Average Error: 59.9 → 0.3
Time: 18.6s
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 r431 = 1.0;
        double r432 = x;
        double r433 = r431 / r432;
        double r434 = tan(r432);
        double r435 = r431 / r434;
        double r436 = r433 - r435;
        return r436;
}
↓
double f(double x) {
        double r437 = 0.022222222222222223;
        double r438 = x;
        double r439 = 3.0;
        double r440 = pow(r438, r439);
        double r441 = r437 * r440;
        double r442 = 0.0021164021164021165;
        double r443 = 5.0;
        double r444 = pow(r438, r443);
        double r445 = r442 * r444;
        double r446 = r441 + r445;
        double r447 = 0.3333333333333333;
        double r448 = r447 * r438;
        double r449 = r446 + r448;
        return r449;
}

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 2020025 
(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))))