Average Error: 37.1 → 15.0
Time: 26.4s
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le \frac{-4516838385750293}{6.518515124270355476059026202910010115365 \cdot 10^{91}}:\\ \;\;\;\;\frac{\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon}{\left(1 - \frac{\tan x}{\frac{\cos \varepsilon}{\sin \varepsilon}}\right) \cdot \left(\cos x \cdot \cos \varepsilon\right)} - \tan x\\ \mathbf{elif}\;\varepsilon \le \frac{4506594759216947}{41538374868278621028243970633760768}:\\ \;\;\;\;\varepsilon + \left(x \cdot \varepsilon\right) \cdot \left(\varepsilon + x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \sin x}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}\\ \end{array}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le \frac{-4516838385750293}{6.518515124270355476059026202910010115365 \cdot 10^{91}}:\\
\;\;\;\;\frac{\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon}{\left(1 - \frac{\tan x}{\frac{\cos \varepsilon}{\sin \varepsilon}}\right) \cdot \left(\cos x \cdot \cos \varepsilon\right)} - \tan x\\

\mathbf{elif}\;\varepsilon \le \frac{4506594759216947}{41538374868278621028243970633760768}:\\
\;\;\;\;\varepsilon + \left(x \cdot \varepsilon\right) \cdot \left(\varepsilon + x\right)\\

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

\end{array}
double f(double x, double eps) {
        double r87490 = x;
        double r87491 = eps;
        double r87492 = r87490 + r87491;
        double r87493 = tan(r87492);
        double r87494 = tan(r87490);
        double r87495 = r87493 - r87494;
        return r87495;
}
↓
double f(double x, double eps) {
        double r87496 = eps;
        double r87497 = -4516838385750293.0;
        double r87498 = 6.5185151242703555e+91;
        double r87499 = r87497 / r87498;
        bool r87500 = r87496 <= r87499;
        double r87501 = x;
        double r87502 = sin(r87501);
        double r87503 = cos(r87496);
        double r87504 = r87502 * r87503;
        double r87505 = cos(r87501);
        double r87506 = sin(r87496);
        double r87507 = r87505 * r87506;
        double r87508 = r87504 + r87507;
        double r87509 = 1.0;
        double r87510 = tan(r87501);
        double r87511 = r87503 / r87506;
        double r87512 = r87510 / r87511;
        double r87513 = r87509 - r87512;
        double r87514 = r87505 * r87503;
        double r87515 = r87513 * r87514;
        double r87516 = r87508 / r87515;
        double r87517 = r87516 - r87510;
        double r87518 = 4506594759216947.0;
        double r87519 = 4.153837486827862e+34;
        double r87520 = r87518 / r87519;
        bool r87521 = r87496 <= r87520;
        double r87522 = r87501 * r87496;
        double r87523 = r87496 + r87501;
        double r87524 = r87522 * r87523;
        double r87525 = r87496 + r87524;
        double r87526 = tan(r87496);
        double r87527 = r87510 + r87526;
        double r87528 = r87527 * r87505;
        double r87529 = r87510 * r87526;
        double r87530 = r87509 - r87529;
        double r87531 = r87530 * r87502;
        double r87532 = r87528 - r87531;
        double r87533 = r87530 * r87505;
        double r87534 = r87532 / r87533;
        double r87535 = r87521 ? r87525 : r87534;
        double r87536 = r87500 ? r87517 : r87535;
        return r87536;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.1
Target15.0
Herbie15.0
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Split input into 3 regimes

  2. if eps < -6.929244313528967e-77

    1. Initial program 30.9

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

    3. Applied tan-sum6.0

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

    5. Applied tan-quot6.1

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

    6. Applied tan-quot6.1

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

    7. Applied frac-add6.1

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

    8. Applied associate-/l/6.1

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

    10. Applied tan-quot6.1

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

    11. Applied associate-*r/6.1

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

    13. Applied associate-/l*6.1

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

    if -6.929244313528967e-77 < eps < 1.084923224249313e-19

    1. Initial program 46.6

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

    3. Applied tan-sum46.6

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

    4. Taylor expanded around 0 30.1

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

    5. Simplified29.9

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

    if 1.084923224249313e-19 < eps

    1. Initial program 29.0

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

    3. Applied tan-quot28.8

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

    4. Applied tan-sum1.2

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

    5. Applied frac-sub1.3

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

  4. Final simplification15.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le \frac{-4516838385750293}{6.518515124270355476059026202910010115365 \cdot 10^{91}}:\\ \;\;\;\;\frac{\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon}{\left(1 - \frac{\tan x}{\frac{\cos \varepsilon}{\sin \varepsilon}}\right) \cdot \left(\cos x \cdot \cos \varepsilon\right)} - \tan x\\ \mathbf{elif}\;\varepsilon \le \frac{4506594759216947}{41538374868278621028243970633760768}:\\ \;\;\;\;\varepsilon + \left(x \cdot \varepsilon\right) \cdot \left(\varepsilon + x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \sin x}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 
(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)))