Average Error: 31.3 → 0.2
Time: 31.9s
Precision: 64
Internal Precision: 128
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.027676068794790603:\\ \;\;\;\;\frac{1}{\frac{x - \tan x}{x - \sin x}}\\ \mathbf{elif}\;x \le 2.4173513547348295:\\ \;\;\;\;\frac{9}{40} \cdot \left(x \cdot x\right) - \left({x}^{4} \cdot \frac{27}{2800} - \frac{-1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{\sin x}{\cos x}}{x} \cdot \frac{\frac{\sin x}{\cos x}}{x} + \frac{\frac{\sin x}{\cos x}}{x}\right) + \left(\left(1 - \frac{\sin x}{x}\right) - \frac{\frac{\sin x}{\cos x}}{x} \cdot \frac{\sin x}{x}\right)\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 3 regimes

  2. if x < -0.027676068794790603

    1. Initial program 0.1

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

    3. Applied *-un-lft-identity0.1

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

    4. Applied associate-/l*0.1

      \[\leadsto \color{blue}{\frac{1}{\frac{x - \tan x}{x - \sin x}}}\]

    if -0.027676068794790603 < x < 2.4173513547348295

    1. Initial program 62.6

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

    3. Applied *-un-lft-identity62.6

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

    4. Applied associate-/l*62.6

      \[\leadsto \color{blue}{\frac{1}{\frac{x - \tan x}{x - \sin x}}}\]

    5. Taylor expanded around 0 0.1

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

    6. Simplified0.1

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

    if 2.4173513547348295 < x

    1. Initial program 0.0

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

    3. Applied *-un-lft-identity0.0

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

    4. Applied associate-/l*0.0

      \[\leadsto \color{blue}{\frac{1}{\frac{x - \tan x}{x - \sin x}}}\]

    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

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.027676068794790603:\\ \;\;\;\;\frac{1}{\frac{x - \tan x}{x - \sin x}}\\ \mathbf{elif}\;x \le 2.4173513547348295:\\ \;\;\;\;\frac{9}{40} \cdot \left(x \cdot x\right) - \left({x}^{4} \cdot \frac{27}{2800} - \frac{-1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{\sin x}{\cos x}}{x} \cdot \frac{\frac{\sin x}{\cos x}}{x} + \frac{\frac{\sin x}{\cos x}}{x}\right) + \left(\left(1 - \frac{\sin x}{x}\right) - \frac{\frac{\sin x}{\cos x}}{x} \cdot \frac{\sin x}{x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019089 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  (/ (- x (sin x)) (- x (tan x))))