Average Error: 31.4 → 0.0
Time: 1.4m
Precision: 64
Internal Precision: 2368
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{9}{40} \cdot {x}^{2} - \left(\frac{1}{2} + \frac{27}{2800} \cdot {x}^{4}\right) \le -0.5111112566485495:\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \mathbf{if}\;\frac{9}{40} \cdot {x}^{2} - \left(\frac{1}{2} + \frac{27}{2800} \cdot {x}^{4}\right) \le -0.49966204113092333:\\ \;\;\;\;\frac{9}{40} \cdot {x}^{2} - \left(\frac{1}{2} + \frac{27}{2800} \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{x - \sin x}{x - \tan x}\right)}^{3}}\\ \end{array}\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if (- (* 9/40 (pow x 2)) (+ 1/2 (* 27/2800 (pow x 4)))) < -0.5111112566485495

    1. Initial program 0.0

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

    if -0.5111112566485495 < (- (* 9/40 (pow x 2)) (+ 1/2 (* 27/2800 (pow x 4)))) < -0.49966204113092333

    1. Initial program 62.8

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

    2. Taylor expanded around 0 0.0

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

    if -0.49966204113092333 < (- (* 9/40 (pow x 2)) (+ 1/2 (* 27/2800 (pow x 4))))

    1. Initial program 0.1

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

    3. Applied add-cbrt-cube61.1

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

    4. Applied add-cbrt-cube61.1

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

    5. Applied cbrt-undiv61.1

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

    6. Applied simplify0.1

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

Runtime

Time bar (total: 1.4m)Debug logProfile

herbie shell --seed 2018207 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  (/ (- x (sin x)) (- x (tan x))))