Average Error: 31.3 → 0.6
Time: 1.8m
Precision: 64
Internal Precision: 2432
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.7387217572195604 \cdot 10^{-16}:\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \mathbf{if}\;x \le 2500116.352981572:\\ \;\;\;\;\frac{9}{40} \cdot {x}^{2} - \left(\frac{1}{2} + \frac{27}{2800} \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -1.7387217572195604e-16 or 2500116.352981572 < x

    1. Initial program 1.2

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

    if -1.7387217572195604e-16 < x < 2500116.352981572

    1. Initial program 63.1

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

    2. Taylor expanded around 0 0.0

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

Runtime

Time bar (total: 1.8m)Debug logProfile

herbie shell --seed '#(1062803647 245428163 493620569 3595423923 1908391097 2390014376)' 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  (/ (- x (sin x)) (- x (tan x))))