invcot (example 3.9)

Average Error: 59.9 → 0.4

Time: 10.6s

Precision: binary64

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

Math

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

↓

\[\mathsf{fma}\left(0.3333333333333333, x, 0.022222222222222223 \cdot {x}^{3}\right) \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (fma 0.3333333333333333 x (* 0.022222222222222223 (pow x 3.0))))

C

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

↓

double code(double x) {
	return fma(0.3333333333333333, x, (0.022222222222222223 * pow(x, 3.0)));
}

Error

Bits error versus x

Target

Original	59.9
Target	0.1
Herbie	0.4

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

Derivation

1. Initial program 59.9

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

2. Taylor expanded in x around 0 0.4

   \[\leadsto \color{blue}{0.022222222222222223 \cdot {x}^{3} + 0.3333333333333333 \cdot x} \]

3. Simplified 0.4

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.3333333333333333, x, 0.022222222222222223 \cdot {x}^{3}\right)} \]

4. Final simplification 0.4

   \[\leadsto \mathsf{fma}\left(0.3333333333333333, x, 0.022222222222222223 \cdot {x}^{3}\right) \]

Reproduce

herbie shell --seed 2022067 
(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))))