Average Error: 36.7 → 9.3
Time: 33.7s
Precision: 64
Internal precision: 2432
\[\tan \left(x + \varepsilon\right) - \tan x\]
⬇
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -4.1660066365355184 \cdot 10^{-24}:\\ \;\;\;\;\frac{\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \sin x}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}\\ \mathbf{if}\;\varepsilon \le 2.6866613583039565 \cdot 10^{-28}:\\ \;\;\;\;\left({x}^2 \cdot {\varepsilon}^3 + {\varepsilon}^{4} \cdot {x}^3\right) + \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}\right)}^2 - {\left(\tan x\right)}^2}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} + \tan x}\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Target

Original36.7
Comparison27.3
Herbie9.3
\[ \frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

Derivation

  1. Split input into 3 regimes.

  2. if eps < -4.1660066365355184e-24

    1. Initial program 30.0

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

    3. Applied tan-quot 29.8

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

    4. Applied tan-sum 1.9

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

    5. Applied frac-sub 1.9

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

    if -4.1660066365355184e-24 < eps < 2.6866613583039565e-28

    1. Initial program 45.5

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

    2. Applied taylor 18.6

      \[\leadsto \varepsilon + \left({\varepsilon}^{4} \cdot {x}^{3} + {\varepsilon}^{3} \cdot {x}^2\right)\]

    3. Taylor expanded around 0 18.6

      \[\leadsto \color{blue}{\varepsilon + \left({\varepsilon}^{4} \cdot {x}^{3} + {\varepsilon}^{3} \cdot {x}^2\right)}\]

    4. Applied simplify 18.6

      \[\leadsto \color{blue}{\left({x}^2 \cdot {\varepsilon}^3 + {\varepsilon}^{4} \cdot {x}^3\right) + \varepsilon}\]

    if 2.6866613583039565e-28 < eps

    1. Initial program 29.5

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

    3. Applied tan-sum 1.8

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

    5. Applied flip-- 1.9

      \[\leadsto \color{blue}{\frac{{\left(\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}\right)}^2 - {\left(\tan x\right)}^2}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} + \tan x}}\]
  3. Recombined 3 regimes into one program.
  4. Removed slow pow expressions

Runtime

Time bar (total: 33.7s) Debug logProfile

Please include this information when filing a bug report:

herbie shell --seed '#(1064875752 1442698706 3150723005 1316518582 2592983078 3835530843)'
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"
  :herbie-expected 28

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

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