Average Error: 36.9 → 14.5
Time: 2.0m
Precision: 64
Internal Precision: 2432
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -6.679506785351335 \cdot 10^{-12}:\\ \;\;\;\;\frac{\frac{{\left(\tan x\right)}^{3} + {\left(\tan \varepsilon\right)}^{3}}{\left(\tan \varepsilon - \tan x\right) \cdot \tan \varepsilon + \tan x \cdot \tan x}}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{if}\;\varepsilon \le 1.5366040845210357 \cdot 10^{-61}:\\ \;\;\;\;\varepsilon + \left({\varepsilon}^{3} \cdot {x}^{2} + {\varepsilon}^{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\tan x + \tan \varepsilon\right) \cdot \cos x\right) \cdot \left(\left(\tan x + \tan \varepsilon\right) \cdot \cos x\right) - \left(\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \sin x\right) \cdot \left(\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \sin x\right)}{\left(\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x\right) \cdot \left(\left(\tan x + \tan \varepsilon\right) \cdot \cos x + \left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \sin x\right)}\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Target

Original36.9
Target14.9
Herbie14.5
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Split input into 3 regimes

  2. if eps < -6.679506785351335e-12

    1. Initial program 29.0

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

    3. Applied tan-sum0.6

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

    5. Applied flip3-+0.8

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

    6. Applied simplify0.8

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

    if -6.679506785351335e-12 < eps < 1.5366040845210357e-61

    1. Initial program 45.8

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

    2. Taylor expanded around 0 28.8

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

    if 1.5366040845210357e-61 < eps

    1. Initial program 30.6

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

    3. Applied tan-quot30.4

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

    4. Applied tan-sum5.0

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

    5. Applied frac-sub5.0

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

    7. Applied flip--5.1

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

    8. Applied associate-/l/5.1

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

Runtime

Time bar (total: 2.0m)Debug logProfile

herbie shell --seed '#(1063027428 1192549564 1443466578 604016274 3637110559 1698629644)' 
(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)))