Average Error: 31.3 → 0.1
Time: 37.5s
Precision: 64
Internal Precision: 128
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.028029302772051696:\\ \;\;\;\;\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}\\ \mathbf{elif}\;x \le 2.4607679221351315:\\ \;\;\;\;\frac{9}{40} \cdot {x}^{2} - \left(\frac{1}{2} + \frac{27}{2800} \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{\sin x}{\cos x} - \sin x\right) \cdot \frac{\frac{\sin x}{\cos x}}{x \cdot x} - \frac{\sin x}{x}\right) + \left(\frac{\frac{\sin x}{\cos x}}{x} + 1\right)\\ \end{array}\]

Error

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Results

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Derivation

  1. Split input into 3 regimes

  2. if x < -0.028029302772051696

    1. Initial program 0.1

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

    2. Initial simplification0.1

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

    4. Applied div-sub0.1

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

    if -0.028029302772051696 < x < 2.4607679221351315

    1. Initial program 62.6

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

    2. Initial simplification62.6

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

    4. Applied div-sub62.5

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

    5. Taylor expanded around 0 0.1

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

    if 2.4607679221351315 < x

    1. Initial program 0.0

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

    2. Initial simplification0.0

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

    4. Applied div-sub0.0

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

    5. Taylor expanded around -inf 0.3

      \[\leadsto \color{blue}{\left(1 + \left(\frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2} \cdot {x}^{2}} + \frac{\sin x}{\cos x \cdot x}\right)\right) - \left(\frac{\sin x}{x} + \frac{{\left(\sin x\right)}^{2}}{\cos x \cdot {x}^{2}}\right)}\]

    6. Simplified0.3

      \[\leadsto \color{blue}{\left(1 + \frac{\frac{\sin x}{\cos x}}{x}\right) + \left(\frac{\frac{\sin x}{\cos x}}{x \cdot x} \cdot \left(\frac{\sin x}{\cos x} - \sin x\right) - \frac{\sin x}{x}\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.028029302772051696:\\ \;\;\;\;\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}\\ \mathbf{elif}\;x \le 2.4607679221351315:\\ \;\;\;\;\frac{9}{40} \cdot {x}^{2} - \left(\frac{1}{2} + \frac{27}{2800} \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{\sin x}{\cos x} - \sin x\right) \cdot \frac{\frac{\sin x}{\cos x}}{x \cdot x} - \frac{\sin x}{x}\right) + \left(\frac{\frac{\sin x}{\cos x}}{x} + 1\right)\\ \end{array}\]

Runtime

Time bar (total: 37.5s)Debug logProfile

herbie shell --seed 2018346 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  (/ (- x (sin x)) (- x (tan x))))