Average Error: 31.0 → 0.2
Time: 42.3s
Precision: 64
Internal Precision: 2368
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.028193395491035257:\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \mathbf{elif}\;x \le 2.487108797492067:\\ \;\;\;\;(\left(x \cdot \frac{9}{40}\right) \cdot x + \left((\frac{-27}{2800} \cdot \left({x}^{4}\right) + \frac{-1}{2})_*\right))_*\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(\frac{{\left(\sin x\right)}^{2}}{{x}^{2} \cdot {\left(\cos x\right)}^{2}} + \frac{\sin x}{\cos x \cdot x}\right)\right) - \left(\frac{{\left(\sin x\right)}^{2}}{{x}^{2} \cdot \cos x} + \frac{\sin x}{x}\right)\\ \end{array}\]

Error

Bits error versus x

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 0 0.1

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

    4. Simplified0.1

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

    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 0.0

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

    4. Taylor expanded around -inf 0.5

      \[\leadsto \color{blue}{\left(1 + \left(\frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2} \cdot {x}^{2}} + \frac{\sin x}{\cos x \cdot x}\right)\right) - \left(\frac{\sin x}{x} + \frac{{\left(\sin x\right)}^{2}}{\cos x \cdot {x}^{2}}\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.2

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

Runtime

Time bar (total: 42.3s)Debug logProfile

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