Average Error: 59.8 → 0.3
Time: 24.3s
Precision: 64
\[-0.0259999999999999988065102485279567190446 \lt x \land x \lt 0.0259999999999999988065102485279567190446\]
\[\frac{1}{x} - \frac{1}{\tan x}\]
↓
\[0.02222222222222222307030925492199457949027 \cdot {x}^{3} + \left(0.002116402116402116544841005563171165704262 \cdot {x}^{5} + 0.3333333333333333148296162562473909929395 \cdot x\right)\]
\frac{1}{x} - \frac{1}{\tan x}
↓
0.02222222222222222307030925492199457949027 \cdot {x}^{3} + \left(0.002116402116402116544841005563171165704262 \cdot {x}^{5} + 0.3333333333333333148296162562473909929395 \cdot x\right)
double f(double x) {
        double r120437 = 1.0;
        double r120438 = x;
        double r120439 = r120437 / r120438;
        double r120440 = tan(r120438);
        double r120441 = r120437 / r120440;
        double r120442 = r120439 - r120441;
        return r120442;
}
↓
double f(double x) {
        double r120443 = 0.022222222222222223;
        double r120444 = x;
        double r120445 = 3.0;
        double r120446 = pow(r120444, r120445);
        double r120447 = r120443 * r120446;
        double r120448 = 0.0021164021164021165;
        double r120449 = 5.0;
        double r120450 = pow(r120444, r120449);
        double r120451 = r120448 * r120450;
        double r120452 = 0.3333333333333333;
        double r120453 = r120452 * r120444;
        double r120454 = r120451 + r120453;
        double r120455 = r120447 + r120454;
        return r120455;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original59.8
Target0.1
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;\left|x\right| \lt 0.0259999999999999988065102485279567190446:\\ \;\;\;\;\frac{x}{3} \cdot \left(1 + \frac{x \cdot x}{15}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} - \frac{1}{\tan x}\\ \end{array}\]

Derivation

  1. Initial program 59.8

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

  2. Taylor expanded around 0 0.3

    \[\leadsto \color{blue}{0.02222222222222222307030925492199457949027 \cdot {x}^{3} + \left(0.002116402116402116544841005563171165704262 \cdot {x}^{5} + 0.3333333333333333148296162562473909929395 \cdot x\right)}\]

  3. Final simplification0.3

    \[\leadsto 0.02222222222222222307030925492199457949027 \cdot {x}^{3} + \left(0.002116402116402116544841005563171165704262 \cdot {x}^{5} + 0.3333333333333333148296162562473909929395 \cdot x\right)\]

Reproduce

herbie shell --seed 2019291 
(FPCore (x)
  :name "invcot (example 3.9)"
  :precision binary64
  :pre (and (< -0.0259999999999999988 x) (< x 0.0259999999999999988))

  :herbie-target
  (if (< (fabs x) 0.0259999999999999988) (* (/ x 3) (+ 1 (/ (* x x) 15))) (- (/ 1 x) (/ 1 (tan x))))

  (- (/ 1 x) (/ 1 (tan x))))