Average Error: 37.6 → 15.5
Time: 10.0s
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -1.90524906925258064 \cdot 10^{-75}:\\ \;\;\;\;\frac{\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \left(1 - \frac{\sin x \cdot \tan \varepsilon}{\cos x}\right) \cdot \sin x}{\left(1 - \frac{\sin x \cdot \tan \varepsilon}{\cos x}\right) \cdot \cos x}\\ \mathbf{elif}\;\varepsilon \le 1.8727168353926166 \cdot 10^{-44}:\\ \;\;\;\;\left(x \cdot \varepsilon\right) \cdot \left(\varepsilon + x\right) + \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\tan x\right)}^{3} + {\left(\tan \varepsilon\right)}^{3}}{\left(\tan \varepsilon \cdot \left(\tan \varepsilon - \tan x\right) + \tan x \cdot \tan x\right) \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)} - \tan x\\ \end{array}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -1.90524906925258064 \cdot 10^{-75}:\\
\;\;\;\;\frac{\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \left(1 - \frac{\sin x \cdot \tan \varepsilon}{\cos x}\right) \cdot \sin x}{\left(1 - \frac{\sin x \cdot \tan \varepsilon}{\cos x}\right) \cdot \cos x}\\

\mathbf{elif}\;\varepsilon \le 1.8727168353926166 \cdot 10^{-44}:\\
\;\;\;\;\left(x \cdot \varepsilon\right) \cdot \left(\varepsilon + x\right) + \varepsilon\\

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

\end{array}
double f(double x, double eps) {
        double r156566 = x;
        double r156567 = eps;
        double r156568 = r156566 + r156567;
        double r156569 = tan(r156568);
        double r156570 = tan(r156566);
        double r156571 = r156569 - r156570;
        return r156571;
}
↓
double f(double x, double eps) {
        double r156572 = eps;
        double r156573 = -1.9052490692525806e-75;
        bool r156574 = r156572 <= r156573;
        double r156575 = x;
        double r156576 = tan(r156575);
        double r156577 = tan(r156572);
        double r156578 = r156576 + r156577;
        double r156579 = cos(r156575);
        double r156580 = r156578 * r156579;
        double r156581 = 1.0;
        double r156582 = sin(r156575);
        double r156583 = r156582 * r156577;
        double r156584 = r156583 / r156579;
        double r156585 = r156581 - r156584;
        double r156586 = r156585 * r156582;
        double r156587 = r156580 - r156586;
        double r156588 = r156585 * r156579;
        double r156589 = r156587 / r156588;
        double r156590 = 1.8727168353926166e-44;
        bool r156591 = r156572 <= r156590;
        double r156592 = r156575 * r156572;
        double r156593 = r156572 + r156575;
        double r156594 = r156592 * r156593;
        double r156595 = r156594 + r156572;
        double r156596 = 3.0;
        double r156597 = pow(r156576, r156596);
        double r156598 = pow(r156577, r156596);
        double r156599 = r156597 + r156598;
        double r156600 = r156577 - r156576;
        double r156601 = r156577 * r156600;
        double r156602 = r156576 * r156576;
        double r156603 = r156601 + r156602;
        double r156604 = r156576 * r156577;
        double r156605 = r156581 - r156604;
        double r156606 = r156603 * r156605;
        double r156607 = r156599 / r156606;
        double r156608 = r156607 - r156576;
        double r156609 = r156591 ? r156595 : r156608;
        double r156610 = r156574 ? r156589 : r156609;
        return r156610;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.6
Target15.4
Herbie15.5
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Split input into 3 regimes

  2. if eps < -1.9052490692525806e-75

    1. Initial program 30.2

      \[\tan \left(x + \varepsilon\right) - \tan x\]
    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-quot6.0

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

    6. Applied associate-*l/6.0

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

    8. Applied tan-quot6.1

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

    9. Applied frac-sub6.1

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

    if -1.9052490692525806e-75 < eps < 1.8727168353926166e-44

    1. Initial program 47.3

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

    3. Applied tan-sum47.3

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

    4. Taylor expanded around 0 31.3

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

    5. Simplified31.1

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

    if 1.8727168353926166e-44 < eps

    1. Initial program 31.4

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

    3. Applied tan-sum2.6

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

    5. Applied flip3-+2.8

      \[\leadsto \frac{\color{blue}{\frac{{\left(\tan x\right)}^{3} + {\left(\tan \varepsilon\right)}^{3}}{\tan x \cdot \tan x + \left(\tan \varepsilon \cdot \tan \varepsilon - \tan x \cdot \tan \varepsilon\right)}}}{1 - \tan x \cdot \tan \varepsilon} - \tan x\]

    6. Applied associate-/l/2.8

      \[\leadsto \color{blue}{\frac{{\left(\tan x\right)}^{3} + {\left(\tan \varepsilon\right)}^{3}}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan x + \left(\tan \varepsilon \cdot \tan \varepsilon - \tan x \cdot \tan \varepsilon\right)\right)}} - \tan x\]

    7. Simplified2.8

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

  4. Final simplification15.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1.90524906925258064 \cdot 10^{-75}:\\ \;\;\;\;\frac{\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \left(1 - \frac{\sin x \cdot \tan \varepsilon}{\cos x}\right) \cdot \sin x}{\left(1 - \frac{\sin x \cdot \tan \varepsilon}{\cos x}\right) \cdot \cos x}\\ \mathbf{elif}\;\varepsilon \le 1.8727168353926166 \cdot 10^{-44}:\\ \;\;\;\;\left(x \cdot \varepsilon\right) \cdot \left(\varepsilon + x\right) + \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\tan x\right)}^{3} + {\left(\tan \varepsilon\right)}^{3}}{\left(\tan \varepsilon \cdot \left(\tan \varepsilon - \tan x\right) + \tan x \cdot \tan x\right) \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)} - \tan x\\ \end{array}\]

Reproduce

herbie shell --seed 2020036 
(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)))