Average Error: 59.9 → 0.3
Time: 35.8s
Precision: 64
Internal Precision: 2368
\[\frac{1}{x} - \frac{1}{\tan x}\]
↓
\[\frac{1}{3} \cdot x + \left(\left(x \cdot x\right) \cdot \left(x \cdot \frac{1}{45}\right) + \frac{2}{945} \cdot {x}^{5}\right)\]

Error

Bits error versus x

Target

Original59.9
Target0.1
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;\left|x\right| \lt 0.026:\\ \;\;\;\;\frac{x}{3} \cdot \left(1 + \frac{x \cdot x}{15}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} - \frac{1}{\tan x}\\ \end{array}\]

Derivation

  1. Initial program 59.9

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

  2. Taylor expanded around 0 0.3

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

  4. Applied add-cube-cbrt1.5

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

  5. Taylor expanded around 0 34.4

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

  6. Applied simplify0.3

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

Runtime

Time bar (total: 35.8s)Debug logProfile

herbie shell --seed '#(1072107073 2127697367 3936270018 2300570620 2134894798 4023771849)' 
(FPCore (x)
  :name "invcot (example 3.9)"
  :pre (and (< -0.026 x) (< x 0.026))

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

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