Average Error: 59.9 → 0.0
Time: 29.9s
Precision: 64
\[-0.026 \lt x \land x \lt 0.026\]
\[\frac{1}{x} - \frac{1}{\tan x}\]
↓
\[{x}^{5} \cdot \frac{2}{945} + \frac{x}{\frac{\frac{1}{9} + \left(\left(\frac{1}{45} \cdot x\right) \cdot x - \frac{1}{3}\right) \cdot \left(\left(\frac{1}{45} \cdot x\right) \cdot x\right)}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{91125} \cdot \left(x \cdot x\right)\right)\right) + \frac{1}{27}}}\]
\frac{1}{x} - \frac{1}{\tan x}
↓
{x}^{5} \cdot \frac{2}{945} + \frac{x}{\frac{\frac{1}{9} + \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\right)}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{91125} \cdot \left(x \cdot x\right)\right)\right) + \frac{1}{27}}}
double f(double x) {
        double r2375619 = 1.0;
        double r2375620 = x;
        double r2375621 = r2375619 / r2375620;
        double r2375622 = tan(r2375620);
        double r2375623 = r2375619 / r2375622;
        double r2375624 = r2375621 - r2375623;
        return r2375624;
}
↓
double f(double x) {
        double r2375625 = x;
        double r2375626 = 5.0;
        double r2375627 = pow(r2375625, r2375626);
        double r2375628 = 0.0021164021164021165;
        double r2375629 = r2375627 * r2375628;
        double r2375630 = 0.1111111111111111;
        double r2375631 = 0.022222222222222223;
        double r2375632 = r2375631 * r2375625;
        double r2375633 = r2375632 * r2375625;
        double r2375634 = 0.3333333333333333;
        double r2375635 = r2375633 - r2375634;
        double r2375636 = r2375635 * r2375633;
        double r2375637 = r2375630 + r2375636;
        double r2375638 = r2375625 * r2375625;
        double r2375639 = 1.0973936899862826e-05;
        double r2375640 = r2375639 * r2375638;
        double r2375641 = r2375638 * r2375640;
        double r2375642 = r2375638 * r2375641;
        double r2375643 = 0.037037037037037035;
        double r2375644 = r2375642 + r2375643;
        double r2375645 = r2375637 / r2375644;
        double r2375646 = r2375625 / r2375645;
        double r2375647 = r2375629 + r2375646;
        return r2375647;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original59.9
Target0.1
Herbie0.0
\[\begin{array}{l} \mathbf{if}\;\left|x\right| \lt 0.026:\\ \;\;\;\;\frac{x}{3} \cdot \left(1 + \frac{x \cdot x}{15}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} - \frac{1}{\tan x}\\ \end{array}\]

Derivation

  1. Initial program 59.9

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

  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

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

  5. Applied flip3-+1.2

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

  6. Applied associate-*l/1.1

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

  7. Simplified0.3

    \[\leadsto \frac{2}{945} \cdot {x}^{5} + \frac{\color{blue}{x \cdot \left(\left(\frac{1}{91125} \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) + \frac{1}{27}\right)}}{\frac{1}{3} \cdot \frac{1}{3} + \left(\left(\left(x \cdot x\right) \cdot \frac{1}{45}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{45}\right) - \frac{1}{3} \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{45}\right)\right)}\]
  8. Using strategy rm

  9. Applied associate-/l*0.0

    \[\leadsto \frac{2}{945} \cdot {x}^{5} + \color{blue}{\frac{x}{\frac{\frac{1}{3} \cdot \frac{1}{3} + \left(\left(\left(x \cdot x\right) \cdot \frac{1}{45}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{45}\right) - \frac{1}{3} \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{45}\right)\right)}{\left(\frac{1}{91125} \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) + \frac{1}{27}}}}\]

  10. Simplified0.0

    \[\leadsto \frac{2}{945} \cdot {x}^{5} + \frac{x}{\color{blue}{\frac{\left(\left(\frac{1}{45} \cdot x\right) \cdot x\right) \cdot \left(\left(\frac{1}{45} \cdot x\right) \cdot x - \frac{1}{3}\right) + \frac{1}{9}}{\frac{1}{27} + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{91125}\right)\right)}}}\]

  11. Final simplification0.0

    \[\leadsto {x}^{5} \cdot \frac{2}{945} + \frac{x}{\frac{\frac{1}{9} + \left(\left(\frac{1}{45} \cdot x\right) \cdot x - \frac{1}{3}\right) \cdot \left(\left(\frac{1}{45} \cdot x\right) \cdot x\right)}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{91125} \cdot \left(x \cdot x\right)\right)\right) + \frac{1}{27}}}\]

Reproduce

herbie shell --seed 2019130 
(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))))