Average Error: 31.4 → 0.1
Time: 40.6s
Precision: 64
Ground Truth: 128
\[\frac{x - \sin x}{x - \tan x}\]
⬇
\[\begin{array}{l} \mathbf{if}\;x \le -4.436569855739804 \cdot 10^{-07}:\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \mathbf{if}\;x \le 1.9330446047513736:\\ \;\;\;\;\frac{9}{40} \cdot {x}^2 - \left(\frac{1}{2} + \frac{27}{2800} \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes.

  2. if x < -4.436569855739804e-07 or 1.9330446047513736 < x

    1. Initial program 0.2

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

    if -4.436569855739804e-07 < x < 1.9330446047513736

    1. Initial program 63.0

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

    2. Applied taylor 0.0

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

    3. 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. Recombined 2 regimes into one program.
  4. Removed slow pow expressions

Runtime

Total time: 40.6s Debug log

Please report a bug with the following info:

herbie --seed '#(2450790218 3769101461 3677668716 1966659313 2958760900 2975588775)'
(FPCore (x)
  :name "NMSE problem 3.4.5"
  (/ (- x (sin x)) (- x (tan x))))