Average Error: 36.9 → 7.3
Time: 1.3m
Precision: 64
Internal precision: 2432
\[\tan \left(x + \varepsilon\right) - \tan x\]
⬇
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -1.2423933280735359 \cdot 10^{-26}:\\ \;\;\;\;\frac{\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \sin x}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}\\ \mathbf{if}\;\varepsilon \le 6.648853381887418 \cdot 10^{-19}:\\ \;\;\;\;{\varepsilon}^{3} \cdot {x}^{2} + \left(\varepsilon + {\varepsilon}^{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \sin x}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Target

Original36.9
Comparison27.9
Herbie7.3
\[ \frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

Derivation

  1. Split input into 2 regimes.

  2. if eps < -1.2423933280735359e-26 or 6.648853381887418e-19 < eps

    1. Initial program 30.3

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

    3. Applied tan-quot 30.2

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

    4. Applied tan-sum 1.5

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

    5. Applied frac-sub 1.6

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

    if -1.2423933280735359e-26 < eps < 6.648853381887418e-19

    1. Initial program 44.8

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

    2. Taylor expanded around 0 14.3

      \[\leadsto \color{blue}{{\varepsilon}^{3} \cdot {x}^{2} + \left(\varepsilon + {\varepsilon}^{2} \cdot x\right)}\]
  3. Recombined 2 regimes into one program.
  4. Removed slow pow expressions

Runtime

Time bar (total: 1.3m) Debug log

Please include this information when filing a bug report:

herbie shell --seed '#(2329929097 3210370195 3111198779 2406363002 3511342718 2136436390)'
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"
  :herbie-expected 28

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

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