Average Error: 31.5 → 0.0
Time: 35.3s
Precision: 64
Internal Precision: 128
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.03058386577751066 \lor \neg \left(x \le 0.027383861466899858\right):\\ \;\;\;\;\frac{1}{\frac{{\left(\frac{x}{x - \sin x}\right)}^{3} - {\left(\frac{\tan x}{x - \sin x}\right)}^{3}}{\frac{x}{x - \sin x} \cdot \frac{x}{x - \sin x} + \left(\frac{\tan x}{x - \sin x} \cdot \frac{\tan x}{x - \sin x} + \frac{\tan x}{x - \sin x} \cdot \frac{x}{x - \sin x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{2} \cdot \frac{9}{40} - \left({x}^{4} \cdot \frac{27}{2800} + \frac{1}{2}\right)\\ \end{array}\]

Error

Bits error versus x

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Results

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Derivation

  1. Split input into 2 regimes

  2. if x < -0.03058386577751066 or 0.027383861466899858 < 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. Using strategy rm

    4. Applied *-un-lft-identity0.0

      \[\leadsto \frac{\color{blue}{1 \cdot \left(x - \sin x\right)}}{x - \tan x}\]

    5. Applied associate-/l*0.0

      \[\leadsto \color{blue}{\frac{1}{\frac{x - \tan x}{x - \sin x}}}\]
    6. Using strategy rm

    7. Applied div-sub0.0

      \[\leadsto \frac{1}{\color{blue}{\frac{x}{x - \sin x} - \frac{\tan x}{x - \sin x}}}\]
    8. Using strategy rm

    9. Applied flip3--0.1

      \[\leadsto \frac{1}{\color{blue}{\frac{{\left(\frac{x}{x - \sin x}\right)}^{3} - {\left(\frac{\tan x}{x - \sin x}\right)}^{3}}{\frac{x}{x - \sin x} \cdot \frac{x}{x - \sin x} + \left(\frac{\tan x}{x - \sin x} \cdot \frac{\tan x}{x - \sin x} + \frac{x}{x - \sin x} \cdot \frac{\tan x}{x - \sin x}\right)}}}\]

    if -0.03058386577751066 < x < 0.027383861466899858

    1. Initial program 62.9

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

    2. Initial simplification62.9

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

    3. Taylor expanded around 0 0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.03058386577751066 \lor \neg \left(x \le 0.027383861466899858\right):\\ \;\;\;\;\frac{1}{\frac{{\left(\frac{x}{x - \sin x}\right)}^{3} - {\left(\frac{\tan x}{x - \sin x}\right)}^{3}}{\frac{x}{x - \sin x} \cdot \frac{x}{x - \sin x} + \left(\frac{\tan x}{x - \sin x} \cdot \frac{\tan x}{x - \sin x} + \frac{\tan x}{x - \sin x} \cdot \frac{x}{x - \sin x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{2} \cdot \frac{9}{40} - \left({x}^{4} \cdot \frac{27}{2800} + \frac{1}{2}\right)\\ \end{array}\]

Runtime

Time bar (total: 35.3s)Debug logProfile

herbie shell --seed 2018304 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  (/ (- x (sin x)) (- x (tan x))))