Average Error: 59.9 → 0.3
Time: 10.6s
Precision: 64
\[-0.0259999999999999988 \lt x \land x \lt 0.0259999999999999988\]
\[\frac{1}{x} - \frac{1}{\tan x}\]
↓
\[\left(0.0222222222222222231 \cdot {x}^{3} + 0.00211640211640211654 \cdot {x}^{5}\right) + 0.333333333333333315 \cdot x\]
\frac{1}{x} - \frac{1}{\tan x}
↓
\left(0.0222222222222222231 \cdot {x}^{3} + 0.00211640211640211654 \cdot {x}^{5}\right) + 0.333333333333333315 \cdot x
double f(double x) {
        double r73714 = 1.0;
        double r73715 = x;
        double r73716 = r73714 / r73715;
        double r73717 = tan(r73715);
        double r73718 = r73714 / r73717;
        double r73719 = r73716 - r73718;
        return r73719;
}
↓
double f(double x) {
        double r73720 = 0.022222222222222223;
        double r73721 = x;
        double r73722 = 3.0;
        double r73723 = pow(r73721, r73722);
        double r73724 = r73720 * r73723;
        double r73725 = 0.0021164021164021165;
        double r73726 = 5.0;
        double r73727 = pow(r73721, r73726);
        double r73728 = r73725 * r73727;
        double r73729 = r73724 + r73728;
        double r73730 = 0.3333333333333333;
        double r73731 = r73730 * r73721;
        double r73732 = r73729 + r73731;
        return r73732;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original59.9
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 59.9

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

  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. Using strategy rm

  4. Applied associate-+r+0.3

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

  5. Final simplification0.3

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

Reproduce

herbie shell --seed 2020056 
(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))))