Average Error: 37.0 → 7.2
Time: 2.7m
Precision: 64
Internal Precision: 2432
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -4.937900603750566 \cdot 10^{-17}:\\ \;\;\;\;\frac{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} \cdot \frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x \cdot \tan x}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} + \tan x}\\ \mathbf{if}\;\varepsilon \le 2.74084816432034 \cdot 10^{-15}:\\ \;\;\;\;\varepsilon + \left({\varepsilon}^{3} \cdot {x}^{2} + {\varepsilon}^{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \sin x}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Target

Original37.0
Target28.0
Herbie7.2
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Split input into 3 regimes

  2. if eps < -4.937900603750566e-17

    1. Initial program 29.4

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

    3. Applied tan-sum0.9

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

    5. Applied flip--1.0

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

    if -4.937900603750566e-17 < eps < 2.74084816432034e-15

    1. Initial program 45.2

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

    2. Taylor expanded around 0 14.4

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

    if 2.74084816432034e-15 < eps

    1. Initial program 30.2

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

    3. Applied tan-quot30.1

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

    4. Applied tan-sum0.8

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

    5. Applied frac-sub0.8

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

Runtime

Time bar (total: 2.7m)Debug log

herbie shell --seed '#(1567391828 2030694642 2833800258 828025724 3004380912 3532991858)' +o setup:early-exit
(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)))