Average Error: 60.0 → 0.3
Time: 18.3s
Precision: 64
\[-0.0259999999999999988065102485279567190446 \lt x \land x \lt 0.0259999999999999988065102485279567190446\]
\[\frac{1}{x} - \frac{1}{\tan x}\]
↓
\[0.02222222222222222307030925492199457949027 \cdot {x}^{3} + \left(0.002116402116402116544841005563171165704262 \cdot {x}^{5} + 0.3333333333333333148296162562473909929395 \cdot x\right)\]
\frac{1}{x} - \frac{1}{\tan x}
↓
0.02222222222222222307030925492199457949027 \cdot {x}^{3} + \left(0.002116402116402116544841005563171165704262 \cdot {x}^{5} + 0.3333333333333333148296162562473909929395 \cdot x\right)
double f(double x) {
        double r134769 = 1.0;
        double r134770 = x;
        double r134771 = r134769 / r134770;
        double r134772 = tan(r134770);
        double r134773 = r134769 / r134772;
        double r134774 = r134771 - r134773;
        return r134774;
}
↓
double f(double x) {
        double r134775 = 0.022222222222222223;
        double r134776 = x;
        double r134777 = 3.0;
        double r134778 = pow(r134776, r134777);
        double r134779 = r134775 * r134778;
        double r134780 = 0.0021164021164021165;
        double r134781 = 5.0;
        double r134782 = pow(r134776, r134781);
        double r134783 = r134780 * r134782;
        double r134784 = 0.3333333333333333;
        double r134785 = r134784 * r134776;
        double r134786 = r134783 + r134785;
        double r134787 = r134779 + r134786;
        return r134787;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original60.0
Target0.1
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;\left|x\right| \lt 0.0259999999999999988065102485279567190446:\\ \;\;\;\;\frac{x}{3} \cdot \left(1 + \frac{x \cdot x}{15}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} - \frac{1}{\tan x}\\ \end{array}\]

Derivation

  1. Initial program 60.0

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

  2. Taylor expanded around 0 0.3

    \[\leadsto \color{blue}{0.02222222222222222307030925492199457949027 \cdot {x}^{3} + \left(0.002116402116402116544841005563171165704262 \cdot {x}^{5} + 0.3333333333333333148296162562473909929395 \cdot x\right)}\]

  3. Final simplification0.3

    \[\leadsto 0.02222222222222222307030925492199457949027 \cdot {x}^{3} + \left(0.002116402116402116544841005563171165704262 \cdot {x}^{5} + 0.3333333333333333148296162562473909929395 \cdot x\right)\]

Reproduce

herbie shell --seed 2019353 
(FPCore (x)
  :name "invcot (example 3.9)"
  :precision binary64
  :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))))