Average Error: 59.9 → 0.3

Time: 47.4s

Precision: 64

Internal Precision: 2368

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

↓

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

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
\[\leadsto \color{blue}{\frac{1}{45} \cdot {x}^{3} + \left(\frac{2}{945} \cdot {x}^{5} + \frac{1}{3} \cdot x\right)}\]
Applied simplify0.3
\[\leadsto \color{blue}{(\left((\left(x \cdot x\right) \cdot \frac{1}{45} + \frac{1}{3})_*\right) \cdot x + \left(\frac{2}{945} \cdot {x}^{5}\right))_*}\]
Using strategy rm
Applied add-sqr-sqrt0.4
\[\leadsto (\color{blue}{\left(\sqrt{(\left(x \cdot x\right) \cdot \frac{1}{45} + \frac{1}{3})_*} \cdot \sqrt{(\left(x \cdot x\right) \cdot \frac{1}{45} + \frac{1}{3})_*}\right)} \cdot x + \left(\frac{2}{945} \cdot {x}^{5}\right))_*\]
Using strategy rm
Applied fma-udef0.4
\[\leadsto \color{blue}{\left(\sqrt{(\left(x \cdot x\right) \cdot \frac{1}{45} + \frac{1}{3})_*} \cdot \sqrt{(\left(x \cdot x\right) \cdot \frac{1}{45} + \frac{1}{3})_*}\right) \cdot x + \frac{2}{945} \cdot {x}^{5}}\]
Applied simplify0.3
\[\leadsto \color{blue}{(\left(x \cdot x\right) \cdot \left(x \cdot \frac{1}{45}\right) + \left(\frac{1}{3} \cdot x\right))_*} + \frac{2}{945} \cdot {x}^{5}\]

Runtime

herbie shell --seed 2018178 +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))))