Average Error: 36.8 → 15.3
Time: 12.4s
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 r108137 = x;
        double r108138 = eps;
        double r108139 = r108137 + r108138;
        double r108140 = tan(r108139);
        double r108141 = tan(r108137);
        double r108142 = r108140 - r108141;
        return r108142;
}
↓
double f(double x, double eps) {
        double r108143 = eps;
        double r108144 = -7.65111219922148e-19;
        bool r108145 = r108143 <= r108144;
        double r108146 = x;
        double r108147 = tan(r108146);
        double r108148 = tan(r108143);
        double r108149 = r108147 + r108148;
        double r108150 = cos(r108146);
        double r108151 = r108149 * r108150;
        double r108152 = 1.0;
        double r108153 = r108147 * r108148;
        double r108154 = r108152 - r108153;
        double r108155 = sin(r108146);
        double r108156 = r108154 * r108155;
        double r108157 = r108151 - r108156;
        double r108158 = r108154 * r108150;
        double r108159 = r108157 / r108158;
        double r108160 = 1.6763362246559567e-31;
        bool r108161 = r108143 <= r108160;
        double r108162 = 2.0;
        double r108163 = pow(r108143, r108162);
        double r108164 = pow(r108146, r108162);
        double r108165 = fma(r108143, r108164, r108143);
        double r108166 = fma(r108163, r108146, r108165);
        double r108167 = 3.0;
        double r108168 = pow(r108147, r108167);
        double r108169 = pow(r108148, r108167);
        double r108170 = r108168 + r108169;
        double r108171 = r108148 - r108147;
        double r108172 = r108148 * r108171;
        double r108173 = fma(r108147, r108147, r108172);
        double r108174 = r108170 / r108173;
        double r108175 = r108174 / r108154;
        double r108176 = r108175 - r108147;
        double r108177 = r108161 ? r108166 : r108176;
        double r108178 = r108145 ? r108159 : r108177;
        return r108178;
}

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)))