Average Error: 30.5 → 0.1
Time: 41.3s
Precision: 64
Internal precision: 2432
\[\frac{x - \sin x}{x - \tan x}\]
⬇
\[\begin{array}{l} \mathbf{if}\;x \le -3.859884253300168 \cdot 10^{-07}:\\ \;\;\;\;\log \left(e^{\frac{x - \sin x}{x - \tan x}}\right)\\ \mathbf{if}\;x \le 0.1441617602023804:\\ \;\;\;\;\frac{9}{40} \cdot {x}^{2} - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\frac{x - \sin x}{x - \tan x}}\right)\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes.

  2. if x < -3.859884253300168e-07 or 0.1441617602023804 < x

    1. Initial program 0.3

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

    3. Applied add-log-exp 0.3

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

    if -3.859884253300168e-07 < x < 0.1441617602023804

    1. Initial program 63.1

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

    2. Taylor expanded around 0 0.0

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

Runtime

Time bar (total: 41.3s) Debug log

Please include this information when filing a bug report:

herbie shell --seed '#(2329929097 3210370195 3111198779 2406363002 3511342718 2136436390)'
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  (/ (- x (sin x)) (- x (tan x))))