 * r146507;
        double r146520 = r146507 + r146500;
        double r146521 = r146519 * r146520;
        double r146522 = r146521 + r146500;
        double r146523 = r146506 ? r146518 : r146522;
        return r146523;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original36.8
Target15.2
Herbie15.0
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Split input into 2 regimes

  2. if eps < -5.9719887302803746e-27 or 2.0534977977042638e-22 < eps

    1. Initial program 29.8

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

    3. Applied tan-sum1.6

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

    5. Applied flip--1.7

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

    6. Simplified1.7

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

    if -5.9719887302803746e-27 < eps < 2.0534977977042638e-22

    1. Initial program 45.1

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

    2. Taylor expanded around 0 30.9

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

    3. Simplified30.7

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

  4. Final simplification15.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -5.97198873028037458 \cdot 10^{-27} \lor \neg \left(\varepsilon \le 2.0534977977042638 \cdot 10^{-22}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{\frac{1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)}{1 + \tan x \cdot \tan \varepsilon}} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot x\right) \cdot \left(x + \varepsilon\right) + \varepsilon\\ \end{array}\]

Reproduce

herbie shell --seed 2020057 
(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)))