Average Error: 37.2 → 15.8
Time: 14.4s
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -3.821283658645314156687993332074134483423 \cdot 10^{-93}:\\ \;\;\;\;\frac{\frac{\frac{\tan x + \tan \varepsilon}{\log \left(e^{1 - \tan x \cdot \tan \varepsilon}\right)} \cdot \left(\tan x + \tan \varepsilon\right)}{1 - \tan x \cdot \tan \varepsilon} + \left(-\tan x \cdot \tan x\right)}{\frac{\tan x + \tan \varepsilon}{\log \left(e^{1 - \tan x \cdot \tan \varepsilon}\right)} + \tan x}\\ \mathbf{elif}\;\varepsilon \le 5.540253431343870974412281715276917814538 \cdot 10^{-28}:\\ \;\;\;\;\left(x \cdot \varepsilon\right) \cdot \left(\varepsilon + x\right) + \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} \cdot \frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x \cdot \tan x}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} + \tan x}\\ \end{array}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -3.821283658645314156687993332074134483423 \cdot 10^{-93}:\\
\;\;\;\;\frac{\frac{\frac{\tan x + \tan \varepsilon}{\log \left(e^{1 - \tan x \cdot \tan \varepsilon}\right)} \cdot \left(\tan x + \tan \varepsilon\right)}{1 - \tan x \cdot \tan \varepsilon} + \left(-\tan x \cdot \tan x\right)}{\frac{\tan x + \tan \varepsilon}{\log \left(e^{1 - \tan x \cdot \tan \varepsilon}\right)} + \tan x}\\

\mathbf{elif}\;\varepsilon \le 5.540253431343870974412281715276917814538 \cdot 10^{-28}:\\
\;\;\;\;\left(x \cdot \varepsilon\right) \cdot \left(\varepsilon + x\right) + \varepsilon\\

\mathbf{else}:\\
\;\;\;\;\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}\\

\end{array}
double f(double x, double eps) {
        double r146117 = x;
        double r146118 = eps;
        double r146119 = r146117 + r146118;
        double r146120 = tan(r146119);
        double r146121 = tan(r146117);
        double r146122 = r146120 - r146121;
        return r146122;
}
↓
double f(double x, double eps) {
        double r146123 = eps;
        double r146124 = -3.821283658645314e-93;
        bool r146125 = r146123 <= r146124;
        double r146126 = x;
        double r146127 = tan(r146126);
        double r146128 = tan(r146123);
        double r146129 = r146127 + r146128;
        double r146130 = 1.0;
        double r146131 = r146127 * r146128;
        double r146132 = r146130 - r146131;
        double r146133 = exp(r146132);
        double r146134 = log(r146133);
        double r146135 = r146129 / r146134;
        double r146136 = r146135 * r146129;
        double r146137 = r146136 / r146132;
        double r146138 = r146127 * r146127;
        double r146139 = -r146138;
        double r146140 = r146137 + r146139;
        double r146141 = r146135 + r146127;
        double r146142 = r146140 / r146141;
        double r146143 = 5.540253431343871e-28;
        bool r146144 = r146123 <= r146143;
        double r146145 = r146126 * r146123;
        double r146146 = r146123 + r146126;
        double r146147 = r146145 * r146146;
        double r146148 = r146147 + r146123;
        double r146149 = r146129 / r146132;
        double r146150 = r146149 * r146149;
        double r146151 = r146150 - r146138;
        double r146152 = r146149 + r146127;
        double r146153 = r146151 / r146152;
        double r146154 = r146144 ? r146148 : r146153;
        double r146155 = r146125 ? r146142 : r146154;
        return r146155;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.2
Target15.1
Herbie15.8
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Split input into 3 regimes

  2. if eps < -3.821283658645314e-93

    1. Initial program 30.5

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

    3. Applied tan-sum7.6

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

    5. Applied add-log-exp7.7

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

    6. Applied add-log-exp7.7

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

    7. Applied diff-log7.8

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

    8. Simplified7.7

      \[\leadsto \frac{\tan x + \tan \varepsilon}{\log \color{blue}{\left(e^{1 - \tan x \cdot \tan \varepsilon}\right)}} - \tan x\]
    9. Using strategy rm

    10. Applied flip--7.8

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

    11. Simplified7.8

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

    if -3.821283658645314e-93 < eps < 5.540253431343871e-28

    1. Initial program 47.4

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

    3. Applied tan-sum47.4

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

    4. Taylor expanded around 0 31.9

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

    5. Simplified31.7

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

    if 5.540253431343871e-28 < eps

    1. Initial program 30.0

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

    3. Applied tan-sum1.9

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

    5. Applied flip--2.1

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

  4. Final simplification15.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -3.821283658645314156687993332074134483423 \cdot 10^{-93}:\\ \;\;\;\;\frac{\frac{\frac{\tan x + \tan \varepsilon}{\log \left(e^{1 - \tan x \cdot \tan \varepsilon}\right)} \cdot \left(\tan x + \tan \varepsilon\right)}{1 - \tan x \cdot \tan \varepsilon} + \left(-\tan x \cdot \tan x\right)}{\frac{\tan x + \tan \varepsilon}{\log \left(e^{1 - \tan x \cdot \tan \varepsilon}\right)} + \tan x}\\ \mathbf{elif}\;\varepsilon \le 5.540253431343870974412281715276917814538 \cdot 10^{-28}:\\ \;\;\;\;\left(x \cdot \varepsilon\right) \cdot \left(\varepsilon + x\right) + \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} \cdot \frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x \cdot \tan x}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} + \tan x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019318 
(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)))