Average Error: 59.9 → 0.0
Time: 37.9s
Precision: 64
\[-0.026 \lt x \land x \lt 0.026\]
\[\frac{1}{x} - \frac{1}{\tan x}\]
↓
\[{x}^{5} \cdot \frac{2}{945} + \frac{x}{\frac{\left(\left(x \cdot x\right) \cdot \frac{1}{45}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{45}\right) + \left(\frac{1}{9} - \left(x \cdot x\right) \cdot \frac{1}{135}\right)}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{1}{91125} \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) + \frac{1}{27}}}\]
\frac{1}{x} - \frac{1}{\tan x}
↓
{x}^{5} \cdot \frac{2}{945} + \frac{x}{\frac{\left(\left(x \cdot x\right) \cdot \frac{1}{45}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{45}\right) + \left(\frac{1}{9} - \left(x \cdot x\right) \cdot \frac{1}{135}\right)}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{1}{91125} \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) + \frac{1}{27}}}
double f(double x) {
        double r2297054 = 1.0;
        double r2297055 = x;
        double r2297056 = r2297054 / r2297055;
        double r2297057 = tan(r2297055);
        double r2297058 = r2297054 / r2297057;
        double r2297059 = r2297056 - r2297058;
        return r2297059;
}
↓
double f(double x) {
        double r2297060 = x;
        double r2297061 = 5.0;
        double r2297062 = pow(r2297060, r2297061);
        double r2297063 = 0.0021164021164021165;
        double r2297064 = r2297062 * r2297063;
        double r2297065 = r2297060 * r2297060;
        double r2297066 = 0.022222222222222223;
        double r2297067 = r2297065 * r2297066;
        double r2297068 = r2297067 * r2297067;
        double r2297069 = 0.1111111111111111;
        double r2297070 = 0.007407407407407408;
        double r2297071 = r2297065 * r2297070;
        double r2297072 = r2297069 - r2297071;
        double r2297073 = r2297068 + r2297072;
        double r2297074 = r2297060 * r2297065;
        double r2297075 = 1.0973936899862826e-05;
        double r2297076 = r2297075 * r2297074;
        double r2297077 = r2297074 * r2297076;
        double r2297078 = 0.037037037037037035;
        double r2297079 = r2297077 + r2297078;
        double r2297080 = r2297073 / r2297079;
        double r2297081 = r2297060 / r2297080;
        double r2297082 = r2297064 + r2297081;
        return r2297082;
}

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

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

  5. Applied flip3-+1.2

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

  6. Applied associate-*r/1.1

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

  7. Simplified0.3

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

  9. Applied associate-/l*0.0

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

  10. Simplified0.0

    \[\leadsto \frac{2}{945} \cdot {x}^{5} + \frac{x}{\color{blue}{\frac{\left(\frac{1}{9} - \frac{1}{135} \cdot \left(x \cdot x\right)\right) + \left(\frac{1}{45} \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{1}{45} \cdot \left(x \cdot x\right)\right)}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{1}{91125}\right) + \frac{1}{27}}}}\]

  11. Final simplification0.0

    \[\leadsto {x}^{5} \cdot \frac{2}{945} + \frac{x}{\frac{\left(\left(x \cdot x\right) \cdot \frac{1}{45}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{45}\right) + \left(\frac{1}{9} - \left(x \cdot x\right) \cdot \frac{1}{135}\right)}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{1}{91125} \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) + \frac{1}{27}}}\]

Reproduce

herbie shell --seed 2019144 
(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))))