Average Error: 31.2 → 0.0
Time: 1.7m
Precision: 64
Internal Precision: 2368
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.02453297975411125 \lor \neg \left(x \le 0.024108214702017676\right):\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left({x}^{2} \cdot \frac{9}{40}\right)}^{3} - {\left(\frac{1}{2} + \frac{27}{2800} \cdot {x}^{4}\right)}^{3}}{\left(\left(\frac{1}{2} + \frac{27}{2800} \cdot {x}^{4}\right) \cdot \left(\frac{1}{2} + \frac{27}{2800} \cdot {x}^{4}\right) + \left({x}^{2} \cdot \frac{9}{40}\right) \cdot \left(\frac{1}{2} + \frac{27}{2800} \cdot {x}^{4}\right)\right) + \left({x}^{2} \cdot \frac{9}{40}\right) \cdot \left({x}^{2} \cdot \frac{9}{40}\right)}\\ \end{array}\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if x < -0.02453297975411125 or 0.024108214702017676 < x

    1. Initial program 0.1

      \[\frac{x - \sin x}{x - \tan x}\]

    if -0.02453297975411125 < x < 0.024108214702017676

    1. Initial program 62.8

      \[\frac{x - \sin x}{x - \tan x}\]

    2. Taylor expanded around 0 0.0

      \[\leadsto \color{blue}{\frac{9}{40} \cdot {x}^{2} - \left(\frac{1}{2} + \frac{27}{2800} \cdot {x}^{4}\right)}\]
    3. Using strategy rm

    4. Applied flip3--0.0

      \[\leadsto \color{blue}{\frac{{\left(\frac{9}{40} \cdot {x}^{2}\right)}^{3} - {\left(\frac{1}{2} + \frac{27}{2800} \cdot {x}^{4}\right)}^{3}}{\left(\frac{9}{40} \cdot {x}^{2}\right) \cdot \left(\frac{9}{40} \cdot {x}^{2}\right) + \left(\left(\frac{1}{2} + \frac{27}{2800} \cdot {x}^{4}\right) \cdot \left(\frac{1}{2} + \frac{27}{2800} \cdot {x}^{4}\right) + \left(\frac{9}{40} \cdot {x}^{2}\right) \cdot \left(\frac{1}{2} + \frac{27}{2800} \cdot {x}^{4}\right)\right)}}\]
  3. Recombined 2 regimes into one program.

  4. Applied simplify0.0

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;x \le -0.02453297975411125 \lor \neg \left(x \le 0.024108214702017676\right):\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left({x}^{2} \cdot \frac{9}{40}\right)}^{3} - {\left(\frac{1}{2} + \frac{27}{2800} \cdot {x}^{4}\right)}^{3}}{\left(\left(\frac{1}{2} + \frac{27}{2800} \cdot {x}^{4}\right) \cdot \left(\frac{1}{2} + \frac{27}{2800} \cdot {x}^{4}\right) + \left({x}^{2} \cdot \frac{9}{40}\right) \cdot \left(\frac{1}{2} + \frac{27}{2800} \cdot {x}^{4}\right)\right) + \left({x}^{2} \cdot \frac{9}{40}\right) \cdot \left({x}^{2} \cdot \frac{9}{40}\right)}\\ \end{array}}\]

Runtime

Time bar (total: 1.7m)Debug logProfile

herbie shell --seed 2018199 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  (/ (- x (sin x)) (- x (tan x))))