invcot (example 3.9)

Average Error: 59.9 → 0.3

Time: 8.9s

Precision: binary64

\[-0.026 < x \land x < 0.026\]

Math

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

↓

\[0.022222222222222223 \cdot {x}^{3} + \left(0.0021164021164021165 \cdot {x}^{5} + x \cdot 0.3333333333333333\right)\]

FPCore

(FPCore (x) :precision binary64 (- (/ 1.0 x) (/ 1.0 (tan x))))

↓

(FPCore (x)
 :precision binary64
 (+
  (* 0.022222222222222223 (pow x 3.0))
  (+ (* 0.0021164021164021165 (pow x 5.0)) (* x 0.3333333333333333))))

C

double code(double x) {
	return ((double) ((1.0 / x) - (1.0 / ((double) tan(x)))));
}

↓

double code(double x) {
	return ((double) (((double) (0.022222222222222223 * ((double) pow(x, 3.0)))) + ((double) (((double) (0.0021164021164021165 * ((double) pow(x, 5.0)))) + ((double) (x * 0.3333333333333333))))));
}

Error

Bits error versus x

Target

Original	59.9
Target	0.1
Herbie	0.3

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

Derivation

1. Initial program 59.9

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

2. Taylor expanded around 0 0.3

   \[\leadsto \color{blue}{0.022222222222222223 \cdot {x}^{3} + \left(0.0021164021164021165 \cdot {x}^{5} + 0.3333333333333333 \cdot x\right)}\]

3. Simplified 0.3

   \[\leadsto \color{blue}{0.022222222222222223 \cdot {x}^{3} + \left(0.0021164021164021165 \cdot {x}^{5} + x \cdot 0.3333333333333333\right)}\]

4. Final simplification 0.3

   \[\leadsto 0.022222222222222223 \cdot {x}^{3} + \left(0.0021164021164021165 \cdot {x}^{5} + x \cdot 0.3333333333333333\right)\]

Reproduce

herbie shell --seed 2020219 
(FPCore (x)
  :name "invcot (example 3.9)"
  :precision binary64
  :pre (and (< -0.026 x) (< x 0.026))

  :herbie-target
  (if (< (fabs x) 0.026) (* (/ x 3.0) (+ 1.0 (/ (* x x) 15.0))) (- (/ 1.0 x) (/ 1.0 (tan x))))

  (- (/ 1.0 x) (/ 1.0 (tan x))))