Average Error: 31.4 → 0.0
Time: 2.5m
Precision: 64
Internal Precision: 2368
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \cdot \left(x \cdot \frac{9}{40}\right) - (\frac{27}{2800} \cdot \left({x}^{4}\right) + \frac{1}{2})_* \le -0.5111112566485495:\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \mathbf{if}\;x \cdot \left(x \cdot \frac{9}{40}\right) - (\frac{27}{2800} \cdot \left({x}^{4}\right) + \frac{1}{2})_* \le -0.49966204113092333:\\ \;\;\;\;x \cdot \left(x \cdot \frac{9}{40}\right) - (\frac{27}{2800} \cdot \left({x}^{4}\right) + \frac{1}{2})_*\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{x - \sin x}{x - \tan x}\right)}^{3}}\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 3 regimes

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

    1. Initial program 0.0

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

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

    1. Initial program 62.8

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

    2. Taylor expanded around 0 0.0

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

    3. Applied simplify0.0

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

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

    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: 2.5m)Debug logProfile

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