Average Error: 30.7 → 0.0
Time: 24.0s
Precision: 64
Internal Precision: 128
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.5852731179020363:\\ \;\;\;\;\sqrt{\frac{x - \sin x}{x - \frac{\sin x}{\cos x}}} \cdot \sqrt{\frac{x - \sin x}{x - \frac{\sin x}{\cos x}}}\\ \mathbf{elif}\;x \le 0.03140441023808357:\\ \;\;\;\;\frac{9}{40} \cdot {x}^{2} - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \sin x}{x - \frac{\sin x}{\cos x}}\\ \end{array}\]

Error

Bits error versus x

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Derivation

  1. Split input into 3 regimes

  2. if x < -1.5852731179020363

    1. Initial program 0.0

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

    2. Taylor expanded around -inf 0.0

      \[\leadsto \frac{x - \sin x}{\color{blue}{x - \frac{\sin x}{\cos x}}}\]
    3. Using strategy rm

    4. Applied add-sqr-sqrt0.0

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

    if -1.5852731179020363 < x < 0.03140441023808357

    1. Initial program 62.6

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

    2. Taylor expanded around -inf 62.7

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

    3. 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 0.03140441023808357 < x

    1. Initial program 0.0

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

    2. Taylor expanded around -inf 0.0

      \[\leadsto \frac{x - \sin x}{\color{blue}{x - \frac{\sin x}{\cos x}}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.5852731179020363:\\ \;\;\;\;\sqrt{\frac{x - \sin x}{x - \frac{\sin x}{\cos x}}} \cdot \sqrt{\frac{x - \sin x}{x - \frac{\sin x}{\cos x}}}\\ \mathbf{elif}\;x \le 0.03140441023808357:\\ \;\;\;\;\frac{9}{40} \cdot {x}^{2} - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \sin x}{x - \frac{\sin x}{\cos x}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019026 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  (/ (- x (sin x)) (- x (tan x))))