Average Error: 31.5 → 0.1
Time: 1.4m
Precision: 64
Internal Precision: 2368
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.028995460494355585 \lor \neg \left(x \le 0.029569308365725668\right):\\ \;\;\;\;\frac{\sqrt[3]{x - \sin x} \cdot \sqrt[3]{x - \sin x}}{\sqrt[3]{x - \tan x} \cdot \sqrt[3]{x - \tan x}} \cdot \frac{\sqrt[3]{x - \sin x}}{\sqrt[3]{x - \tan x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{9}{40} \cdot {x}^{2} - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)\\ \end{array}\]

Error

Bits error versus x

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Results

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Derivation

  1. Split input into 2 regimes

  2. if x < -0.028995460494355585 or 0.029569308365725668 < x

    1. Initial program 0.0

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

    3. Applied add-cube-cbrt1.4

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

    4. Applied add-cube-cbrt0.1

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

    5. Applied times-frac0.1

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

    if -0.028995460494355585 < x < 0.029569308365725668

    1. Initial program 62.6

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

    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 simplify0.1

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;x \le -0.028995460494355585 \lor \neg \left(x \le 0.029569308365725668\right):\\ \;\;\;\;\frac{\sqrt[3]{x - \sin x} \cdot \sqrt[3]{x - \sin x}}{\sqrt[3]{x - \tan x} \cdot \sqrt[3]{x - \tan x}} \cdot \frac{\sqrt[3]{x - \sin x}}{\sqrt[3]{x - \tan x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{9}{40} \cdot {x}^{2} - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)\\ \end{array}}\]

Runtime

Time bar (total: 1.4m)Debug logProfile

herbie shell --seed 2018206 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  (/ (- x (sin x)) (- x (tan x))))