Average Error: 60.0 → 0.4

Time: 29.3s

Precision: 64

Internal Precision: 2368

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

↓

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

Error

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

Target

Original	60.0
Target	0.1
Herbie	0.4

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

Derivation

Initial program 60.0
\[\frac{1}{x} - \frac{1}{\tan x}\]
Initial simplification60.0
\[\leadsto \frac{1}{x} - \frac{1}{\tan x}\]
Taylor expanded around 0 0.3
\[\leadsto \color{blue}{\frac{1}{3} \cdot x + \left(\frac{1}{45} \cdot {x}^{3} + \frac{2}{945} \cdot {x}^{5}\right)}\]
Simplified0.4
\[\leadsto \color{blue}{(\left((\frac{1}{45} \cdot \left(x \cdot x\right) + \frac{1}{3})_*\right) \cdot x + \left({x}^{5} \cdot \frac{2}{945}\right))_*}\]
Using strategy rm
Applied expm1-log1p-u0.4
\[\leadsto (\color{blue}{\left((e^{\log_* (1 + (\frac{1}{45} \cdot \left(x \cdot x\right) + \frac{1}{3})_*)} - 1)^*\right)} \cdot x + \left({x}^{5} \cdot \frac{2}{945}\right))_*\]
Using strategy rm
Applied expm1-log1p-u0.4
\[\leadsto (\left((e^{\color{blue}{(e^{\log_* (1 + \log_* (1 + (\frac{1}{45} \cdot \left(x \cdot x\right) + \frac{1}{3})_*))} - 1)^*}} - 1)^*\right) \cdot x + \left({x}^{5} \cdot \frac{2}{945}\right))_*\]
Final simplification0.4
\[\leadsto (\left((e^{(e^{\log_* (1 + \log_* (1 + (\frac{1}{45} \cdot \left(x \cdot x\right) + \frac{1}{3})_*))} - 1)^*} - 1)^*\right) \cdot x + \left({x}^{5} \cdot \frac{2}{945}\right))_*\]

Runtime

herbie shell --seed 2018254 +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))))