Average Error: 36.5 → 14.5
Time: 1.1m
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -1.0188842112755097 \cdot 10^{-17}:\\ \;\;\;\;(\left(\frac{\tan \varepsilon + \tan x}{1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)}\right) \cdot \left(\tan x \cdot \tan \varepsilon + 1\right) + \left(-\tan x\right))_*\\ \mathbf{elif}\;\varepsilon \le 2.896438129997832 \cdot 10^{-56}:\\ \;\;\;\;(\left(x \cdot \varepsilon\right) \cdot \left(x + \varepsilon\right) + \varepsilon)_*\\ \mathbf{else}:\\ \;\;\;\;(\left(\frac{\tan \varepsilon + \tan x}{1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)}\right) \cdot \left(\tan x \cdot \tan \varepsilon + 1\right) + \left(-\tan x\right))_*\\ \end{array}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -1.0188842112755097 \cdot 10^{-17}:\\
\;\;\;\;(\left(\frac{\tan \varepsilon + \tan x}{1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)}\right) \cdot \left(\tan x \cdot \tan \varepsilon + 1\right) + \left(-\tan x\right))_*\\

\mathbf{elif}\;\varepsilon \le 2.896438129997832 \cdot 10^{-56}:\\
\;\;\;\;(\left(x \cdot \varepsilon\right) \cdot \left(x + \varepsilon\right) + \varepsilon)_*\\

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

\end{array}
double f(double x, double eps) {
        double r15817498 = x;
        double r15817499 = eps;
        double r15817500 = r15817498 + r15817499;
        double r15817501 = tan(r15817500);
        double r15817502 = tan(r15817498);
        double r15817503 = r15817501 - r15817502;
        return r15817503;
}
↓
double f(double x, double eps) {
        double r15817504 = eps;
        double r15817505 = -1.0188842112755097e-17;
        bool r15817506 = r15817504 <= r15817505;
        double r15817507 = tan(r15817504);
        double r15817508 = x;
        double r15817509 = tan(r15817508);
        double r15817510 = r15817507 + r15817509;
        double r15817511 = 1.0;
        double r15817512 = r15817509 * r15817507;
        double r15817513 = r15817512 * r15817512;
        double r15817514 = r15817511 - r15817513;
        double r15817515 = r15817510 / r15817514;
        double r15817516 = r15817512 + r15817511;
        double r15817517 = -r15817509;
        double r15817518 = fma(r15817515, r15817516, r15817517);
        double r15817519 = 2.896438129997832e-56;
        bool r15817520 = r15817504 <= r15817519;
        double r15817521 = r15817508 * r15817504;
        double r15817522 = r15817508 + r15817504;
        double r15817523 = fma(r15817521, r15817522, r15817504);
        double r15817524 = r15817520 ? r15817523 : r15817518;
        double r15817525 = r15817506 ? r15817518 : r15817524;
        return r15817525;
}

Error

Bits error versus x

Bits error versus eps

Target

Original36.5
Target15.1
Herbie14.5
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Split input into 2 regimes

  2. if eps < -1.0188842112755097e-17 or 2.896438129997832e-56 < eps

    1. Initial program 29.5

      \[\tan \left(x + \varepsilon\right) - \tan x\]
    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 flip--2.6

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

    6. Applied associate-/r/2.6

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

    7. Applied fma-neg2.6

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

    if -1.0188842112755097e-17 < eps < 2.896438129997832e-56

    1. Initial program 45.5

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

    2. Taylor expanded around 0 29.8

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

    3. Simplified29.8

      \[\leadsto \color{blue}{(\left(x \cdot \varepsilon\right) \cdot \left(\varepsilon + x\right) + \varepsilon)_*}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification14.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1.0188842112755097 \cdot 10^{-17}:\\ \;\;\;\;(\left(\frac{\tan \varepsilon + \tan x}{1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)}\right) \cdot \left(\tan x \cdot \tan \varepsilon + 1\right) + \left(-\tan x\right))_*\\ \mathbf{elif}\;\varepsilon \le 2.896438129997832 \cdot 10^{-56}:\\ \;\;\;\;(\left(x \cdot \varepsilon\right) \cdot \left(x + \varepsilon\right) + \varepsilon)_*\\ \mathbf{else}:\\ \;\;\;\;(\left(\frac{\tan \varepsilon + \tan x}{1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)}\right) \cdot \left(\tan x \cdot \tan \varepsilon + 1\right) + \left(-\tan x\right))_*\\ \end{array}\]

Reproduce

herbie shell --seed 2019112 +o rules:numerics
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"

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

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