Average Error: 30.8 → 0.4
Time: 39.6s
Precision: 64
Internal Precision: 2368
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -2.503034322186462 \lor \neg \left(x \le 2.4435472739275776\right):\\ \;\;\;\;\left(1 + \frac{\frac{\sin x}{\cos x}}{x}\right) + \left(\frac{\frac{\sin x}{\cos x}}{x \cdot x} \cdot \left(\frac{\sin x}{\cos x} - \sin x\right) - \frac{\sin x}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left({x}^{2} \cdot \frac{9}{40} - {x}^{4} \cdot \frac{27}{2800}\right) - \frac{1}{2}\\ \end{array}\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if x < -2.503034322186462 or 2.4435472739275776 < x

    1. Initial program 0.0

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

    2. 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. Simplified0.5

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

    if -2.503034322186462 < x < 2.4435472739275776

    1. Initial program 62.4

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

    2. Taylor expanded around 0 0.3

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

    4. Applied associate--r+0.2

      \[\leadsto \color{blue}{\left(\frac{9}{40} \cdot {x}^{2} - \frac{27}{2800} \cdot {x}^{4}\right) - \frac{1}{2}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.503034322186462 \lor \neg \left(x \le 2.4435472739275776\right):\\ \;\;\;\;\left(1 + \frac{\frac{\sin x}{\cos x}}{x}\right) + \left(\frac{\frac{\sin x}{\cos x}}{x \cdot x} \cdot \left(\frac{\sin x}{\cos x} - \sin x\right) - \frac{\sin x}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left({x}^{2} \cdot \frac{9}{40} - {x}^{4} \cdot \frac{27}{2800}\right) - \frac{1}{2}\\ \end{array}\]

Runtime

Time bar (total: 39.6s)Debug logProfile

herbie shell --seed 2018254 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  (/ (- x (sin x)) (- x (tan x))))