Average Error: 31.8 → 0.1
Time: 28.4s
Precision: 64
Internal precision: 2432
\[\frac{x - \sin x}{x - \tan x}\]
⬇
\[\begin{array}{l} \mathbf{if}\;x \le -6.667433711053936 \cdot 10^{-08}:\\ \;\;\;\;\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}\\ \mathbf{if}\;x \le 1183236732512159.2:\\ \;\;\;\;\frac{9}{40} \cdot {x}^2 - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes.

  2. if x < -6.667433711053936e-08 or 1183236732512159.2 < x

    1. Initial program 0.3

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

    3. Applied div-sub 0.2

      \[\leadsto \color{blue}{\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}}\]

    if -6.667433711053936e-08 < x < 1183236732512159.2

    1. Initial program 63.1

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

    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: 28.4s) Debug log

Please include this information when filing a bug report:

herbie shell --seed '#(3052192724 3812927732 3686175817 630908657 2373248591 511094450)'
(FPCore (x)
  :name "NMSE problem 3.4.5"
  :pre (!= x 0)
  (/ (- x (sin x)) (- x (tan x))))