Average Error: 31.7 → 0.1
Time: 19.7s
Precision: 64
Internal precision: 2432
\[\frac{x - \sin x}{x - \tan x}\]
⬇
\[\begin{array}{l} \mathbf{if}\;x \le -6.005412638913283 \cdot 10^{-09}:\\ \;\;\;\;{\left(\sqrt[3]{\frac{x - \sin x}{x - \tan x}}\right)}^3\\ \mathbf{if}\;x \le 0.22614890659906178:\\ \;\;\;\;\frac{9}{40} \cdot {x}^2 - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{\frac{x - \sin x}{x - \tan x}}\right)}^3\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes.

  2. if x < -6.005412638913283e-09 or 0.22614890659906178 < x

    1. Initial program 0.2

      \[\frac{x - \sin x}{x - \tan x}\]
    2. Using strategy rm

    3. Applied add-cube-cbrt 0.3

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{x - \sin x}{x - \tan x}}\right)}^3}\]

    if -6.005412638913283e-09 < x < 0.22614890659906178

    1. Initial program 63.2

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

    2. Applied taylor 0.0

      \[\leadsto \frac{9}{40} \cdot {x}^2 - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)\]

    3. Taylor expanded around 0 0.0

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

Runtime

Time bar (total: 19.7s) Debug logProfile

Please include this information when filing a bug report:

herbie shell --seed '#(1065003094 2074156664 2352254222 753858891 3745550101 3374585842)'
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  (/ (- x (sin x)) (- x (tan x))))