Average Error: 31.0 → 0.0
Time: 45.6s
Precision: 64
Internal Precision: 2368
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.028805839143768792:\\ \;\;\;\;\log_* (1 + (e^{\frac{x - \sin x}{x - \tan x}} - 1)^*)\\ \mathbf{elif}\;x \le 0.02136162486376371:\\ \;\;\;\;(\left(\frac{9}{40} \cdot x\right) \cdot x + \left((\frac{-27}{2800} \cdot \left({x}^{4}\right) + \frac{-1}{2})_*\right))_*\\ \mathbf{else}:\\ \;\;\;\;\log_* (1 + (e^{\frac{x - \sin x}{x - \tan x}} - 1)^*)\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -0.028805839143768792 or 0.02136162486376371 < x

    1. Initial program 0.0

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

    2. Initial simplification0.0

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

    3. Taylor expanded around inf 0.0

      \[\leadsto \frac{\color{blue}{x - \sin x}}{x - \tan x}\]
    4. Using strategy rm

    5. Applied log1p-expm1-u0.0

      \[\leadsto \color{blue}{\log_* (1 + (e^{\frac{x - \sin x}{x - \tan x}} - 1)^*)}\]

    if -0.028805839143768792 < x < 0.02136162486376371

    1. Initial program 62.8

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

    2. Initial simplification62.8

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

    3. Taylor expanded around 0 0.0

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

    4. Simplified0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.028805839143768792:\\ \;\;\;\;\log_* (1 + (e^{\frac{x - \sin x}{x - \tan x}} - 1)^*)\\ \mathbf{elif}\;x \le 0.02136162486376371:\\ \;\;\;\;(\left(\frac{9}{40} \cdot x\right) \cdot x + \left((\frac{-27}{2800} \cdot \left({x}^{4}\right) + \frac{-1}{2})_*\right))_*\\ \mathbf{else}:\\ \;\;\;\;\log_* (1 + (e^{\frac{x - \sin x}{x - \tan x}} - 1)^*)\\ \end{array}\]

Runtime

Time bar (total: 45.6s)Debug logProfile

herbie shell --seed 2018234 +o rules:numerics
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  (/ (- x (sin x)) (- x (tan x))))