Average Error: 31.2 → 0.0
Time: 4.4m
Precision: 64
Internal Precision: 2368
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.03977511220299969 \lor \neg \left(x \le 0.03999707453083916\right):\\ \;\;\;\;\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(\frac{351}{22400} \cdot \frac{{x}^{4}}{e^{\frac{1}{2}}} + \left(x \cdot x\right) \cdot \frac{\frac{9}{40}}{e^{\frac{1}{2}}}\right) + e^{-\frac{1}{2}}\right)\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -0.03977511220299969 or 0.03999707453083916 < x

    1. Initial program 0.0

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

    3. Applied div-sub0.0

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

    if -0.03977511220299969 < x < 0.03999707453083916

    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. Using strategy rm

    4. Applied add-log-exp0.0

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

    5. Applied add-log-exp0.0

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

    6. Applied diff-log0.0

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

    7. Taylor expanded around 0 0.0

      \[\leadsto \log \color{blue}{\left(\frac{351}{22400} \cdot \frac{{x}^{4}}{e^{\frac{1}{2}}} + \left(\frac{9}{40} \cdot \frac{{x}^{2}}{e^{\frac{1}{2}}} + \frac{1}{e^{\frac{1}{2}}}\right)\right)}\]

    8. Applied simplify0.0

      \[\leadsto \color{blue}{\log \left(\left(\left(x \cdot x\right) \cdot \frac{\frac{9}{40}}{e^{\frac{1}{2}}} + \frac{351}{22400} \cdot \frac{{x}^{4}}{e^{\frac{1}{2}}}\right) + e^{-\frac{1}{2}}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Applied simplify0.0

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;x \le -0.03977511220299969 \lor \neg \left(x \le 0.03999707453083916\right):\\ \;\;\;\;\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(\frac{351}{22400} \cdot \frac{{x}^{4}}{e^{\frac{1}{2}}} + \left(x \cdot x\right) \cdot \frac{\frac{9}{40}}{e^{\frac{1}{2}}}\right) + e^{-\frac{1}{2}}\right)\\ \end{array}}\]

Runtime

Time bar (total: 4.4m)Debug logProfile

herbie shell --seed '#(1071373924 2949776965 1885069702 3247780810 90874544 2263903749)' 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  (/ (- x (sin x)) (- x (tan x))))