Average Error: 60.0 → 0.0
Time: 23.9s
Precision: 64
\[-0.026 \lt x \land x \lt 0.026\]
\[\frac{1}{x} - \frac{1}{\tan x}\]
↓
\[{x}^{5} \cdot \frac{2}{945} + \frac{x}{\frac{\frac{1}{9} + \left(x \cdot \left(\frac{1}{45} \cdot x\right) + \frac{-1}{3}\right) \cdot \left(x \cdot \left(\frac{1}{45} \cdot x\right)\right)}{\frac{1}{91125} \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) + \frac{1}{27}}}\]
\frac{1}{x} - \frac{1}{\tan x}
↓
{x}^{5} \cdot \frac{2}{945} + \frac{x}{\frac{\frac{1}{9} + \left(x \cdot \left(\frac{1}{45} \cdot x\right) + \frac{-1}{3}\right) \cdot \left(x \cdot \left(\frac{1}{45} \cdot x\right)\right)}{\frac{1}{91125} \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) + \frac{1}{27}}}
double f(double x) {
        double r4191694 = 1.0;
        double r4191695 = x;
        double r4191696 = r4191694 / r4191695;
        double r4191697 = tan(r4191695);
        double r4191698 = r4191694 / r4191697;
        double r4191699 = r4191696 - r4191698;
        return r4191699;
}
↓
double f(double x) {
        double r4191700 = x;
        double r4191701 = 5.0;
        double r4191702 = pow(r4191700, r4191701);
        double r4191703 = 0.0021164021164021165;
        double r4191704 = r4191702 * r4191703;
        double r4191705 = 0.1111111111111111;
        double r4191706 = 0.022222222222222223;
        double r4191707 = r4191706 * r4191700;
        double r4191708 = r4191700 * r4191707;
        double r4191709 = -0.3333333333333333;
        double r4191710 = r4191708 + r4191709;
        double r4191711 = r4191710 * r4191708;
        double r4191712 = r4191705 + r4191711;
        double r4191713 = 1.0973936899862826e-05;
        double r4191714 = r4191700 * r4191700;
        double r4191715 = r4191700 * r4191714;
        double r4191716 = r4191715 * r4191715;
        double r4191717 = r4191713 * r4191716;
        double r4191718 = 0.037037037037037035;
        double r4191719 = r4191717 + r4191718;
        double r4191720 = r4191712 / r4191719;
        double r4191721 = r4191700 / r4191720;
        double r4191722 = r4191704 + r4191721;
        return r4191722;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 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

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

  5. Applied flip3-+1.2

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

  6. Applied associate-*r/1.1

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

  7. Simplified0.3

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

  9. Applied associate-/l*0.0

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

  10. Simplified0.0

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

  11. Final simplification0.0

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

Reproduce

herbie shell --seed 2019164 
(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))))