Average Error: 59.9 → 0.3
Time: 1.5m
Precision: 64
Internal Precision: 128
\[\frac{1}{x} - \frac{1}{\tan x}\]
↓
\[(\left({x}^{5}\right) \cdot \frac{2}{945} + \left((\frac{1}{45} \cdot \left(x \cdot x\right) + \frac{1}{3})_* \cdot x\right))_*\]

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

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

  3. Simplified0.3

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

  5. Applied add-sqr-sqrt0.3

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

  6. Taylor expanded around 0 0.3

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

  7. Simplified0.3

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

  8. Final simplification0.3

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

Reproduce

herbie shell --seed 2019088 +o rules:numerics
(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))))