Average Error: 36.6 → 15.8
Time: 9.9s
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -2.2637252434053613 \cdot 10^{-90}:\\ \;\;\;\;\frac{1}{\frac{1 - \tan x \cdot \tan \varepsilon}{\tan x + \tan \varepsilon}} - \tan x\\ \mathbf{elif}\;\varepsilon \le 5.7485271720269307 \cdot 10^{-167}:\\ \;\;\;\;\left(x \cdot \varepsilon\right) \cdot \left(\varepsilon + x\right) + \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}} \cdot \left(1 \cdot 1 + \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right) + 1 \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)\right) - \tan x\\ \end{array}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -2.2637252434053613 \cdot 10^{-90}:\\
\;\;\;\;\frac{1}{\frac{1 - \tan x \cdot \tan \varepsilon}{\tan x + \tan \varepsilon}} - \tan x\\

\mathbf{elif}\;\varepsilon \le 5.7485271720269307 \cdot 10^{-167}:\\
\;\;\;\;\left(x \cdot \varepsilon\right) \cdot \left(\varepsilon + x\right) + \varepsilon\\

\mathbf{else}:\\
\;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}} \cdot \left(1 \cdot 1 + \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right) + 1 \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)\right) - \tan x\\

\end{array}
double f(double x, double eps) {
        double r131597 = x;
        double r131598 = eps;
        double r131599 = r131597 + r131598;
        double r131600 = tan(r131599);
        double r131601 = tan(r131597);
        double r131602 = r131600 - r131601;
        return r131602;
}
↓
double f(double x, double eps) {
        double r131603 = eps;
        double r131604 = -2.2637252434053613e-90;
        bool r131605 = r131603 <= r131604;
        double r131606 = 1.0;
        double r131607 = x;
        double r131608 = tan(r131607);
        double r131609 = tan(r131603);
        double r131610 = r131608 * r131609;
        double r131611 = r131606 - r131610;
        double r131612 = r131608 + r131609;
        double r131613 = r131611 / r131612;
        double r131614 = r131606 / r131613;
        double r131615 = r131614 - r131608;
        double r131616 = 5.748527172026931e-167;
        bool r131617 = r131603 <= r131616;
        double r131618 = r131607 * r131603;
        double r131619 = r131603 + r131607;
        double r131620 = r131618 * r131619;
        double r131621 = r131620 + r131603;
        double r131622 = 3.0;
        double r131623 = pow(r131610, r131622);
        double r131624 = r131606 - r131623;
        double r131625 = r131612 / r131624;
        double r131626 = r131606 * r131606;
        double r131627 = r131610 * r131610;
        double r131628 = r131606 * r131610;
        double r131629 = r131627 + r131628;
        double r131630 = r131626 + r131629;
        double r131631 = r131625 * r131630;
        double r131632 = r131631 - r131608;
        double r131633 = r131617 ? r131621 : r131632;
        double r131634 = r131605 ? r131615 : r131633;
        return r131634;
}

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
Target15.4
Herbie15.8
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Split input into 3 regimes

  2. if eps < -2.2637252434053613e-90

    1. Initial program 30.9

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

    3. Applied tan-sum6.8

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

    5. Applied clear-num6.9

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

    if -2.2637252434053613e-90 < eps < 5.748527172026931e-167

    1. Initial program 48.9

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

    3. Applied tan-sum48.9

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

    4. Taylor expanded around 0 30.0

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

    5. Simplified29.7

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

    if 5.748527172026931e-167 < eps

    1. Initial program 32.4

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

    3. Applied tan-sum13.0

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

    5. Applied flip3--13.1

      \[\leadsto \frac{\tan x + \tan \varepsilon}{\color{blue}{\frac{{1}^{3} - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}}{1 \cdot 1 + \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right) + 1 \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)}}} - \tan x\]

    6. Applied associate-/r/13.1

      \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{{1}^{3} - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}} \cdot \left(1 \cdot 1 + \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right) + 1 \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)\right)} - \tan x\]

    7. Simplified13.1

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

  4. Final simplification15.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -2.2637252434053613 \cdot 10^{-90}:\\ \;\;\;\;\frac{1}{\frac{1 - \tan x \cdot \tan \varepsilon}{\tan x + \tan \varepsilon}} - \tan x\\ \mathbf{elif}\;\varepsilon \le 5.7485271720269307 \cdot 10^{-167}:\\ \;\;\;\;\left(x \cdot \varepsilon\right) \cdot \left(\varepsilon + x\right) + \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}} \cdot \left(1 \cdot 1 + \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right) + 1 \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)\right) - \tan x\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 
(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)))