Average Error: 59.8 → 0.3
Time: 8.8s
Precision: binary64
\[-0.026 < x \land x < 0.026\]
\[\frac{1}{x} - \frac{1}{\tan x}\]
↓
\[\left(0.0021164021164021165 \cdot {x}^{5} + {x}^{3} \cdot 0.022222222222222223\right) + x \cdot 0.3333333333333333\]
\frac{1}{x} - \frac{1}{\tan x}
↓
\left(0.0021164021164021165 \cdot {x}^{5} + {x}^{3} \cdot 0.022222222222222223\right) + x \cdot 0.3333333333333333
(FPCore (x) :precision binary64 (- (/ 1.0 x) (/ 1.0 (tan x))))
↓
(FPCore (x)
 :precision binary64
 (+
  (+
   (* 0.0021164021164021165 (pow x 5.0))
   (* (pow x 3.0) 0.022222222222222223))
  (* x 0.3333333333333333)))
double code(double x) {
	return (1.0 / x) - (1.0 / tan(x));
}
↓
double code(double x) {
	return ((0.0021164021164021165 * pow(x, 5.0)) + (pow(x, 3.0) * 0.022222222222222223)) + (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
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;\left|x\right| < 0.026:\\ \;\;\;\;\frac{x}{3} \cdot \left(1 + \frac{x \cdot x}{15}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} - \frac{1}{\tan x}\\ \end{array}\]

Derivation

  1. Initial program 59.8

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

  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. Simplified0.3

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

  5. Applied associate-+r+_binary640.3

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

  6. Simplified0.3

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

  7. Final simplification0.3

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

Reproduce

herbie shell --seed 2020224 
(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))))