Average Error: 37.3 → 15.2
Time: 28.6s
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -2.1070254757403724 \cdot 10^{-22}:\\ \;\;\;\;\frac{1}{1 - \tan x \cdot \tan \varepsilon} \cdot \left(\tan \varepsilon + \tan x\right) - \tan x\\ \mathbf{elif}\;\varepsilon \le 2.7293458866662548 \cdot 10^{-21}:\\ \;\;\;\;\left(x \cdot \varepsilon\right) \cdot \left(x + \varepsilon\right) + \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\tan \varepsilon + \tan x\right) \cdot \cos x - \sin x \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)}{\cos x \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)}\\ \end{array}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -2.1070254757403724 \cdot 10^{-22}:\\
\;\;\;\;\frac{1}{1 - \tan x \cdot \tan \varepsilon} \cdot \left(\tan \varepsilon + \tan x\right) - \tan x\\

\mathbf{elif}\;\varepsilon \le 2.7293458866662548 \cdot 10^{-21}:\\
\;\;\;\;\left(x \cdot \varepsilon\right) \cdot \left(x + \varepsilon\right) + \varepsilon\\

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

\end{array}
double f(double x, double eps) {
        double r1314798 = x;
        double r1314799 = eps;
        double r1314800 = r1314798 + r1314799;
        double r1314801 = tan(r1314800);
        double r1314802 = tan(r1314798);
        double r1314803 = r1314801 - r1314802;
        return r1314803;
}
↓
double f(double x, double eps) {
        double r1314804 = eps;
        double r1314805 = -2.1070254757403724e-22;
        bool r1314806 = r1314804 <= r1314805;
        double r1314807 = 1.0;
        double r1314808 = x;
        double r1314809 = tan(r1314808);
        double r1314810 = tan(r1314804);
        double r1314811 = r1314809 * r1314810;
        double r1314812 = r1314807 - r1314811;
        double r1314813 = r1314807 / r1314812;
        double r1314814 = r1314810 + r1314809;
        double r1314815 = r1314813 * r1314814;
        double r1314816 = r1314815 - r1314809;
        double r1314817 = 2.7293458866662548e-21;
        bool r1314818 = r1314804 <= r1314817;
        double r1314819 = r1314808 * r1314804;
        double r1314820 = r1314808 + r1314804;
        double r1314821 = r1314819 * r1314820;
        double r1314822 = r1314821 + r1314804;
        double r1314823 = cos(r1314808);
        double r1314824 = r1314814 * r1314823;
        double r1314825 = sin(r1314808);
        double r1314826 = r1314825 * r1314812;
        double r1314827 = r1314824 - r1314826;
        double r1314828 = r1314823 * r1314812;
        double r1314829 = r1314827 / r1314828;
        double r1314830 = r1314818 ? r1314822 : r1314829;
        double r1314831 = r1314806 ? r1314816 : r1314830;
        return r1314831;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.3
Target15.2
Herbie15.2
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Split input into 3 regimes

  2. if eps < -2.1070254757403724e-22

    1. Initial program 31.3

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

    3. Applied tan-sum1.6

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

    5. Applied div-inv1.6

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

    7. Applied *-un-lft-identity1.6

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

    8. Applied *-un-lft-identity1.6

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

    9. Applied times-frac1.6

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

    10. Applied associate-*r*1.6

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

    11. Simplified1.6

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

    if -2.1070254757403724e-22 < eps < 2.7293458866662548e-21

    1. Initial program 44.9

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

    2. Taylor expanded around 0 30.6

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

    3. Simplified30.6

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

    if 2.7293458866662548e-21 < eps

    1. Initial program 29.8

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

    3. Applied tan-sum1.4

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

    5. Applied div-inv1.5

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

    7. Applied tan-quot1.5

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

    8. Applied un-div-inv1.5

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

    9. Applied frac-sub1.6

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

  4. Final simplification15.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -2.1070254757403724 \cdot 10^{-22}:\\ \;\;\;\;\frac{1}{1 - \tan x \cdot \tan \varepsilon} \cdot \left(\tan \varepsilon + \tan x\right) - \tan x\\ \mathbf{elif}\;\varepsilon \le 2.7293458866662548 \cdot 10^{-21}:\\ \;\;\;\;\left(x \cdot \varepsilon\right) \cdot \left(x + \varepsilon\right) + \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\tan \varepsilon + \tan x\right) \cdot \cos x - \sin x \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)}{\cos x \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019151 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"

  :herbie-target
  (/ (sin eps) (* (cos x) (cos (+ x eps))))

  (- (tan (+ x eps)) (tan x)))