Average Error: 59.9 → 0.3
Time: 10.3s
Precision: binary64
\[-0.026 < x \land x < 0.026\]
\[\frac{1}{x} - \frac{1}{\tan x}\]
↓
\[x \cdot 0.3333333333333333 + \left(0.022222222222222223 \cdot {x}^{3} + 0.0021164021164021165 \cdot {x}^{5}\right)\]
\frac{1}{x} - \frac{1}{\tan x}
↓
x \cdot 0.3333333333333333 + \left(0.022222222222222223 \cdot {x}^{3} + 0.0021164021164021165 \cdot {x}^{5}\right)
(FPCore (x) :precision binary64 (- (/ 1.0 x) (/ 1.0 (tan x))))
↓
(FPCore (x)
 :precision binary64
 (+
  (* x 0.3333333333333333)
  (+
   (* 0.022222222222222223 (pow x 3.0))
   (* 0.0021164021164021165 (pow x 5.0)))))
double code(double x) {
	return (1.0 / x) - (1.0 / tan(x));
}
↓
double code(double x) {
	return (x * 0.3333333333333333) + ((0.022222222222222223 * pow(x, 3.0)) + (0.0021164021164021165 * pow(x, 5.0)));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original59.9
Target0.1
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;\left|x\right| < 0.026:\\ \;\;\;\;\frac{x}{3} \cdot \left(1 + \frac{x \cdot x}{15}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} - \frac{1}{\tan x}\\ \end{array}\]

Derivation

  1. Initial program 59.9

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

  2. Taylor expanded around 0 0.3

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

  3. Simplified0.3

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

  4. Final simplification0.3

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

Reproduce

herbie shell --seed 2020253 
(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))))