Average Error: 37.0 → 15.5
Time: 35.1s
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -6.248907778173893262746741592331412477694 \cdot 10^{-50}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\tan x + \tan \varepsilon}{1 - \frac{\left(\tan x \cdot \sin \varepsilon\right) \cdot \left(\tan x \cdot \sin \varepsilon\right)}{\cos \varepsilon \cdot \cos \varepsilon}}, 1 + \tan x \cdot \tan \varepsilon, -\tan x\right)\\ \mathbf{elif}\;\varepsilon \le 3.496400468760211769169676099038170360789 \cdot 10^{-74}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon \cdot x, x + \varepsilon, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\tan x + \tan \varepsilon}{1 - \frac{\left(\tan x \cdot \sin \varepsilon\right) \cdot \left(\tan x \cdot \sin \varepsilon\right)}{\cos \varepsilon \cdot \cos \varepsilon}}, 1 + \tan x \cdot \tan \varepsilon, -\tan x\right)\\ \end{array}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -6.248907778173893262746741592331412477694 \cdot 10^{-50}:\\
\;\;\;\;\mathsf{fma}\left(\frac{\tan x + \tan \varepsilon}{1 - \frac{\left(\tan x \cdot \sin \varepsilon\right) \cdot \left(\tan x \cdot \sin \varepsilon\right)}{\cos \varepsilon \cdot \cos \varepsilon}}, 1 + \tan x \cdot \tan \varepsilon, -\tan x\right)\\

\mathbf{elif}\;\varepsilon \le 3.496400468760211769169676099038170360789 \cdot 10^{-74}:\\
\;\;\;\;\mathsf{fma}\left(\varepsilon \cdot x, x + \varepsilon, \varepsilon\right)\\

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

\end{array}
double f(double x, double eps) {
        double r12586626 = x;
        double r12586627 = eps;
        double r12586628 = r12586626 + r12586627;
        double r12586629 = tan(r12586628);
        double r12586630 = tan(r12586626);
        double r12586631 = r12586629 - r12586630;
        return r12586631;
}
↓
double f(double x, double eps) {
        double r12586632 = eps;
        double r12586633 = -6.248907778173893e-50;
        bool r12586634 = r12586632 <= r12586633;
        double r12586635 = x;
        double r12586636 = tan(r12586635);
        double r12586637 = tan(r12586632);
        double r12586638 = r12586636 + r12586637;
        double r12586639 = 1.0;
        double r12586640 = sin(r12586632);
        double r12586641 = r12586636 * r12586640;
        double r12586642 = r12586641 * r12586641;
        double r12586643 = cos(r12586632);
        double r12586644 = r12586643 * r12586643;
        double r12586645 = r12586642 / r12586644;
        double r12586646 = r12586639 - r12586645;
        double r12586647 = r12586638 / r12586646;
        double r12586648 = r12586636 * r12586637;
        double r12586649 = r12586639 + r12586648;
        double r12586650 = -r12586636;
        double r12586651 = fma(r12586647, r12586649, r12586650);
        double r12586652 = 3.4964004687602118e-74;
        bool r12586653 = r12586632 <= r12586652;
        double r12586654 = r12586632 * r12586635;
        double r12586655 = r12586635 + r12586632;
        double r12586656 = fma(r12586654, r12586655, r12586632);
        double r12586657 = r12586653 ? r12586656 : r12586651;
        double r12586658 = r12586634 ? r12586651 : r12586657;
        return r12586658;
}

Error

Bits error versus x

Bits error versus eps

Target

Original37.0
Target15.1
Herbie15.5
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Split input into 2 regimes

  2. if eps < -6.248907778173893e-50 or 3.4964004687602118e-74 < eps

    1. Initial program 30.3

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

    3. Applied tan-sum4.9

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

    5. Applied flip--4.9

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

    6. Applied associate-/r/4.9

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

    7. Applied fma-neg4.9

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

    9. Applied tan-quot4.9

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

    10. Applied associate-*r/4.9

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

    11. Applied tan-quot4.9

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

    12. Applied associate-*r/4.9

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

    13. Applied frac-times4.9

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

    if -6.248907778173893e-50 < eps < 3.4964004687602118e-74

    1. Initial program 47.0

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

    2. Taylor expanded around 0 31.4

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

    3. Simplified31.2

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

  4. Final simplification15.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -6.248907778173893262746741592331412477694 \cdot 10^{-50}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\tan x + \tan \varepsilon}{1 - \frac{\left(\tan x \cdot \sin \varepsilon\right) \cdot \left(\tan x \cdot \sin \varepsilon\right)}{\cos \varepsilon \cdot \cos \varepsilon}}, 1 + \tan x \cdot \tan \varepsilon, -\tan x\right)\\ \mathbf{elif}\;\varepsilon \le 3.496400468760211769169676099038170360789 \cdot 10^{-74}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon \cdot x, x + \varepsilon, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\tan x + \tan \varepsilon}{1 - \frac{\left(\tan x \cdot \sin \varepsilon\right) \cdot \left(\tan x \cdot \sin \varepsilon\right)}{\cos \varepsilon \cdot \cos \varepsilon}}, 1 + \tan x \cdot \tan \varepsilon, -\tan x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019173 +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)))