Average Error: 59.9 → 0.3
Time: 42.2s
Precision: 64
\[-0.0259999999999999988065102485279567190446 \lt x \land x \lt 0.0259999999999999988065102485279567190446\]
\[\frac{1}{x} - \frac{1}{\tan x}\]
↓
\[0.002116402116402116544841005563171165704262 \cdot {x}^{5} + \left(x \cdot 0.3333333333333333148296162562473909929395 + {x}^{3} \cdot 0.02222222222222222307030925492199457949027\right)\]
\frac{1}{x} - \frac{1}{\tan x}
↓
0.002116402116402116544841005563171165704262 \cdot {x}^{5} + \left(x \cdot 0.3333333333333333148296162562473909929395 + {x}^{3} \cdot 0.02222222222222222307030925492199457949027\right)
double f(double x) {
        double r130677 = 1.0;
        double r130678 = x;
        double r130679 = r130677 / r130678;
        double r130680 = tan(r130678);
        double r130681 = r130677 / r130680;
        double r130682 = r130679 - r130681;
        return r130682;
}
↓
double f(double x) {
        double r130683 = 0.0021164021164021165;
        double r130684 = x;
        double r130685 = 5.0;
        double r130686 = pow(r130684, r130685);
        double r130687 = r130683 * r130686;
        double r130688 = 0.3333333333333333;
        double r130689 = r130684 * r130688;
        double r130690 = 3.0;
        double r130691 = pow(r130684, r130690);
        double r130692 = 0.022222222222222223;
        double r130693 = r130691 * r130692;
        double r130694 = r130689 + r130693;
        double r130695 = r130687 + r130694;
        return r130695;
}

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

Derivation

  1. Initial program 59.9

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

  2. Taylor expanded around 0 0.3

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

  3. Simplified0.3

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

  4. Final simplification0.3

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

Reproduce

herbie shell --seed 2019194 
(FPCore (x)
  :name "invcot (example 3.9)"
  :pre (and (< -0.026 x) (< x 0.026))

  :herbie-target
  (if (< (fabs x) 0.026) (* (/ x 3.0) (+ 1.0 (/ (* x x) 15.0))) (- (/ 1.0 x) (/ 1.0 (tan x))))

  (- (/ 1.0 x) (/ 1.0 (tan x))))