Average Error: 60.0 → 0.3
Time: 13.0s
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 r103399 = 1.0;
        double r103400 = x;
        double r103401 = r103399 / r103400;
        double r103402 = tan(r103400);
        double r103403 = r103399 / r103402;
        double r103404 = r103401 - r103403;
        return r103404;
}
↓
double f(double x) {
        double r103405 = 0.022222222222222223;
        double r103406 = x;
        double r103407 = 3.0;
        double r103408 = pow(r103406, r103407);
        double r103409 = r103405 * r103408;
        double r103410 = 0.0021164021164021165;
        double r103411 = 5.0;
        double r103412 = pow(r103406, r103411);
        double r103413 = r103410 * r103412;
        double r103414 = 0.3333333333333333;
        double r103415 = r103414 * r103406;
        double r103416 = r103413 + r103415;
        double r103417 = r103409 + r103416;
        return r103417;
}

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 2020089 
(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))))