Average Error: 59.8 → 0.0
Time: 28.5s
Precision: 64
\[-0.026 \lt x \land x \lt 0.026\]
\[\frac{1}{x} - \frac{1}{\tan x}\]
↓
\[{x}^{5} \cdot \frac{2}{945} + \frac{x}{\frac{\frac{\frac{-1}{3} + \frac{1}{45} \cdot \left(x \cdot x\right)}{\frac{1}{3} + \frac{1}{45} \cdot \left(x \cdot x\right)}}{\frac{-1}{3} + \frac{1}{45} \cdot \left(x \cdot x\right)}}\]
\frac{1}{x} - \frac{1}{\tan x}
↓
{x}^{5} \cdot \frac{2}{945} + \frac{x}{\frac{\frac{\frac{-1}{3} + \frac{1}{45} \cdot \left(x \cdot x\right)}{\frac{1}{3} + \frac{1}{45} \cdot \left(x \cdot x\right)}}{\frac{-1}{3} + \frac{1}{45} \cdot \left(x \cdot x\right)}}
double f(double x) {
        double r2755834 = 1.0;
        double r2755835 = x;
        double r2755836 = r2755834 / r2755835;
        double r2755837 = tan(r2755835);
        double r2755838 = r2755834 / r2755837;
        double r2755839 = r2755836 - r2755838;
        return r2755839;
}
↓
double f(double x) {
        double r2755840 = x;
        double r2755841 = 5.0;
        double r2755842 = pow(r2755840, r2755841);
        double r2755843 = 0.0021164021164021165;
        double r2755844 = r2755842 * r2755843;
        double r2755845 = -0.3333333333333333;
        double r2755846 = 0.022222222222222223;
        double r2755847 = r2755840 * r2755840;
        double r2755848 = r2755846 * r2755847;
        double r2755849 = r2755845 + r2755848;
        double r2755850 = 0.3333333333333333;
        double r2755851 = r2755850 + r2755848;
        double r2755852 = r2755849 / r2755851;
        double r2755853 = r2755852 / r2755849;
        double r2755854 = r2755840 / r2755853;
        double r2755855 = r2755844 + r2755854;
        return r2755855;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original59.8
Target0.1
Herbie0.0
\[\begin{array}{l} \mathbf{if}\;\left|x\right| \lt 0.026:\\ \;\;\;\;\frac{x}{3} \cdot \left(1 + \frac{x \cdot x}{15}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} - \frac{1}{\tan x}\\ \end{array}\]

Derivation

  1. Initial program 59.8

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

  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

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

  5. Applied flip-+0.3

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

  6. Applied associate-*r/0.3

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

  8. Applied associate-/l*0.0

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

  9. Simplified0.0

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

  10. Final simplification0.0

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

Reproduce

herbie shell --seed 2019162 
(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))))