Average Error: 59.9 → 0.3
Time: 15.8s
Precision: 64
\[-0.0259999999999999988065102485279567190446 \lt x \land x \lt 0.0259999999999999988065102485279567190446\]
\[\frac{1}{x} - \frac{1}{\tan x}\]
↓
\[\left(0.02222222222222222307030925492199457949027 \cdot \left(x \cdot x\right) + 0.3333333333333333148296162562473909929395\right) \cdot x + {x}^{5} \cdot 0.002116402116402116544841005563171165704262\]
\frac{1}{x} - \frac{1}{\tan x}
↓
\left(0.02222222222222222307030925492199457949027 \cdot \left(x \cdot x\right) + 0.3333333333333333148296162562473909929395\right) \cdot x + {x}^{5} \cdot 0.002116402116402116544841005563171165704262
double f(double x) {
        double r3932336 = 1.0;
        double r3932337 = x;
        double r3932338 = r3932336 / r3932337;
        double r3932339 = tan(r3932337);
        double r3932340 = r3932336 / r3932339;
        double r3932341 = r3932338 - r3932340;
        return r3932341;
}
↓
double f(double x) {
        double r3932342 = 0.022222222222222223;
        double r3932343 = x;
        double r3932344 = r3932343 * r3932343;
        double r3932345 = r3932342 * r3932344;
        double r3932346 = 0.3333333333333333;
        double r3932347 = r3932345 + r3932346;
        double r3932348 = r3932347 * r3932343;
        double r3932349 = 5.0;
        double r3932350 = pow(r3932343, r3932349);
        double r3932351 = 0.0021164021164021165;
        double r3932352 = r3932350 * r3932351;
        double r3932353 = r3932348 + r3932352;
        return r3932353;
}

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

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

  4. Final simplification0.3

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

Reproduce

herbie shell --seed 2019172 
(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))))