Average Error: 60.0 → 0.3
Time: 27.1s
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 r68414 = 1.0;
        double r68415 = x;
        double r68416 = r68414 / r68415;
        double r68417 = tan(r68415);
        double r68418 = r68414 / r68417;
        double r68419 = r68416 - r68418;
        return r68419;
}
↓
double f(double x) {
        double r68420 = 0.0021164021164021165;
        double r68421 = x;
        double r68422 = 5.0;
        double r68423 = pow(r68421, r68422);
        double r68424 = r68420 * r68423;
        double r68425 = 0.3333333333333333;
        double r68426 = r68421 * r68425;
        double r68427 = 3.0;
        double r68428 = pow(r68421, r68427);
        double r68429 = 0.022222222222222223;
        double r68430 = r68428 * r68429;
        double r68431 = r68426 + r68430;
        double r68432 = r68424 + r68431;
        return r68432;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original60.0
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 60.0

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

  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 2019195 
(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))))