Average Error: 36.9 → 15.0
Time: 1.0m
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -6.004417074507704 \cdot 10^{-21}:\\ \;\;\;\;\frac{\cos x \cdot \left(\tan \varepsilon + \tan x\right) - \left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \sin x}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}\\ \mathbf{elif}\;\varepsilon \le 3.132510926694067 \cdot 10^{-23}:\\ \;\;\;\;\varepsilon + \left(\varepsilon \cdot \left(x + \varepsilon\right)\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\tan \varepsilon + \tan x}{1 - \log \left(e^{\tan x \cdot \tan \varepsilon}\right)} \cdot \frac{\tan \varepsilon + \tan x}{1 - \log \left(e^{\tan x \cdot \tan \varepsilon}\right)} - \tan x \cdot \tan x}{\frac{\tan \varepsilon + \tan x}{1 - \log \left(e^{\tan x \cdot \tan \varepsilon}\right)} + \tan x}\\ \end{array}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -6.004417074507704 \cdot 10^{-21}:\\
\;\;\;\;\frac{\cos x \cdot \left(\tan \varepsilon + \tan x\right) - \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 3.132510926694067 \cdot 10^{-23}:\\
\;\;\;\;\varepsilon + \left(\varepsilon \cdot \left(x + \varepsilon\right)\right) \cdot x\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\tan \varepsilon + \tan x}{1 - \log \left(e^{\tan x \cdot \tan \varepsilon}\right)} \cdot \frac{\tan \varepsilon + \tan x}{1 - \log \left(e^{\tan x \cdot \tan \varepsilon}\right)} - \tan x \cdot \tan x}{\frac{\tan \varepsilon + \tan x}{1 - \log \left(e^{\tan x \cdot \tan \varepsilon}\right)} + \tan x}\\

\end{array}
double f(double x, double eps) {
        double r8656371 = x;
        double r8656372 = eps;
        double r8656373 = r8656371 + r8656372;
        double r8656374 = tan(r8656373);
        double r8656375 = tan(r8656371);
        double r8656376 = r8656374 - r8656375;
        return r8656376;
}
↓
double f(double x, double eps) {
        double r8656377 = eps;
        double r8656378 = -6.004417074507704e-21;
        bool r8656379 = r8656377 <= r8656378;
        double r8656380 = x;
        double r8656381 = cos(r8656380);
        double r8656382 = tan(r8656377);
        double r8656383 = tan(r8656380);
        double r8656384 = r8656382 + r8656383;
        double r8656385 = r8656381 * r8656384;
        double r8656386 = 1.0;
        double r8656387 = r8656383 * r8656382;
        double r8656388 = r8656386 - r8656387;
        double r8656389 = sin(r8656380);
        double r8656390 = r8656388 * r8656389;
        double r8656391 = r8656385 - r8656390;
        double r8656392 = r8656388 * r8656381;
        double r8656393 = r8656391 / r8656392;
        double r8656394 = 3.132510926694067e-23;
        bool r8656395 = r8656377 <= r8656394;
        double r8656396 = r8656380 + r8656377;
        double r8656397 = r8656377 * r8656396;
        double r8656398 = r8656397 * r8656380;
        double r8656399 = r8656377 + r8656398;
        double r8656400 = exp(r8656387);
        double r8656401 = log(r8656400);
        double r8656402 = r8656386 - r8656401;
        double r8656403 = r8656384 / r8656402;
        double r8656404 = r8656403 * r8656403;
        double r8656405 = r8656383 * r8656383;
        double r8656406 = r8656404 - r8656405;
        double r8656407 = r8656403 + r8656383;
        double r8656408 = r8656406 / r8656407;
        double r8656409 = r8656395 ? r8656399 : r8656408;
        double r8656410 = r8656379 ? r8656393 : r8656409;
        return r8656410;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original36.9
Target15.1
Herbie15.0
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Split input into 3 regimes

  2. if eps < -6.004417074507704e-21

    1. Initial program 29.5

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

    3. Applied tan-quot29.4

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

    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

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

    if -6.004417074507704e-21 < eps < 3.132510926694067e-23

    1. Initial program 45.3

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

    3. Applied tan-sum45.3

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

    4. Taylor expanded around 0 30.5

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

    5. Simplified30.4

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

    if 3.132510926694067e-23 < eps

    1. Initial program 30.2

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

    3. Applied tan-sum2.0

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

    5. Applied add-log-exp2.3

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

    7. Applied flip--2.4

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

  4. Final simplification15.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -6.004417074507704 \cdot 10^{-21}:\\ \;\;\;\;\frac{\cos x \cdot \left(\tan \varepsilon + \tan x\right) - \left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \sin x}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}\\ \mathbf{elif}\;\varepsilon \le 3.132510926694067 \cdot 10^{-23}:\\ \;\;\;\;\varepsilon + \left(\varepsilon \cdot \left(x + \varepsilon\right)\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\tan \varepsilon + \tan x}{1 - \log \left(e^{\tan x \cdot \tan \varepsilon}\right)} \cdot \frac{\tan \varepsilon + \tan x}{1 - \log \left(e^{\tan x \cdot \tan \varepsilon}\right)} - \tan x \cdot \tan x}{\frac{\tan \varepsilon + \tan x}{1 - \log \left(e^{\tan x \cdot \tan \varepsilon}\right)} + \tan x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019104 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"

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

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