Average Error: 31.3 → 0.2
Time: 47.1s
Precision: 64
Internal Precision: 128
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.027676068794790603:\\ \;\;\;\;\log_* (1 + (e^{\frac{x - \sin x}{x - \tan x}} - 1)^*)\\ \mathbf{elif}\;x \le 2.4173513547348295:\\ \;\;\;\;(\left(x \cdot x\right) \cdot \frac{9}{40} + \left((\frac{-27}{2800} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) + \frac{-1}{2})_*\right))_*\\ \mathbf{else}:\\ \;\;\;\;\log_* (1 + (e^{\left(\left(\frac{\frac{\sin x}{\cos x}}{x} - \frac{\sin x}{x}\right) \cdot \frac{\frac{\sin x}{\cos x}}{x} - \frac{\sin x}{x}\right) + \left(\frac{\frac{\sin x}{\cos x}}{x} + 1\right)} - 1)^*)\\ \end{array}\]

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 log1p-expm1-u0.1

      \[\leadsto \color{blue}{\log_* (1 + (e^{\frac{x - \sin x}{x - \tan x}} - 1)^*)}\]

    if -0.027676068794790603 < x < 2.4173513547348295

    1. Initial program 62.6

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

    2. Taylor expanded around 0 0.1

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

    3. Simplified0.1

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

    if 2.4173513547348295 < x

    1. Initial program 0.0

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

    3. Applied log1p-expm1-u0.0

      \[\leadsto \color{blue}{\log_* (1 + (e^{\frac{x - \sin x}{x - \tan x}} - 1)^*)}\]

    4. Taylor expanded around inf 0.3

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

    5. Simplified0.3

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

  4. Final simplification0.2

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

Reproduce

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