Average Error: 59.8 → 0.5
Time: 9.5s
Precision: 64
\[-0.0259999999999999988 \lt x \land x \lt 0.0259999999999999988\]
\[\frac{1}{x} - \frac{1}{\tan x}\]
↓
\[\left(\log \left(\sqrt{e^{0.0222222222222222231 \cdot {x}^{3}}}\right) + \log \left(\sqrt{e^{0.0222222222222222231 \cdot {x}^{3}}}\right)\right) + \left(\log \left(e^{0.00211640211640211654 \cdot {x}^{5}}\right) + 0.333333333333333315 \cdot x\right)\]
\frac{1}{x} - \frac{1}{\tan x}
↓
\left(\log \left(\sqrt{e^{0.0222222222222222231 \cdot {x}^{3}}}\right) + \log \left(\sqrt{e^{0.0222222222222222231 \cdot {x}^{3}}}\right)\right) + \left(\log \left(e^{0.00211640211640211654 \cdot {x}^{5}}\right) + 0.333333333333333315 \cdot x\right)
double code(double x) {
	return ((1.0 / x) - (1.0 / tan(x)));
}
↓
double code(double x) {
	return ((log(sqrt(exp((0.022222222222222223 * pow(x, 3.0))))) + log(sqrt(exp((0.022222222222222223 * pow(x, 3.0)))))) + (log(exp((0.0021164021164021165 * pow(x, 5.0)))) + (0.3333333333333333 * x)));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original59.8
Target0.1
Herbie0.5
\[\begin{array}{l} \mathbf{if}\;\left|x\right| \lt 0.0259999999999999988:\\ \;\;\;\;\frac{x}{3} \cdot \left(1 + \frac{x \cdot x}{15}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} - \frac{1}{\tan x}\\ \end{array}\]

Derivation

  1. Initial program 59.8

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

  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 add-log-exp0.4

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

  6. Applied add-log-exp0.5

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

  8. Applied add-sqr-sqrt0.5

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

  9. Applied log-prod0.5

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

  10. Final simplification0.5

    \[\leadsto \left(\log \left(\sqrt{e^{0.0222222222222222231 \cdot {x}^{3}}}\right) + \log \left(\sqrt{e^{0.0222222222222222231 \cdot {x}^{3}}}\right)\right) + \left(\log \left(e^{0.00211640211640211654 \cdot {x}^{5}}\right) + 0.333333333333333315 \cdot x\right)\]

Reproduce

herbie shell --seed 2020102 
(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))))