Average Error: 36.7 → 6.3
Time: 39.6s
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -2.880748992750146398620281343784345499703 \cdot 10^{-51} \lor \neg \left(\varepsilon \le 1.342409282000481224159305634141017423042 \cdot 10^{-20}\right):\\ \;\;\;\;\frac{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} \cdot \frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x \cdot \tan x}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} + \tan x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \cdot {\varepsilon}^{2} + \varepsilon\right) - \frac{1}{6} \cdot {\varepsilon}^{3}}{\left(\cos x \cdot \cos \varepsilon\right) \cdot \cos x}\\ \end{array}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -2.880748992750146398620281343784345499703 \cdot 10^{-51} \lor \neg \left(\varepsilon \le 1.342409282000481224159305634141017423042 \cdot 10^{-20}\right):\\
\;\;\;\;\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}\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(x \cdot {\varepsilon}^{2} + \varepsilon\right) - \frac{1}{6} \cdot {\varepsilon}^{3}}{\left(\cos x \cdot \cos \varepsilon\right) \cdot \cos x}\\

\end{array}
double f(double x, double eps) {
        double r67578 = x;
        double r67579 = eps;
        double r67580 = r67578 + r67579;
        double r67581 = tan(r67580);
        double r67582 = tan(r67578);
        double r67583 = r67581 - r67582;
        return r67583;
}
↓
double f(double x, double eps) {
        double r67584 = eps;
        double r67585 = -2.8807489927501464e-51;
        bool r67586 = r67584 <= r67585;
        double r67587 = 1.3424092820004812e-20;
        bool r67588 = r67584 <= r67587;
        double r67589 = !r67588;
        bool r67590 = r67586 || r67589;
        double r67591 = x;
        double r67592 = tan(r67591);
        double r67593 = tan(r67584);
        double r67594 = r67592 + r67593;
        double r67595 = 1.0;
        double r67596 = r67592 * r67593;
        double r67597 = r67595 - r67596;
        double r67598 = r67594 / r67597;
        double r67599 = r67598 * r67598;
        double r67600 = r67592 * r67592;
        double r67601 = r67599 - r67600;
        double r67602 = r67598 + r67592;
        double r67603 = r67601 / r67602;
        double r67604 = 2.0;
        double r67605 = pow(r67584, r67604);
        double r67606 = r67591 * r67605;
        double r67607 = r67606 + r67584;
        double r67608 = 0.16666666666666666;
        double r67609 = 3.0;
        double r67610 = pow(r67584, r67609);
        double r67611 = r67608 * r67610;
        double r67612 = r67607 - r67611;
        double r67613 = cos(r67591);
        double r67614 = cos(r67584);
        double r67615 = r67613 * r67614;
        double r67616 = r67615 * r67613;
        double r67617 = r67612 / r67616;
        double r67618 = r67590 ? r67603 : r67617;
        return r67618;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original36.7
Target14.9
Herbie6.3
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Split input into 2 regimes

  2. if eps < -2.8807489927501464e-51 or 1.3424092820004812e-20 < eps

    1. Initial program 29.4

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

    3. Applied tan-sum2.8

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

    5. Applied flip--2.9

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

    if -2.8807489927501464e-51 < eps < 1.3424092820004812e-20

    1. Initial program 45.8

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

    3. Applied tan-sum45.8

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

    5. Applied div-inv45.8

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

    7. Applied tan-quot46.0

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

    8. Applied tan-quot46.0

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

    9. Applied tan-quot45.8

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

    10. Applied frac-add45.8

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

    11. Applied associate-*l/45.8

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

    12. Applied frac-sub45.8

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

    13. Simplified45.8

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

    14. Taylor expanded around 0 10.4

      \[\leadsto \frac{\color{blue}{\left(x \cdot {\varepsilon}^{2} + \varepsilon\right) - \frac{1}{6} \cdot {\varepsilon}^{3}}}{\left(\cos x \cdot \cos \varepsilon\right) \cdot \cos x}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification6.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -2.880748992750146398620281343784345499703 \cdot 10^{-51} \lor \neg \left(\varepsilon \le 1.342409282000481224159305634141017423042 \cdot 10^{-20}\right):\\ \;\;\;\;\frac{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} \cdot \frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x \cdot \tan x}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} + \tan x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \cdot {\varepsilon}^{2} + \varepsilon\right) - \frac{1}{6} \cdot {\varepsilon}^{3}}{\left(\cos x \cdot \cos \varepsilon\right) \cdot \cos x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 
(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)))