Average Error: 37.1 → 13.5
Time: 26.6s
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -9.642672794131257633584719535235763032688 \cdot 10^{-5} \lor \neg \left(\varepsilon \le 4.975666270883717105063888360742797507804 \cdot 10^{-20}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x, \tan x + \tan \varepsilon, \sin x \cdot \left(-1 + \tan x \cdot \tan \varepsilon\right)\right)}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({\varepsilon}^{2}, \mathsf{fma}\left(\varepsilon, \frac{1}{3}, x\right), \varepsilon\right)\\ \end{array}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -9.642672794131257633584719535235763032688 \cdot 10^{-5} \lor \neg \left(\varepsilon \le 4.975666270883717105063888360742797507804 \cdot 10^{-20}\right):\\
\;\;\;\;\frac{\mathsf{fma}\left(\cos x, \tan x + \tan \varepsilon, \sin x \cdot \left(-1 + \tan x \cdot \tan \varepsilon\right)\right)}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left({\varepsilon}^{2}, \mathsf{fma}\left(\varepsilon, \frac{1}{3}, x\right), \varepsilon\right)\\

\end{array}
double f(double x, double eps) {
        double r150166 = x;
        double r150167 = eps;
        double r150168 = r150166 + r150167;
        double r150169 = tan(r150168);
        double r150170 = tan(r150166);
        double r150171 = r150169 - r150170;
        return r150171;
}
↓
double f(double x, double eps) {
        double r150172 = eps;
        double r150173 = -9.642672794131258e-05;
        bool r150174 = r150172 <= r150173;
        double r150175 = 4.975666270883717e-20;
        bool r150176 = r150172 <= r150175;
        double r150177 = !r150176;
        bool r150178 = r150174 || r150177;
        double r150179 = x;
        double r150180 = cos(r150179);
        double r150181 = tan(r150179);
        double r150182 = tan(r150172);
        double r150183 = r150181 + r150182;
        double r150184 = sin(r150179);
        double r150185 = -1.0;
        double r150186 = r150181 * r150182;
        double r150187 = r150185 + r150186;
        double r150188 = r150184 * r150187;
        double r150189 = fma(r150180, r150183, r150188);
        double r150190 = 1.0;
        double r150191 = r150190 - r150186;
        double r150192 = r150191 * r150180;
        double r150193 = r150189 / r150192;
        double r150194 = 2.0;
        double r150195 = pow(r150172, r150194);
        double r150196 = 0.3333333333333333;
        double r150197 = fma(r150172, r150196, r150179);
        double r150198 = fma(r150195, r150197, r150172);
        double r150199 = r150178 ? r150193 : r150198;
        return r150199;
}

Error

Bits error versus x

Bits error versus eps

Target

Original37.1
Target15.0
Herbie13.5
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Split input into 2 regimes

  2. if eps < -9.642672794131258e-05 or 4.975666270883717e-20 < eps

    1. Initial program 29.6

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

    3. Applied tan-sum0.8

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

    5. Applied tan-quot0.8

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

    6. Applied frac-sub0.9

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

    7. Simplified0.9

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

    if -9.642672794131258e-05 < eps < 4.975666270883717e-20

    1. Initial program 45.1

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

    3. Applied tan-sum44.8

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

    4. Taylor expanded around inf 44.9

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

    6. Applied div-inv44.9

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

    7. Applied fma-neg44.9

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

    8. Taylor expanded around 0 27.0

      \[\leadsto \color{blue}{x \cdot {\varepsilon}^{2} + \left(\frac{1}{3} \cdot {\varepsilon}^{3} + \varepsilon\right)}\]

    9. Simplified27.0

      \[\leadsto \color{blue}{\mathsf{fma}\left({\varepsilon}^{2}, \mathsf{fma}\left(\varepsilon, \frac{1}{3}, x\right), \varepsilon\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification13.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -9.642672794131257633584719535235763032688 \cdot 10^{-5} \lor \neg \left(\varepsilon \le 4.975666270883717105063888360742797507804 \cdot 10^{-20}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x, \tan x + \tan \varepsilon, \sin x \cdot \left(-1 + \tan x \cdot \tan \varepsilon\right)\right)}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({\varepsilon}^{2}, \mathsf{fma}\left(\varepsilon, \frac{1}{3}, x\right), \varepsilon\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 +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)))