Average Error: 60.0 → 0.3
Time: 35.0s
Precision: 64
\[-0.0259999999999999988 \lt x \land x \lt 0.0259999999999999988\]
\[\frac{1}{x} - \frac{1}{\tan x}\]
↓
\[0.0222222222222222231 \cdot {x}^{3} + \left(0.00211640211640211654 \cdot {x}^{5} + 0.333333333333333315 \cdot x\right)\]
\frac{1}{x} - \frac{1}{\tan x}
↓
0.0222222222222222231 \cdot {x}^{3} + \left(0.00211640211640211654 \cdot {x}^{5} + 0.333333333333333315 \cdot x\right)
double f(double x) {
        double r127774 = 1.0;
        double r127775 = x;
        double r127776 = r127774 / r127775;
        double r127777 = tan(r127775);
        double r127778 = r127774 / r127777;
        double r127779 = r127776 - r127778;
        return r127779;
}
↓
double f(double x) {
        double r127780 = 0.022222222222222223;
        double r127781 = x;
        double r127782 = 3.0;
        double r127783 = pow(r127781, r127782);
        double r127784 = r127780 * r127783;
        double r127785 = 0.0021164021164021165;
        double r127786 = 5.0;
        double r127787 = pow(r127781, r127786);
        double r127788 = r127785 * r127787;
        double r127789 = 0.3333333333333333;
        double r127790 = r127789 * r127781;
        double r127791 = r127788 + r127790;
        double r127792 = r127784 + r127791;
        return r127792;
}

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
\[\begin{array}{l} \mathbf{if}\;\left|x\right| \lt 0.0259999999999999988:\\ \;\;\;\;\frac{x}{3} \cdot \left(1 + \frac{x \cdot x}{15}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} - \frac{1}{\tan x}\\ \end{array}\]

Derivation

  1. Initial program 60.0

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

  2. Taylor expanded around 0 0.3

    \[\leadsto \color{blue}{0.0222222222222222231 \cdot {x}^{3} + \left(0.00211640211640211654 \cdot {x}^{5} + 0.333333333333333315 \cdot x\right)}\]

  3. Final simplification0.3

    \[\leadsto 0.0222222222222222231 \cdot {x}^{3} + \left(0.00211640211640211654 \cdot {x}^{5} + 0.333333333333333315 \cdot x\right)\]

Reproduce

herbie shell --seed 2019198 
(FPCore (x)
  :name "invcot (example 3.9)"
  :pre (and (< -0.026 x) (< x 0.026))

  :herbie-target
  (if (< (fabs x) 0.026) (* (/ x 3.0) (+ 1.0 (/ (* x x) 15.0))) (- (/ 1.0 x) (/ 1.0 (tan x))))

  (- (/ 1.0 x) (/ 1.0 (tan x))))