Average Error: 59.9 → 0.3

Time: 31.9s

Precision: 64

Internal Precision: 2368

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

↓

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

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
\[\frac{1}{x} - \frac{1}{\tan x}\]
Initial simplification59.9
\[\leadsto \frac{1}{x} - \frac{1}{\tan x}\]
Taylor expanded around 0 0.3
\[\leadsto \color{blue}{\frac{1}{3} \cdot x + \left(\frac{1}{45} \cdot {x}^{3} + \frac{2}{945} \cdot {x}^{5}\right)}\]
Simplified0.3
\[\leadsto \color{blue}{(\left((\frac{1}{45} \cdot \left(x \cdot x\right) + \frac{1}{3})_*\right) \cdot x + \left({x}^{5} \cdot \frac{2}{945}\right))_*}\]
Using strategy rm
Applied expm1-log1p-u0.3
\[\leadsto (\color{blue}{\left((e^{\log_* (1 + (\frac{1}{45} \cdot \left(x \cdot x\right) + \frac{1}{3})_*)} - 1)^*\right)} \cdot x + \left({x}^{5} \cdot \frac{2}{945}\right))_*\]
Final simplification0.3
\[\leadsto (\left((e^{\log_* (1 + (\frac{1}{45} \cdot \left(x \cdot x\right) + \frac{1}{3})_*)} - 1)^*\right) \cdot x + \left(\frac{2}{945} \cdot {x}^{5}\right))_*\]

Runtime

	Baseline	Herbie	Oracle	Span	%
Regimes	0.3	0.3	0.2	0.1	0%

herbie shell --seed 2018290 +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))))