Average Error: 59.8 → 0.4
Time: 26.7s
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 r88296 = 1.0;
        double r88297 = x;
        double r88298 = r88296 / r88297;
        double r88299 = tan(r88297);
        double r88300 = r88296 / r88299;
        double r88301 = r88298 - r88300;
        return r88301;
}
↓
double f(double x) {
        double r88302 = 0.0021164021164021165;
        double r88303 = x;
        double r88304 = 5.0;
        double r88305 = pow(r88303, r88304);
        double r88306 = r88302 * r88305;
        double r88307 = 0.3333333333333333;
        double r88308 = r88303 * r88307;
        double r88309 = 3.0;
        double r88310 = pow(r88303, r88309);
        double r88311 = 0.022222222222222223;
        double r88312 = r88310 * r88311;
        double r88313 = r88308 + r88312;
        double r88314 = r88306 + r88313;
        return r88314;
}

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

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

  4. Final simplification0.4

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

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))))