Average Error: 37.0 → 15.4
Time: 7.9s
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -6.4253650413344779 \cdot 10^{-20}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\tan x + \tan \varepsilon}{{1}^{3} - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}}, 1 \cdot 1 + \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right) + 1 \cdot \left(\tan x \cdot \tan \varepsilon\right)\right), -\tan x\right)\\ \mathbf{elif}\;\varepsilon \le 7.60268013552436 \cdot 10^{-69}:\\ \;\;\;\;\mathsf{fma}\left({\varepsilon}^{2}, x, \mathsf{fma}\left(\varepsilon, {x}^{2}, \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\tan x + \tan \varepsilon}{1 \cdot 1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}}, 1 + \tan x \cdot \tan \varepsilon, -\tan x\right)\\ \end{array}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -6.4253650413344779 \cdot 10^{-20}:\\
\;\;\;\;\mathsf{fma}\left(\frac{\tan x + \tan \varepsilon}{{1}^{3} - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}}, 1 \cdot 1 + \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right) + 1 \cdot \left(\tan x \cdot \tan \varepsilon\right)\right), -\tan x\right)\\

\mathbf{elif}\;\varepsilon \le 7.60268013552436 \cdot 10^{-69}:\\
\;\;\;\;\mathsf{fma}\left({\varepsilon}^{2}, x, \mathsf{fma}\left(\varepsilon, {x}^{2}, \varepsilon\right)\right)\\

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

\end{array}
double f(double x, double eps) {
        double r104612 = x;
        double r104613 = eps;
        double r104614 = r104612 + r104613;
        double r104615 = tan(r104614);
        double r104616 = tan(r104612);
        double r104617 = r104615 - r104616;
        return r104617;
}
↓
double f(double x, double eps) {
        double r104618 = eps;
        double r104619 = -6.425365041334478e-20;
        bool r104620 = r104618 <= r104619;
        double r104621 = x;
        double r104622 = tan(r104621);
        double r104623 = tan(r104618);
        double r104624 = r104622 + r104623;
        double r104625 = 1.0;
        double r104626 = 3.0;
        double r104627 = pow(r104625, r104626);
        double r104628 = r104622 * r104623;
        double r104629 = pow(r104628, r104626);
        double r104630 = r104627 - r104629;
        double r104631 = r104624 / r104630;
        double r104632 = r104625 * r104625;
        double r104633 = r104628 * r104628;
        double r104634 = r104625 * r104628;
        double r104635 = r104633 + r104634;
        double r104636 = r104632 + r104635;
        double r104637 = -r104622;
        double r104638 = fma(r104631, r104636, r104637);
        double r104639 = 7.60268013552436e-69;
        bool r104640 = r104618 <= r104639;
        double r104641 = 2.0;
        double r104642 = pow(r104618, r104641);
        double r104643 = pow(r104621, r104641);
        double r104644 = fma(r104618, r104643, r104618);
        double r104645 = fma(r104642, r104621, r104644);
        double r104646 = sin(r104618);
        double r104647 = r104622 * r104646;
        double r104648 = cos(r104618);
        double r104649 = r104647 / r104648;
        double r104650 = r104628 * r104649;
        double r104651 = r104632 - r104650;
        double r104652 = r104624 / r104651;
        double r104653 = r104625 + r104628;
        double r104654 = fma(r104652, r104653, r104637);
        double r104655 = r104640 ? r104645 : r104654;
        double r104656 = r104620 ? r104638 : r104655;
        return r104656;
}

Error

Bits error versus x

Bits error versus eps

Target

Original37.0
Target15.0
Herbie15.4
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Split input into 3 regimes

  2. if eps < -6.425365041334478e-20

    1. Initial program 30.0

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

    3. Applied tan-sum1.3

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

    5. Applied flip3--1.4

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

    6. Applied associate-/r/1.4

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

    7. Applied fma-neg1.3

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

    if -6.425365041334478e-20 < eps < 7.60268013552436e-69

    1. Initial program 46.4

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

    2. Taylor expanded around 0 31.2

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

    3. Simplified31.2

      \[\leadsto \color{blue}{\mathsf{fma}\left({\varepsilon}^{2}, x, \mathsf{fma}\left(\varepsilon, {x}^{2}, \varepsilon\right)\right)}\]

    if 7.60268013552436e-69 < eps

    1. Initial program 29.7

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

    3. Applied tan-sum5.0

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

    5. Applied flip--5.1

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

    6. Applied associate-/r/5.1

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

    7. Applied fma-neg5.1

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

    9. Applied tan-quot5.1

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

    10. Applied associate-*r/5.1

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

  4. Final simplification15.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -6.4253650413344779 \cdot 10^{-20}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\tan x + \tan \varepsilon}{{1}^{3} - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}}, 1 \cdot 1 + \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right) + 1 \cdot \left(\tan x \cdot \tan \varepsilon\right)\right), -\tan x\right)\\ \mathbf{elif}\;\varepsilon \le 7.60268013552436 \cdot 10^{-69}:\\ \;\;\;\;\mathsf{fma}\left({\varepsilon}^{2}, x, \mathsf{fma}\left(\varepsilon, {x}^{2}, \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\tan x + \tan \varepsilon}{1 \cdot 1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}}, 1 + \tan x \cdot \tan \varepsilon, -\tan x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020062 +o rules:numerics
(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)))