Average Error: 60.0 → 0.3
Time: 8.2s
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 r92655 = 1.0;
        double r92656 = x;
        double r92657 = r92655 / r92656;
        double r92658 = tan(r92656);
        double r92659 = r92655 / r92658;
        double r92660 = r92657 - r92659;
        return r92660;
}
↓
double f(double x) {
        double r92661 = 0.022222222222222223;
        double r92662 = x;
        double r92663 = 3.0;
        double r92664 = pow(r92662, r92663);
        double r92665 = r92661 * r92664;
        double r92666 = 0.0021164021164021165;
        double r92667 = 5.0;
        double r92668 = pow(r92662, r92667);
        double r92669 = r92666 * r92668;
        double r92670 = 0.3333333333333333;
        double r92671 = r92670 * r92662;
        double r92672 = r92669 + r92671;
        double r92673 = r92665 + r92672;
        return r92673;
}

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 2020062 
(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))))