Average Error: 59.9 → 0.3
Time: 10.8s
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 r122249 = 1.0;
        double r122250 = x;
        double r122251 = r122249 / r122250;
        double r122252 = tan(r122250);
        double r122253 = r122249 / r122252;
        double r122254 = r122251 - r122253;
        return r122254;
}
↓
double f(double x) {
        double r122255 = 0.022222222222222223;
        double r122256 = x;
        double r122257 = 3.0;
        double r122258 = pow(r122256, r122257);
        double r122259 = r122255 * r122258;
        double r122260 = 0.0021164021164021165;
        double r122261 = 5.0;
        double r122262 = pow(r122256, r122261);
        double r122263 = r122260 * r122262;
        double r122264 = r122259 + r122263;
        double r122265 = 0.3333333333333333;
        double r122266 = r122265 * r122256;
        double r122267 = r122264 + r122266;
        return r122267;
}

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))))