Average Error: 36.8 → 15.3
Time: 11.9s
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -7.651112199221479645004847613563331532866 \cdot 10^{-19}:\\ \;\;\;\;\frac{\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \sin x}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}\\ \mathbf{elif}\;\varepsilon \le 1.676336224655956653828090193012541335235 \cdot 10^{-31}:\\ \;\;\;\;\mathsf{fma}\left({\varepsilon}^{2}, x, \mathsf{fma}\left(\varepsilon, {x}^{2}, \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\left(\tan x\right)}^{3} + {\left(\tan \varepsilon\right)}^{3}}{\mathsf{fma}\left(\tan x, \tan x, \tan \varepsilon \cdot \left(\tan \varepsilon - \tan x\right)\right)}}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \end{array}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -7.651112199221479645004847613563331532866 \cdot 10^{-19}:\\
\;\;\;\;\frac{\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \sin x}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}\\

\mathbf{elif}\;\varepsilon \le 1.676336224655956653828090193012541335235 \cdot 10^{-31}:\\
\;\;\;\;\mathsf{fma}\left({\varepsilon}^{2}, x, \mathsf{fma}\left(\varepsilon, {x}^{2}, \varepsilon\right)\right)\\

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

\end{array}
double f(double x, double eps) {
        double r113507 = x;
        double r113508 = eps;
        double r113509 = r113507 + r113508;
        double r113510 = tan(r113509);
        double r113511 = tan(r113507);
        double r113512 = r113510 - r113511;
        return r113512;
}
↓
double f(double x, double eps) {
        double r113513 = eps;
        double r113514 = -7.65111219922148e-19;
        bool r113515 = r113513 <= r113514;
        double r113516 = x;
        double r113517 = tan(r113516);
        double r113518 = tan(r113513);
        double r113519 = r113517 + r113518;
        double r113520 = cos(r113516);
        double r113521 = r113519 * r113520;
        double r113522 = 1.0;
        double r113523 = r113517 * r113518;
        double r113524 = r113522 - r113523;
        double r113525 = sin(r113516);
        double r113526 = r113524 * r113525;
        double r113527 = r113521 - r113526;
        double r113528 = r113524 * r113520;
        double r113529 = r113527 / r113528;
        double r113530 = 1.6763362246559567e-31;
        bool r113531 = r113513 <= r113530;
        double r113532 = 2.0;
        double r113533 = pow(r113513, r113532);
        double r113534 = pow(r113516, r113532);
        double r113535 = fma(r113513, r113534, r113513);
        double r113536 = fma(r113533, r113516, r113535);
        double r113537 = 3.0;
        double r113538 = pow(r113517, r113537);
        double r113539 = pow(r113518, r113537);
        double r113540 = r113538 + r113539;
        double r113541 = r113518 - r113517;
        double r113542 = r113518 * r113541;
        double r113543 = fma(r113517, r113517, r113542);
        double r113544 = r113540 / r113543;
        double r113545 = r113544 / r113524;
        double r113546 = r113545 - r113517;
        double r113547 = r113531 ? r113536 : r113546;
        double r113548 = r113515 ? r113529 : r113547;
        return r113548;
}

Error

Bits error versus x

Bits error versus eps

Target

Original36.8
Target15.2
Herbie15.3
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Split input into 3 regimes

  2. if eps < -7.65111219922148e-19

    1. Initial program 29.9

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

    3. Applied tan-quot29.7

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

    4. Applied tan-sum1.0

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

    5. Applied frac-sub1.0

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

    if -7.65111219922148e-19 < eps < 1.6763362246559567e-31

    1. Initial program 45.0

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

    2. Taylor expanded around 0 31.0

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

    3. Simplified31.0

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

    if 1.6763362246559567e-31 < eps

    1. Initial program 29.9

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

    3. Applied tan-sum2.5

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

    5. Applied flip3-+2.7

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

    6. Simplified2.7

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

  4. Final simplification15.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -7.651112199221479645004847613563331532866 \cdot 10^{-19}:\\ \;\;\;\;\frac{\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \sin x}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}\\ \mathbf{elif}\;\varepsilon \le 1.676336224655956653828090193012541335235 \cdot 10^{-31}:\\ \;\;\;\;\mathsf{fma}\left({\varepsilon}^{2}, x, \mathsf{fma}\left(\varepsilon, {x}^{2}, \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\left(\tan x\right)}^{3} + {\left(\tan \varepsilon\right)}^{3}}{\mathsf{fma}\left(\tan x, \tan x, \tan \varepsilon \cdot \left(\tan \varepsilon - \tan x\right)\right)}}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 +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)))