Average Error: 59.9 → 0.3
Time: 24.0s
Precision: 64
\[-0.0259999999999999988065102485279567190446 \lt x \land x \lt 0.0259999999999999988065102485279567190446\]
\[\frac{1}{x} - \frac{1}{\tan x}\]
↓
\[\left(0.02222222222222222307030925492199457949027 \cdot {x}^{3} + 0.002116402116402116544841005563171165704262 \cdot {x}^{5}\right) + 0.3333333333333333148296162562473909929395 \cdot x\]
\frac{1}{x} - \frac{1}{\tan x}
↓
\left(0.02222222222222222307030925492199457949027 \cdot {x}^{3} + 0.002116402116402116544841005563171165704262 \cdot {x}^{5}\right) + 0.3333333333333333148296162562473909929395 \cdot x
double f(double x) {
        double r55701 = 1.0;
        double r55702 = x;
        double r55703 = r55701 / r55702;
        double r55704 = tan(r55702);
        double r55705 = r55701 / r55704;
        double r55706 = r55703 - r55705;
        return r55706;
}
↓
double f(double x) {
        double r55707 = 0.022222222222222223;
        double r55708 = x;
        double r55709 = 3.0;
        double r55710 = pow(r55708, r55709);
        double r55711 = r55707 * r55710;
        double r55712 = 0.0021164021164021165;
        double r55713 = 5.0;
        double r55714 = pow(r55708, r55713);
        double r55715 = r55712 * r55714;
        double r55716 = r55711 + r55715;
        double r55717 = 0.3333333333333333;
        double r55718 = r55717 * r55708;
        double r55719 = r55716 + r55718;
        return r55719;
}

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

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

  4. Applied associate-+r+0.3

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

  5. Final simplification0.3

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

Reproduce

herbie shell --seed 2019212 
(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))))