Average Error: 36.7 → 15.5
Time: 27.6s
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -1.757922927393181967231988612780291928877 \cdot 10^{-27} \lor \neg \left(\varepsilon \le 1.760250544683835501772756411202209200384 \cdot 10^{-79}\right):\\ \;\;\;\;\frac{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} \cdot \frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x \cdot \tan x}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} + \tan x}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \varepsilon\right) \cdot \left(\varepsilon + x\right) + \varepsilon\\ \end{array}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -1.757922927393181967231988612780291928877 \cdot 10^{-27} \lor \neg \left(\varepsilon \le 1.760250544683835501772756411202209200384 \cdot 10^{-79}\right):\\
\;\;\;\;\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}\\

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

\end{array}
double f(double x, double eps) {
        double r106493 = x;
        double r106494 = eps;
        double r106495 = r106493 + r106494;
        double r106496 = tan(r106495);
        double r106497 = tan(r106493);
        double r106498 = r106496 - r106497;
        return r106498;
}
↓
double f(double x, double eps) {
        double r106499 = eps;
        double r106500 = -1.757922927393182e-27;
        bool r106501 = r106499 <= r106500;
        double r106502 = 1.7602505446838355e-79;
        bool r106503 = r106499 <= r106502;
        double r106504 = !r106503;
        bool r106505 = r106501 || r106504;
        double r106506 = x;
        double r106507 = tan(r106506);
        double r106508 = tan(r106499);
        double r106509 = r106507 + r106508;
        double r106510 = 1.0;
        double r106511 = r106507 * r106508;
        double r106512 = r106510 - r106511;
        double r106513 = r106509 / r106512;
        double r106514 = r106513 * r106513;
        double r106515 = r106507 * r106507;
        double r106516 = r106514 - r106515;
        double r106517 = r106513 + r106507;
        double r106518 = r106516 / r106517;
        double r106519 = r106506 * r106499;
        double r106520 = r106499 + r106506;
        double r106521 = r106519 * r106520;
        double r106522 = r106521 + r106499;
        double r106523 = r106505 ? r106518 : r106522;
        return r106523;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original36.7
Target14.9
Herbie15.5
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Split input into 2 regimes

  2. if eps < -1.757922927393182e-27 or 1.7602505446838355e-79 < eps

    1. Initial program 29.6

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

    3. Applied tan-sum4.5

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

    5. Applied flip--4.6

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

    if -1.757922927393182e-27 < eps < 1.7602505446838355e-79

    1. Initial program 46.8

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

    3. Applied tan-sum46.8

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

    4. Taylor expanded around 0 31.4

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

    5. Simplified31.2

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

  4. Final simplification15.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1.757922927393181967231988612780291928877 \cdot 10^{-27} \lor \neg \left(\varepsilon \le 1.760250544683835501772756411202209200384 \cdot 10^{-79}\right):\\ \;\;\;\;\frac{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} \cdot \frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x \cdot \tan x}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} + \tan x}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \varepsilon\right) \cdot \left(\varepsilon + x\right) + \varepsilon\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 
(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)))