Average Error: 31.4 → 0.1
Time: 55.7s
Precision: 64
Ground Truth: 128
\[\frac{x - \sin x}{x - \tan x}\]
⬇
\[\begin{array}{l} \mathbf{if}\;x \le -8.658073314327725 \cdot 10^{-05}:\\ \;\;\;\;\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}\\ \mathbf{if}\;x \le 0.10844290020720185:\\ \;\;\;\;\frac{9}{40} \cdot {x}^2 - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes.

  2. if x < -8.658073314327725e-05 or 0.10844290020720185 < x

    1. Initial program 0.1

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

    3. Applied div-sub 0.1

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

    if -8.658073314327725e-05 < x < 0.10844290020720185

    1. Initial program 63.1

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

    2. Applied taylor 0.0

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

    3. Taylor expanded around 0 0.0

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

Runtime

Total time: 55.7s Debug log

Please include this information when filing a bug report:

herbie --seed '#(925454495 2186793034 1557910699 2722386145 337930805 3187467667)'
(FPCore (x)
  :name "NMSE problem 3.4.5"
  (/ (- x (sin x)) (- x (tan x))))