Average Error: 31.0 → 0.0
Time: 1.0m
Precision: 64
Internal Precision: 2368
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.024523646600168456 \lor \neg \left(x \le 0.02608532585401385\right):\\ \;\;\;\;\log_* (1 + (e^{\frac{x - \sin x}{x - \tan x}} - 1)^*)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \frac{9}{40}\right) \cdot x - (\frac{27}{2800} \cdot \left({x}^{4}\right) + \frac{1}{2})_*\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -0.024523646600168456 or 0.02608532585401385 < x

    1. Initial program 0.0

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

    3. Applied log1p-expm1-u0.0

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

    if -0.024523646600168456 < x < 0.02608532585401385

    1. Initial program 62.8

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

    2. Taylor expanded around 0 0.0

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

    3. Applied simplify0.0

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

  4. Applied simplify0.0

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;x \le -0.024523646600168456 \lor \neg \left(x \le 0.02608532585401385\right):\\ \;\;\;\;\log_* (1 + (e^{\frac{x - \sin x}{x - \tan x}} - 1)^*)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \frac{9}{40}\right) \cdot x - (\frac{27}{2800} \cdot \left({x}^{4}\right) + \frac{1}{2})_*\\ \end{array}}\]

Runtime

Time bar (total: 1.0m)Debug logProfile

herbie shell --seed 2018178 +o rules:numerics
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  (/ (- x (sin x)) (- x (tan x))))