Average Error: 31.3 → 0.0
Time: 58.0s
Precision: 64
Internal Precision: 2368
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.025833890558333857 \lor \neg \left(x \le 0.030216794869512068\right):\\ \;\;\;\;\sqrt[3]{\frac{x - \sin x}{x - \tan x}} \cdot \left(\sqrt[3]{\frac{x - \sin x}{x - \tan x}} \cdot \sqrt[3]{\frac{x - \sin x}{x - \tan x}}\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{2} \cdot \frac{9}{40} - \left({x}^{4} \cdot \frac{27}{2800} + \frac{1}{2}\right)\\ \end{array}\]

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if x < -0.025833890558333857 or 0.030216794869512068 < x

    1. Initial program 0.0

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

    3. Applied add-cube-cbrt0.1

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

    if -0.025833890558333857 < x < 0.030216794869512068

    1. Initial program 62.8

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

    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. Applied simplify0.0

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;x \le -0.025833890558333857 \lor \neg \left(x \le 0.030216794869512068\right):\\ \;\;\;\;\sqrt[3]{\frac{x - \sin x}{x - \tan x}} \cdot \left(\sqrt[3]{\frac{x - \sin x}{x - \tan x}} \cdot \sqrt[3]{\frac{x - \sin x}{x - \tan x}}\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{2} \cdot \frac{9}{40} - \left({x}^{4} \cdot \frac{27}{2800} + \frac{1}{2}\right)\\ \end{array}}\]

Runtime

Time bar (total: 58.0s)Debug logProfile

herbie shell --seed 2018201 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  (/ (- x (sin x)) (- x (tan x))))