Average Error: 59.8 → 0.4
Time: 29.9s
Precision: 64
\[-0.0259999999999999988065102485279567190446 \lt x \land x \lt 0.0259999999999999988065102485279567190446\]
\[\frac{1}{x} - \frac{1}{\tan x}\]
↓
\[{x}^{5} \cdot 0.002116402116402116544841005563171165704262 + \left(0.3333333333333333148296162562473909929395 + \left(x \cdot x\right) \cdot 0.02222222222222222307030925492199457949027\right) \cdot x\]
\frac{1}{x} - \frac{1}{\tan x}
↓
{x}^{5} \cdot 0.002116402116402116544841005563171165704262 + \left(0.3333333333333333148296162562473909929395 + \left(x \cdot x\right) \cdot 0.02222222222222222307030925492199457949027\right) \cdot x
double f(double x) {
        double r5001695 = 1.0;
        double r5001696 = x;
        double r5001697 = r5001695 / r5001696;
        double r5001698 = tan(r5001696);
        double r5001699 = r5001695 / r5001698;
        double r5001700 = r5001697 - r5001699;
        return r5001700;
}
↓
double f(double x) {
        double r5001701 = x;
        double r5001702 = 5.0;
        double r5001703 = pow(r5001701, r5001702);
        double r5001704 = 0.0021164021164021165;
        double r5001705 = r5001703 * r5001704;
        double r5001706 = 0.3333333333333333;
        double r5001707 = r5001701 * r5001701;
        double r5001708 = 0.022222222222222223;
        double r5001709 = r5001707 * r5001708;
        double r5001710 = r5001706 + r5001709;
        double r5001711 = r5001710 * r5001701;
        double r5001712 = r5001705 + r5001711;
        return r5001712;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original59.8
Target0.1
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;\left|x\right| \lt 0.0259999999999999988065102485279567190446:\\ \;\;\;\;\frac{x}{3} \cdot \left(1 + \frac{x \cdot x}{15}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} - \frac{1}{\tan x}\\ \end{array}\]

Derivation

  1. Initial program 59.8

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

  2. Taylor expanded around 0 0.4

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

  3. Simplified0.4

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

  4. Final simplification0.4

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

Reproduce

herbie shell --seed 2019174 
(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))))