Average Error: 59.9 → 0.3
Time: 25.2s
Precision: 64
\[-0.0259999999999999988065102485279567190446 \lt x \land x \lt 0.0259999999999999988065102485279567190446\]
\[\frac{1}{x} - \frac{1}{\tan x}\]
↓
\[{x}^{5} \cdot 0.002116402116402116544841005563171165704262 + \left(0.02222222222222222307030925492199457949027 \cdot \left(x \cdot x\right) + 0.3333333333333333148296162562473909929395\right) \cdot x\]
\frac{1}{x} - \frac{1}{\tan x}
↓
{x}^{5} \cdot 0.002116402116402116544841005563171165704262 + \left(0.02222222222222222307030925492199457949027 \cdot \left(x \cdot x\right) + 0.3333333333333333148296162562473909929395\right) \cdot x
double f(double x) {
        double r4985967 = 1.0;
        double r4985968 = x;
        double r4985969 = r4985967 / r4985968;
        double r4985970 = tan(r4985968);
        double r4985971 = r4985967 / r4985970;
        double r4985972 = r4985969 - r4985971;
        return r4985972;
}
↓
double f(double x) {
        double r4985973 = x;
        double r4985974 = 5.0;
        double r4985975 = pow(r4985973, r4985974);
        double r4985976 = 0.0021164021164021165;
        double r4985977 = r4985975 * r4985976;
        double r4985978 = 0.022222222222222223;
        double r4985979 = r4985973 * r4985973;
        double r4985980 = r4985978 * r4985979;
        double r4985981 = 0.3333333333333333;
        double r4985982 = r4985980 + r4985981;
        double r4985983 = r4985982 * r4985973;
        double r4985984 = r4985977 + r4985983;
        return r4985984;
}

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

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

  4. Final simplification0.3

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

Reproduce

herbie shell --seed 2019170 
(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))))