Average Error: 31.4 → 0.3
Time: 39.6s
Precision: 64
Internal Precision: 2368
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x - \tan x \le -115.11133042569915:\\ \;\;\;\;\sqrt[3]{\frac{x - \sin x}{x - \tan x} \cdot \left(\frac{x - \sin x}{x - \tan x} \cdot \frac{x - \sin x}{x - \tan x}\right)}\\ \mathbf{elif}\;x - \tan x \le 0.0:\\ \;\;\;\;\frac{9}{40} \cdot {x}^{2} - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{x - \sin x}{x - \tan x} \cdot \left(\frac{1}{\frac{x - \tan x}{x - \sin x}} \cdot \frac{x - \sin x}{x - \tan x}\right)}\\ \end{array}\]

Error

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Derivation

  1. Split input into 3 regimes

  2. if (- x (tan x)) < -115.11133042569915

    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 add-cbrt-cube0.0

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

    if -115.11133042569915 < (- x (tan x)) < 0.0

    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 0.2

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

    if 0.0 < (- x (tan x))

    1. Initial program 0.5

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

    2. Initial simplification0.5

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

    4. Applied add-cbrt-cube0.5

      \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{x - \sin x}{x - \tan x} \cdot \frac{x - \sin x}{x - \tan x}\right) \cdot \frac{x - \sin x}{x - \tan x}}}\]
    5. Using strategy rm

    6. Applied clear-num0.6

      \[\leadsto \sqrt[3]{\left(\frac{x - \sin x}{x - \tan x} \cdot \color{blue}{\frac{1}{\frac{x - \tan x}{x - \sin x}}}\right) \cdot \frac{x - \sin x}{x - \tan x}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x - \tan x \le -115.11133042569915:\\ \;\;\;\;\sqrt[3]{\frac{x - \sin x}{x - \tan x} \cdot \left(\frac{x - \sin x}{x - \tan x} \cdot \frac{x - \sin x}{x - \tan x}\right)}\\ \mathbf{elif}\;x - \tan x \le 0.0:\\ \;\;\;\;\frac{9}{40} \cdot {x}^{2} - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{x - \sin x}{x - \tan x} \cdot \left(\frac{1}{\frac{x - \tan x}{x - \sin x}} \cdot \frac{x - \sin x}{x - \tan x}\right)}\\ \end{array}\]

Runtime

Time bar (total: 39.6s)Debug logProfile

herbie shell --seed 2018274 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  (/ (- x (sin x)) (- x (tan x))))