Average Error: 59.9 → 0.3

Time: 32.0s

Precision: 64

Internal Precision: 2432

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

↓

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

Error

The X axis uses an exponential scale — Bits error versus `x`

Target

Original	59.9
Target	0.1
Herbie	0.3

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

Derivation

Initial program 59.9
\[\frac{1}{x} - \frac{1}{\tan x}\]
Taylor expanded around 0 0.3
\[\leadsto \color{blue}{\frac{1}{45} \cdot {x}^{3} + \left(\frac{2}{945} \cdot {x}^{5} + \frac{1}{3} \cdot x\right)}\]
Using strategy rm
Applied add-cube-cbrt1.5
\[\leadsto \frac{1}{45} \cdot {x}^{3} + \left(\frac{2}{945} \cdot {x}^{5} + \color{blue}{\left(\sqrt[3]{\frac{1}{3} \cdot x} \cdot \sqrt[3]{\frac{1}{3} \cdot x}\right) \cdot \sqrt[3]{\frac{1}{3} \cdot x}}\right)\]
Taylor expanded around 0 34.6
\[\leadsto \frac{1}{45} \cdot {x}^{3} + \left(\frac{2}{945} \cdot {x}^{5} + \color{blue}{{\left(e^{\frac{1}{3} \cdot \left(\log \frac{1}{3} + \log x\right)}\right)}^{2}} \cdot \sqrt[3]{\frac{1}{3} \cdot x}\right)\]
Applied simplify1.5
\[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{1}{3} \cdot x}\right)}^{3} + \left(\left(x \cdot x\right) \cdot \left(x \cdot \frac{1}{45}\right) + \frac{2}{945} \cdot {x}^{5}\right)}\]
Applied simplify0.3
\[\leadsto \color{blue}{x \cdot \frac{1}{3}} + \left(\left(x \cdot x\right) \cdot \left(x \cdot \frac{1}{45}\right) + \frac{2}{945} \cdot {x}^{5}\right)\]
Removed slow pow expressions.

Runtime

herbie shell --seed '#(1063027428 1192549564 1443466578 604016274 3637110559 1698629644)' 
(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))))