Average Error: 59.9 → 0.3
Time: 9.7s
Precision: binary64
\[-0.026 < x \land x < 0.026\]
\[\frac{1}{x} - \frac{1}{\tan x}\]
↓
\[\left(x \cdot 0.3333333333333333 + {x}^{3} \cdot 0.022222222222222223\right) + 0.0021164021164021165 \cdot {x}^{5}\]
\frac{1}{x} - \frac{1}{\tan x}
↓
\left(x \cdot 0.3333333333333333 + {x}^{3} \cdot 0.022222222222222223\right) + 0.0021164021164021165 \cdot {x}^{5}
(FPCore (x) :precision binary64 (- (/ 1.0 x) (/ 1.0 (tan x))))
↓
(FPCore (x)
 :precision binary64
 (+
  (+ (* x 0.3333333333333333) (* (pow x 3.0) 0.022222222222222223))
  (* 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) + (pow(x, 3.0) * 0.022222222222222223)) + (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. Using strategy rm

  5. Applied associate-+r+_binary64_13870.3

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

  6. Simplified0.3

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

  7. Final simplification0.3

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

Reproduce

herbie shell --seed 2020277 
(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))))