Average Error: 31.3 → 0.3
Time: 1.2m
Precision: 64
Internal Precision: 128
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -2.4836564107365167:\\ \;\;\;\;\left(\left(\frac{\frac{\sin x}{\cos x}}{x} - \frac{\sin x}{x}\right) - \left(-1 + \frac{\sin x}{x} \cdot \frac{\frac{\sin x}{\cos x}}{x}\right)\right) + \frac{\frac{\sin x}{\cos x}}{x} \cdot \frac{\frac{\sin x}{\cos x}}{x}\\ \mathbf{elif}\;x \le 2.4173513547348295:\\ \;\;\;\;\left(\frac{9}{40} \cdot \left(x \cdot x\right) - \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{27}{2800}\right) - \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{\frac{\sin x}{\cos x}}{x} - \frac{\sin x}{x}\right) - \left(-1 + \frac{\sin x}{x} \cdot \frac{\frac{\sin x}{\cos x}}{x}\right)\right) + \frac{\frac{\sin x}{\cos x}}{x} \cdot \frac{\frac{\sin x}{\cos x}}{x}\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -2.4836564107365167 or 2.4173513547348295 < x

    1. Initial program 0.0

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

    2. Taylor expanded around inf 0.4

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

    3. Simplified0.4

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

    if -2.4836564107365167 < x < 2.4173513547348295

    1. Initial program 62.5

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

    2. Taylor expanded around 0 0.2

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

    3. Simplified0.2

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

    5. Applied associate--r+0.2

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.4836564107365167:\\ \;\;\;\;\left(\left(\frac{\frac{\sin x}{\cos x}}{x} - \frac{\sin x}{x}\right) - \left(-1 + \frac{\sin x}{x} \cdot \frac{\frac{\sin x}{\cos x}}{x}\right)\right) + \frac{\frac{\sin x}{\cos x}}{x} \cdot \frac{\frac{\sin x}{\cos x}}{x}\\ \mathbf{elif}\;x \le 2.4173513547348295:\\ \;\;\;\;\left(\frac{9}{40} \cdot \left(x \cdot x\right) - \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{27}{2800}\right) - \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{\frac{\sin x}{\cos x}}{x} - \frac{\sin x}{x}\right) - \left(-1 + \frac{\sin x}{x} \cdot \frac{\frac{\sin x}{\cos x}}{x}\right)\right) + \frac{\frac{\sin x}{\cos x}}{x} \cdot \frac{\frac{\sin x}{\cos x}}{x}\\ \end{array}\]

Reproduce

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