Average Error: 37.6 → 15.5
Time: 10.6s
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 r166437 = x;
        double r166438 = eps;
        double r166439 = r166437 + r166438;
        double r166440 = tan(r166439);
        double r166441 = tan(r166437);
        double r166442 = r166440 - r166441;
        return r166442;
}
↓
double f(double x, double eps) {
        double r166443 = eps;
        double r166444 = -1.9052490692525806e-75;
        bool r166445 = r166443 <= r166444;
        double r166446 = x;
        double r166447 = tan(r166446);
        double r166448 = tan(r166443);
        double r166449 = r166447 + r166448;
        double r166450 = cos(r166446);
        double r166451 = r166449 * r166450;
        double r166452 = 1.0;
        double r166453 = sin(r166446);
        double r166454 = r166453 * r166448;
        double r166455 = r166454 / r166450;
        double r166456 = r166452 - r166455;
        double r166457 = r166456 * r166453;
        double r166458 = r166451 - r166457;
        double r166459 = r166456 * r166450;
        double r166460 = r166458 / r166459;
        double r166461 = 1.8727168353926166e-44;
        bool r166462 = r166443 <= r166461;
        double r166463 = r166446 * r166443;
        double r166464 = r166443 + r166446;
        double r166465 = r166463 * r166464;
        double r166466 = r166465 + r166443;
        double r166467 = 3.0;
        double r166468 = pow(r166447, r166467);
        double r166469 = pow(r166448, r166467);
        double r166470 = r166468 + r166469;
        double r166471 = r166448 - r166447;
        double r166472 = r166448 * r166471;
        double r166473 = r166447 * r166447;
        double r166474 = r166472 + r166473;
        double r166475 = r166447 * r166448;
        double r166476 = r166452 - r166475;
        double r166477 = r166474 * r166476;
        double r166478 = r166470 / r166477;
        double r166479 = r166478 - r166447;
        double r166480 = r166462 ? r166466 : r166479;
        double r166481 = r166445 ? r166460 : r166480;
        return r166481;
}

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)))