Average Error: 59.8 → 0.4
Time: 7.7s
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 r94865 = 1.0;
        double r94866 = x;
        double r94867 = r94865 / r94866;
        double r94868 = tan(r94866);
        double r94869 = r94865 / r94868;
        double r94870 = r94867 - r94869;
        return r94870;
}
↓
double f(double x) {
        double r94871 = 0.022222222222222223;
        double r94872 = x;
        double r94873 = 3.0;
        double r94874 = pow(r94872, r94873);
        double r94875 = r94871 * r94874;
        double r94876 = 0.0021164021164021165;
        double r94877 = 5.0;
        double r94878 = pow(r94872, r94877);
        double r94879 = r94876 * r94878;
        double r94880 = 0.3333333333333333;
        double r94881 = r94880 * r94872;
        double r94882 = r94879 + r94881;
        double r94883 = r94875 + r94882;
        return r94883;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original59.8
Target0.1
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;\left|x\right| \lt 0.0259999999999999988:\\ \;\;\;\;\frac{x}{3} \cdot \left(1 + \frac{x \cdot x}{15}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} - \frac{1}{\tan x}\\ \end{array}\]

Derivation

  1. Initial program 59.8

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

  2. Taylor expanded around 0 0.4

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

  3. Final simplification0.4

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

Reproduce

herbie shell --seed 2020049 
(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))))