Average Error: 59.8 → 0.4
Time: 15.8s
Precision: binary64
\[-0.026 < x \land x < 0.026\]
\[\frac{1}{x} - \frac{1}{\tan x} \]
↓
\[0.022222222222222223 \cdot {x}^{3} + x \cdot 0.3333333333333333 \]
\frac{1}{x} - \frac{1}{\tan x}
↓
0.022222222222222223 \cdot {x}^{3} + x \cdot 0.3333333333333333

(FPCore (x) :precision binary64 (- (/ 1.0 x) (/ 1.0 (tan x))))
↓
(FPCore (x)
 :precision binary64
 (+ (* 0.022222222222222223 (pow x 3.0)) (* x 0.3333333333333333)))
double code(double x) {
	return (1.0 / x) - (1.0 / tan(x));
}
↓
double code(double x) {
	return (0.022222222222222223 * pow(x, 3.0)) + (x * 0.3333333333333333);
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original59.8
Target0.1
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;\left|x\right| < 0.026:\\ \;\;\;\;\frac{x}{3} \cdot \left(1 + \frac{x \cdot x}{15}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} - \frac{1}{\tan x}\\ \end{array} \]

Derivation

  1. Initial program 59.8

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

  2. Taylor expanded in x around 0 0.4

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

  3. Final simplification0.4

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

Reproduce

herbie shell --seed 2021275 
(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))))