Average Error: 31.0 → 0.2
Time: 37.8s
Precision: 64
Internal Precision: 2368
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.028193395491035257:\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \mathbf{elif}\;x \le 2.487108797492067:\\ \;\;\;\;{x}^{2} \cdot \frac{9}{40} - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{\frac{\sin x}{\cos x}}{x}\right) + \left(\frac{\frac{\sin x}{\cos x}}{x \cdot x} \cdot \left(\frac{\sin x}{\cos x} - \sin x\right) - \frac{\sin x}{x}\right)\\ \end{array}\]

Error

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Results

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Derivation

  1. Split input into 3 regimes

  2. if x < -0.028193395491035257

    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. Taylor expanded around -inf 0.1

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

    if -0.028193395491035257 < x < 2.487108797492067

    1. Initial program 62.4

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

    2. Initial simplification62.4

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

    3. Taylor expanded around -inf 62.4

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

    4. 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 2.487108797492067 < 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. Taylor expanded around -inf 0.5

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

    4. Simplified0.5

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.028193395491035257:\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \mathbf{elif}\;x \le 2.487108797492067:\\ \;\;\;\;{x}^{2} \cdot \frac{9}{40} - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{\frac{\sin x}{\cos x}}{x}\right) + \left(\frac{\frac{\sin x}{\cos x}}{x \cdot x} \cdot \left(\frac{\sin x}{\cos x} - \sin x\right) - \frac{\sin x}{x}\right)\\ \end{array}\]

Runtime

Time bar (total: 37.8s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes31.00.20.030.999.4%
herbie shell --seed 2018296 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  (/ (- x (sin x)) (- x (tan x))))