Average Error: 31.4 → 0.0
Time: 3.4m
Precision: 64
Internal Precision: 2368
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{{\left(x \cdot \left(x \cdot \frac{9}{40}\right)\right)}^{3} - {\left((\frac{27}{2800} \cdot \left({x}^{4}\right) + \frac{1}{2})_*\right)}^{3}}{(\left((\frac{27}{2800} \cdot \left({x}^{4}\right) + \frac{1}{2})_*\right) \cdot \left((\left(x \cdot x\right) \cdot \frac{9}{40} + \left((\frac{27}{2800} \cdot \left({x}^{4}\right) + \frac{1}{2})_*\right))_*\right) + \left(\left(x \cdot \left(\frac{9}{40} \cdot x\right)\right) \cdot \left(x \cdot \left(\frac{9}{40} \cdot x\right)\right)\right))_*} \le -0.5111112566485495:\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \mathbf{if}\;\frac{{\left(x \cdot \left(x \cdot \frac{9}{40}\right)\right)}^{3} - {\left((\frac{27}{2800} \cdot \left({x}^{4}\right) + \frac{1}{2})_*\right)}^{3}}{(\left((\frac{27}{2800} \cdot \left({x}^{4}\right) + \frac{1}{2})_*\right) \cdot \left((\left(x \cdot x\right) \cdot \frac{9}{40} + \left((\frac{27}{2800} \cdot \left({x}^{4}\right) + \frac{1}{2})_*\right))_*\right) + \left(\left(x \cdot \left(\frac{9}{40} \cdot x\right)\right) \cdot \left(x \cdot \left(\frac{9}{40} \cdot x\right)\right)\right))_*} \le -0.49966204113092333:\\ \;\;\;\;\frac{{\left(x \cdot \left(x \cdot \frac{9}{40}\right)\right)}^{3} - {\left((\frac{27}{2800} \cdot \left({x}^{4}\right) + \frac{1}{2})_*\right)}^{3}}{(\left((\frac{27}{2800} \cdot \left({x}^{4}\right) + \frac{1}{2})_*\right) \cdot \left((\left(x \cdot x\right) \cdot \frac{9}{40} + \left((\frac{27}{2800} \cdot \left({x}^{4}\right) + \frac{1}{2})_*\right))_*\right) + \left(\left(x \cdot \left(\frac{9}{40} \cdot x\right)\right) \cdot \left(x \cdot \left(\frac{9}{40} \cdot x\right)\right)\right))_*}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{x - \sin x}{x - \tan x}\right)}^{3}}\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 3 regimes

  2. if (/ (- (pow (* x (* x 9/40)) 3) (pow (fma 27/2800 (pow x 4) 1/2) 3)) (fma (fma 27/2800 (pow x 4) 1/2) (fma (* x x) 9/40 (fma 27/2800 (pow x 4) 1/2)) (* (* x (* 9/40 x)) (* x (* 9/40 x))))) < -0.5111112566485495

    1. Initial program 0.1

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

    if -0.5111112566485495 < (/ (- (pow (* x (* x 9/40)) 3) (pow (fma 27/2800 (pow x 4) 1/2) 3)) (fma (fma 27/2800 (pow x 4) 1/2) (fma (* x x) 9/40 (fma 27/2800 (pow x 4) 1/2)) (* (* x (* 9/40 x)) (* x (* 9/40 x))))) < -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

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

    5. Applied flip3--0.0

      \[\leadsto \color{blue}{\frac{{\left(x \cdot \left(x \cdot \frac{9}{40}\right)\right)}^{3} - {\left((\frac{27}{2800} \cdot \left({x}^{4}\right) + \frac{1}{2})_*\right)}^{3}}{\left(x \cdot \left(x \cdot \frac{9}{40}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{9}{40}\right)\right) + \left((\frac{27}{2800} \cdot \left({x}^{4}\right) + \frac{1}{2})_* \cdot (\frac{27}{2800} \cdot \left({x}^{4}\right) + \frac{1}{2})_* + \left(x \cdot \left(x \cdot \frac{9}{40}\right)\right) \cdot (\frac{27}{2800} \cdot \left({x}^{4}\right) + \frac{1}{2})_*\right)}}\]

    6. Applied simplify0.0

      \[\leadsto \frac{{\left(x \cdot \left(x \cdot \frac{9}{40}\right)\right)}^{3} - {\left((\frac{27}{2800} \cdot \left({x}^{4}\right) + \frac{1}{2})_*\right)}^{3}}{\color{blue}{(\left((\frac{27}{2800} \cdot \left({x}^{4}\right) + \frac{1}{2})_*\right) \cdot \left((\left(x \cdot x\right) \cdot \frac{9}{40} + \left((\frac{27}{2800} \cdot \left({x}^{4}\right) + \frac{1}{2})_*\right))_*\right) + \left(\left(x \cdot \left(\frac{9}{40} \cdot x\right)\right) \cdot \left(x \cdot \left(\frac{9}{40} \cdot x\right)\right)\right))_*}}\]

    if -0.49966204113092333 < (/ (- (pow (* x (* x 9/40)) 3) (pow (fma 27/2800 (pow x 4) 1/2) 3)) (fma (fma 27/2800 (pow x 4) 1/2) (fma (* x x) 9/40 (fma 27/2800 (pow x 4) 1/2)) (* (* x (* 9/40 x)) (* x (* 9/40 x)))))

    1. Initial program 0.0

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

    3. Applied add-cbrt-cube47.0

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

    4. Applied add-cbrt-cube46.9

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

    5. Applied cbrt-undiv46.9

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

    6. Applied simplify0.0

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

Runtime

Time bar (total: 3.4m)Debug logProfile

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