Average Error: 37.1 → 15.5
Time: 1.2m
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -6.095793693328762137298872934344524249228 \cdot 10^{-10}:\\ \;\;\;\;\frac{\cos x \cdot \left(\tan \varepsilon + \tan x\right) - \left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \sin x}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}\\ \mathbf{elif}\;\varepsilon \le 1.066275681806479256320193537654211119112 \cdot 10^{-47}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \left(x \cdot \left(x + \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\tan \varepsilon + \tan x\right) \cdot \left(1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)\right) \cdot \cos x - \sin x \cdot \left(\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \left(1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)\right)}{\left(\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \left(1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)\right) \cdot \cos x}\\ \end{array}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -6.095793693328762137298872934344524249228 \cdot 10^{-10}:\\
\;\;\;\;\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 1.066275681806479256320193537654211119112 \cdot 10^{-47}:\\
\;\;\;\;\varepsilon + \varepsilon \cdot \left(x \cdot \left(x + \varepsilon\right)\right)\\

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

\end{array}
double f(double x, double eps) {
        double r7032785 = x;
        double r7032786 = eps;
        double r7032787 = r7032785 + r7032786;
        double r7032788 = tan(r7032787);
        double r7032789 = tan(r7032785);
        double r7032790 = r7032788 - r7032789;
        return r7032790;
}
↓
double f(double x, double eps) {
        double r7032791 = eps;
        double r7032792 = -6.095793693328762e-10;
        bool r7032793 = r7032791 <= r7032792;
        double r7032794 = x;
        double r7032795 = cos(r7032794);
        double r7032796 = tan(r7032791);
        double r7032797 = tan(r7032794);
        double r7032798 = r7032796 + r7032797;
        double r7032799 = r7032795 * r7032798;
        double r7032800 = 1.0;
        double r7032801 = r7032797 * r7032796;
        double r7032802 = r7032800 - r7032801;
        double r7032803 = sin(r7032794);
        double r7032804 = r7032802 * r7032803;
        double r7032805 = r7032799 - r7032804;
        double r7032806 = r7032802 * r7032795;
        double r7032807 = r7032805 / r7032806;
        double r7032808 = 1.0662756818064793e-47;
        bool r7032809 = r7032791 <= r7032808;
        double r7032810 = r7032794 + r7032791;
        double r7032811 = r7032794 * r7032810;
        double r7032812 = r7032791 * r7032811;
        double r7032813 = r7032791 + r7032812;
        double r7032814 = r7032801 * r7032801;
        double r7032815 = r7032800 - r7032814;
        double r7032816 = r7032798 * r7032815;
        double r7032817 = r7032816 * r7032795;
        double r7032818 = r7032802 * r7032815;
        double r7032819 = r7032803 * r7032818;
        double r7032820 = r7032817 - r7032819;
        double r7032821 = r7032818 * r7032795;
        double r7032822 = r7032820 / r7032821;
        double r7032823 = r7032809 ? r7032813 : r7032822;
        double r7032824 = r7032793 ? r7032807 : r7032823;
        return r7032824;
}

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
Target15.1
Herbie15.5
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Split input into 3 regimes

  2. if eps < -6.095793693328762e-10

    1. Initial program 30.2

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

    3. Applied tan-quot30.1

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

    4. Applied tan-sum0.6

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

    5. Applied frac-sub0.6

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

    if -6.095793693328762e-10 < eps < 1.0662756818064793e-47

    1. Initial program 45.5

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

    2. Taylor expanded around 0 31.3

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

    3. Simplified31.3

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

    if 1.0662756818064793e-47 < eps

    1. Initial program 30.0

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

    3. Applied tan-sum3.7

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

    5. Applied flip--3.7

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

    6. Applied associate-/r/3.7

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

    8. Applied tan-quot3.8

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

    9. Applied flip-+3.8

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

    10. Applied frac-times3.8

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

    11. Applied frac-sub3.8

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

  4. Final simplification15.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -6.095793693328762137298872934344524249228 \cdot 10^{-10}:\\ \;\;\;\;\frac{\cos x \cdot \left(\tan \varepsilon + \tan x\right) - \left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \sin x}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}\\ \mathbf{elif}\;\varepsilon \le 1.066275681806479256320193537654211119112 \cdot 10^{-47}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \left(x \cdot \left(x + \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\tan \varepsilon + \tan x\right) \cdot \left(1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)\right) \cdot \cos x - \sin x \cdot \left(\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \left(1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)\right)}{\left(\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \left(1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)\right) \cdot \cos x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"

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

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