Average Error: 60.0 → 0.2
Time: 33.5s
Precision: 64
\[-0.026 \lt x \land x \lt 0.026\]
\[\frac{1}{x} - \frac{1}{\tan x}\]
↓
\[{x}^{5} \cdot \frac{2}{945} + \left(\left(\left(\frac{1}{45} \cdot x\right) \cdot x + \frac{1}{3}\right) \cdot \left(\left(\frac{1}{45} \cdot x\right) \cdot x - \frac{1}{3}\right)\right) \cdot \frac{x}{\left(\frac{1}{45} \cdot x\right) \cdot x - \frac{1}{3}}\]
\frac{1}{x} - \frac{1}{\tan x}
↓
{x}^{5} \cdot \frac{2}{945} + \left(\left(\left(\frac{1}{45} \cdot x\right) \cdot x + \frac{1}{3}\right) \cdot \left(\left(\frac{1}{45} \cdot x\right) \cdot x - \frac{1}{3}\right)\right) \cdot \frac{x}{\left(\frac{1}{45} \cdot x\right) \cdot x - \frac{1}{3}}
double f(double x) {
        double r2242151 = 1.0;
        double r2242152 = x;
        double r2242153 = r2242151 / r2242152;
        double r2242154 = tan(r2242152);
        double r2242155 = r2242151 / r2242154;
        double r2242156 = r2242153 - r2242155;
        return r2242156;
}
↓
double f(double x) {
        double r2242157 = x;
        double r2242158 = 5.0;
        double r2242159 = pow(r2242157, r2242158);
        double r2242160 = 0.0021164021164021165;
        double r2242161 = r2242159 * r2242160;
        double r2242162 = 0.022222222222222223;
        double r2242163 = r2242162 * r2242157;
        double r2242164 = r2242163 * r2242157;
        double r2242165 = 0.3333333333333333;
        double r2242166 = r2242164 + r2242165;
        double r2242167 = r2242164 - r2242165;
        double r2242168 = r2242166 * r2242167;
        double r2242169 = r2242157 / r2242167;
        double r2242170 = r2242168 * r2242169;
        double r2242171 = r2242161 + r2242170;
        return r2242171;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original60.0
Target0.1
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;\left|x\right| \lt 0.026:\\ \;\;\;\;\frac{x}{3} \cdot \left(1 + \frac{x \cdot x}{15}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} - \frac{1}{\tan x}\\ \end{array}\]

Derivation

  1. Initial program 60.0

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

  2. Taylor expanded around 0 0.3

    \[\leadsto \color{blue}{\frac{1}{3} \cdot x + \left(\frac{1}{45} \cdot {x}^{3} + \frac{2}{945} \cdot {x}^{5}\right)}\]

  3. Simplified0.3

    \[\leadsto \color{blue}{\frac{2}{945} \cdot {x}^{5} + \left(\left(\frac{1}{45} \cdot x\right) \cdot x + \frac{1}{3}\right) \cdot x}\]
  4. Using strategy rm

  5. Applied flip-+0.3

    \[\leadsto \frac{2}{945} \cdot {x}^{5} + \color{blue}{\frac{\left(\left(\frac{1}{45} \cdot x\right) \cdot x\right) \cdot \left(\left(\frac{1}{45} \cdot x\right) \cdot x\right) - \frac{1}{3} \cdot \frac{1}{3}}{\left(\frac{1}{45} \cdot x\right) \cdot x - \frac{1}{3}}} \cdot x\]

  6. Applied associate-*l/0.3

    \[\leadsto \frac{2}{945} \cdot {x}^{5} + \color{blue}{\frac{\left(\left(\left(\frac{1}{45} \cdot x\right) \cdot x\right) \cdot \left(\left(\frac{1}{45} \cdot x\right) \cdot x\right) - \frac{1}{3} \cdot \frac{1}{3}\right) \cdot x}{\left(\frac{1}{45} \cdot x\right) \cdot x - \frac{1}{3}}}\]
  7. Using strategy rm

  8. Applied *-un-lft-identity0.3

    \[\leadsto \frac{2}{945} \cdot {x}^{5} + \frac{\left(\left(\left(\frac{1}{45} \cdot x\right) \cdot x\right) \cdot \left(\left(\frac{1}{45} \cdot x\right) \cdot x\right) - \frac{1}{3} \cdot \frac{1}{3}\right) \cdot x}{\color{blue}{1 \cdot \left(\left(\frac{1}{45} \cdot x\right) \cdot x - \frac{1}{3}\right)}}\]

  9. Applied times-frac0.2

    \[\leadsto \frac{2}{945} \cdot {x}^{5} + \color{blue}{\frac{\left(\left(\frac{1}{45} \cdot x\right) \cdot x\right) \cdot \left(\left(\frac{1}{45} \cdot x\right) \cdot x\right) - \frac{1}{3} \cdot \frac{1}{3}}{1} \cdot \frac{x}{\left(\frac{1}{45} \cdot x\right) \cdot x - \frac{1}{3}}}\]

  10. Simplified0.2

    \[\leadsto \frac{2}{945} \cdot {x}^{5} + \color{blue}{\left(\left(\left(\frac{1}{45} \cdot x\right) \cdot x + \frac{1}{3}\right) \cdot \left(\left(\frac{1}{45} \cdot x\right) \cdot x - \frac{1}{3}\right)\right)} \cdot \frac{x}{\left(\frac{1}{45} \cdot x\right) \cdot x - \frac{1}{3}}\]

  11. Final simplification0.2

    \[\leadsto {x}^{5} \cdot \frac{2}{945} + \left(\left(\left(\frac{1}{45} \cdot x\right) \cdot x + \frac{1}{3}\right) \cdot \left(\left(\frac{1}{45} \cdot x\right) \cdot x - \frac{1}{3}\right)\right) \cdot \frac{x}{\left(\frac{1}{45} \cdot x\right) \cdot x - \frac{1}{3}}\]

Reproduce

herbie shell --seed 2019134 
(FPCore (x)
  :name "invcot (example 3.9)"
  :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))))