Average Error: 59.7 → 0.3
Time: 10.0s
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 r80026 = 1.0;
        double r80027 = x;
        double r80028 = r80026 / r80027;
        double r80029 = tan(r80027);
        double r80030 = r80026 / r80029;
        double r80031 = r80028 - r80030;
        return r80031;
}
↓
double f(double x) {
        double r80032 = 0.022222222222222223;
        double r80033 = x;
        double r80034 = 3.0;
        double r80035 = pow(r80033, r80034);
        double r80036 = r80032 * r80035;
        double r80037 = 0.0021164021164021165;
        double r80038 = 5.0;
        double r80039 = pow(r80033, r80038);
        double r80040 = r80037 * r80039;
        double r80041 = 0.3333333333333333;
        double r80042 = r80041 * r80033;
        double r80043 = r80040 + r80042;
        double r80044 = r80036 + r80043;
        return r80044;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original59.7
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 59.7

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

  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 2020045 
(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))))