Average Error: 59.9 → 0.3
Time: 9.3s
Precision: 64
\[-0.0259999999999999988 \lt x \land x \lt 0.0259999999999999988\]
\[\frac{1}{x} - \frac{1}{\tan x}\]
↓
\[0.0222222222222222231 \cdot {x}^{3} + \frac{1}{\left(1.20937263794406686 \cdot 10^{-4} \cdot {x}^{7} - 0.019047619047619049 \cdot {x}^{3}\right) + \frac{3}{x}}\]
\frac{1}{x} - \frac{1}{\tan x}
↓
0.0222222222222222231 \cdot {x}^{3} + \frac{1}{\left(1.20937263794406686 \cdot 10^{-4} \cdot {x}^{7} - 0.019047619047619049 \cdot {x}^{3}\right) + \frac{3}{x}}
double code(double x) {
	return ((1.0 / x) - (1.0 / tan(x)));
}
↓
double code(double x) {
	return ((0.022222222222222223 * pow(x, 3.0)) + (1.0 / (((0.00012093726379440669 * pow(x, 7.0)) - (0.01904761904761905 * pow(x, 3.0))) + (3.0 / x))));
}

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
\[\begin{array}{l} \mathbf{if}\;\left|x\right| \lt 0.0259999999999999988:\\ \;\;\;\;\frac{x}{3} \cdot \left(1 + \frac{x \cdot x}{15}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} - \frac{1}{\tan x}\\ \end{array}\]

Derivation

  1. Initial program 59.9

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

  2. Taylor expanded around 0 0.3

    \[\leadsto \color{blue}{0.0222222222222222231 \cdot {x}^{3} + \left(0.00211640211640211654 \cdot {x}^{5} + 0.333333333333333315 \cdot x\right)}\]
  3. Using strategy rm

  4. Applied flip-+29.5

    \[\leadsto 0.0222222222222222231 \cdot {x}^{3} + \color{blue}{\frac{\left(0.00211640211640211654 \cdot {x}^{5}\right) \cdot \left(0.00211640211640211654 \cdot {x}^{5}\right) - \left(0.333333333333333315 \cdot x\right) \cdot \left(0.333333333333333315 \cdot x\right)}{0.00211640211640211654 \cdot {x}^{5} - 0.333333333333333315 \cdot x}}\]

  5. Simplified29.5

    \[\leadsto 0.0222222222222222231 \cdot {x}^{3} + \frac{\color{blue}{\left(-\left(0.333333333333333315 \cdot x\right) \cdot \left(0.333333333333333315 \cdot x\right)\right) + 0.00211640211640211654 \cdot \left(0.00211640211640211654 \cdot {x}^{10}\right)}}{0.00211640211640211654 \cdot {x}^{5} - 0.333333333333333315 \cdot x}\]
  6. Using strategy rm

  7. Applied clear-num29.6

    \[\leadsto 0.0222222222222222231 \cdot {x}^{3} + \color{blue}{\frac{1}{\frac{0.00211640211640211654 \cdot {x}^{5} - 0.333333333333333315 \cdot x}{\left(-\left(0.333333333333333315 \cdot x\right) \cdot \left(0.333333333333333315 \cdot x\right)\right) + 0.00211640211640211654 \cdot \left(0.00211640211640211654 \cdot {x}^{10}\right)}}}\]

  8. Simplified0.4

    \[\leadsto 0.0222222222222222231 \cdot {x}^{3} + \frac{1}{\color{blue}{\frac{1}{0.333333333333333315 \cdot x + 0.00211640211640211654 \cdot {x}^{5}}}}\]

  9. Taylor expanded around 0 0.4

    \[\leadsto 0.0222222222222222231 \cdot {x}^{3} + \frac{1}{\color{blue}{\left(1.20937263794406686 \cdot 10^{-4} \cdot {x}^{7} + 3 \cdot \frac{1}{x}\right) - 0.019047619047619049 \cdot {x}^{3}}}\]

  10. Simplified0.3

    \[\leadsto 0.0222222222222222231 \cdot {x}^{3} + \frac{1}{\color{blue}{\left(1.20937263794406686 \cdot 10^{-4} \cdot {x}^{7} - 0.019047619047619049 \cdot {x}^{3}\right) + \frac{3}{x}}}\]

  11. Final simplification0.3

    \[\leadsto 0.0222222222222222231 \cdot {x}^{3} + \frac{1}{\left(1.20937263794406686 \cdot 10^{-4} \cdot {x}^{7} - 0.019047619047619049 \cdot {x}^{3}\right) + \frac{3}{x}}\]

Reproduce

herbie shell --seed 2020078 
(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) (+ 1 (/ (* x x) 15))) (- (/ 1 x) (/ 1 (tan x))))

  (- (/ 1 x) (/ 1 (tan x))))