Average Error: 59.9 → 0.4
Time: 7.6s
Precision: 64
Internal precision: 2432
\[\frac{1}{x} - \frac{1}{\tan x}\]
⬇
\[\frac{1}{45} \cdot {x}^{3} + \left(\log \left(e^{\frac{2}{945} \cdot {x}^{5}}\right) + \frac{1}{3} \cdot x\right)\]

Error

Bits error versus x

Target

Original59.9
Comparison0.1
Herbie0.4
\[ \begin{array}{l} \mathbf{if}\;\left|x\right| \lt 0.026:\\ \;\;\;\;\frac{x}{3} \cdot \left(1 + \frac{{x}^2}{15}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} - \frac{1}{\tan x}\\ \end{array} \]

Derivation

  1. Initial program 59.9

    \[\frac{1}{x} - \frac{1}{\tan x}\]

  2. Applied taylor 0.3

    \[\leadsto \frac{1}{45} \cdot {x}^{3} + \left(\frac{2}{945} \cdot {x}^{5} + \frac{1}{3} \cdot x\right)\]

  3. Taylor expanded around 0 0.3

    \[\leadsto \color{blue}{\frac{1}{45} \cdot {x}^{3} + \left(\frac{2}{945} \cdot {x}^{5} + \frac{1}{3} \cdot x\right)}\]
  4. Using strategy rm

  5. Applied add-log-exp 0.4

    \[\leadsto \frac{1}{45} \cdot {x}^{3} + \left(\color{blue}{\log \left(e^{\frac{2}{945} \cdot {x}^{5}}\right)} + \frac{1}{3} \cdot x\right)\]
  6. Removed slow pow expressions

Runtime

Time bar (total: 7.6s) Debug logProfile

Please include this information when filing a bug report:

herbie shell --seed '#(1066785882 2324371342 4059510649 1466361199 2701357084 1216585281)'
(FPCore (x)
  :name "invcot (example 3.9)"
  :pre (and (< -0.026 x) (< x 0.026))

  :target
  (if (< (fabs x) 0.026) (* (/ x 3) (+ 1 (/ (sqr x) 15))) (- (/ 1 x) (/ 1 (tan x))))

  (- (/ 1 x) (/ 1 (tan x))))