Average Error: 36.5 → 13.8
Time: 1.3m
Precision: 64
Internal Precision: 2432
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(x \cdot {\varepsilon}^{2} + {\varepsilon}^{3} \cdot {x}^{2}\right) + \varepsilon \le -7.377007838795717 \cdot 10^{-15}:\\ \;\;\;\;\frac{\tan \varepsilon + \tan x}{1 - \log \left(e^{\tan x \cdot \tan \varepsilon}\right)} - \tan x\\ \mathbf{if}\;\left(x \cdot {\varepsilon}^{2} + {\varepsilon}^{3} \cdot {x}^{2}\right) + \varepsilon \le 5.060445514815462 \cdot 10^{-26}:\\ \;\;\;\;\left(x \cdot {\varepsilon}^{2} + {\varepsilon}^{3} \cdot {x}^{2}\right) + \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan \varepsilon + \tan x}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}} \cdot \left(\left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right) + \tan x \cdot \tan \varepsilon\right) + 1\right) - \tan x\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Target

Original36.5
Target14.8
Herbie13.8
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Split input into 3 regimes

  2. if (+ eps (+ (* (pow eps 3) (pow x 2)) (* (pow eps 2) x))) < -7.377007838795717e-15

    1. Initial program 36.1

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

    3. Applied tan-sum10.1

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

    5. Applied add-log-exp10.2

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

    if -7.377007838795717e-15 < (+ eps (+ (* (pow eps 3) (pow x 2)) (* (pow eps 2) x))) < 5.060445514815462e-26

    1. Initial program 38.7

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

    2. Taylor expanded around 0 15.5

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

    if 5.060445514815462e-26 < (+ eps (+ (* (pow eps 3) (pow x 2)) (* (pow eps 2) x)))

    1. Initial program 35.1

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

    3. Applied tan-sum14.1

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

    5. Applied flip3--14.1

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

    6. Applied associate-/r/14.1

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

    7. Applied simplify14.1

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

  4. Applied simplify13.8

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\left(x \cdot {\varepsilon}^{2} + {\varepsilon}^{3} \cdot {x}^{2}\right) + \varepsilon \le -7.377007838795717 \cdot 10^{-15}:\\ \;\;\;\;\frac{\tan \varepsilon + \tan x}{1 - \log \left(e^{\tan x \cdot \tan \varepsilon}\right)} - \tan x\\ \mathbf{if}\;\left(x \cdot {\varepsilon}^{2} + {\varepsilon}^{3} \cdot {x}^{2}\right) + \varepsilon \le 5.060445514815462 \cdot 10^{-26}:\\ \;\;\;\;\left(x \cdot {\varepsilon}^{2} + {\varepsilon}^{3} \cdot {x}^{2}\right) + \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan \varepsilon + \tan x}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}} \cdot \left(\left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right) + \tan x \cdot \tan \varepsilon\right) + 1\right) - \tan x\\ \end{array}}\]

Runtime

Time bar (total: 1.3m)Debug logProfile

herbie shell --seed '#(1070991898 1055468627 4280279443 640792587 928206309 3646738750)' 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"

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

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