Average Error: 59.9 → 0.4
Time: 17.5s
Precision: 64
\[-0.0259999999999999988065102485279567190446 \lt x \land x \lt 0.0259999999999999988065102485279567190446\]
\[\frac{1}{x} - \frac{1}{\tan x}\]
↓
\[\frac{6405119470038039}{288230376151711744} \cdot {x}^{3} + \left(\frac{4880091024790887}{2305843009213693952} \cdot {x}^{5} + \frac{6004799503160661}{18014398509481984} \cdot x\right)\]
\frac{1}{x} - \frac{1}{\tan x}
↓
\frac{6405119470038039}{288230376151711744} \cdot {x}^{3} + \left(\frac{4880091024790887}{2305843009213693952} \cdot {x}^{5} + \frac{6004799503160661}{18014398509481984} \cdot x\right)
double f(double x) {
        double r367198 = 1.0;
        double r367199 = x;
        double r367200 = r367198 / r367199;
        double r367201 = tan(r367199);
        double r367202 = r367198 / r367201;
        double r367203 = r367200 - r367202;
        return r367203;
}
↓
double f(double x) {
        double r367204 = 6405119470038039.0;
        double r367205 = 2.8823037615171174e+17;
        double r367206 = r367204 / r367205;
        double r367207 = x;
        double r367208 = 3.0;
        double r367209 = pow(r367207, r367208);
        double r367210 = r367206 * r367209;
        double r367211 = 4880091024790887.0;
        double r367212 = 2.305843009213694e+18;
        double r367213 = r367211 / r367212;
        double r367214 = 5.0;
        double r367215 = pow(r367207, r367214);
        double r367216 = r367213 * r367215;
        double r367217 = 6004799503160661.0;
        double r367218 = 18014398509481984.0;
        double r367219 = r367217 / r367218;
        double r367220 = r367219 * r367207;
        double r367221 = r367216 + r367220;
        double r367222 = r367210 + r367221;
        return r367222;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original59.9
Target0.1
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;\left|x\right| \lt 0.0259999999999999988065102485279567190446:\\ \;\;\;\;\frac{x}{3} \cdot \left(1 + \frac{x \cdot x}{15}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} - \frac{1}{\tan x}\\ \end{array}\]

Derivation

  1. Initial program 59.9

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

  2. Taylor expanded around 0 0.4

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

  3. Simplified0.4

    \[\leadsto \color{blue}{\frac{6405119470038039}{288230376151711744} \cdot {x}^{3} + \left(\frac{4880091024790887}{2305843009213693952} \cdot {x}^{5} + \frac{6004799503160661}{18014398509481984} \cdot x\right)}\]

  4. Final simplification0.4

    \[\leadsto \frac{6405119470038039}{288230376151711744} \cdot {x}^{3} + \left(\frac{4880091024790887}{2305843009213693952} \cdot {x}^{5} + \frac{6004799503160661}{18014398509481984} \cdot x\right)\]

Reproduce

herbie shell --seed 350497007 
(FPCore (x)
  :name "invcot (example 3.9)"
  :precision binary64
  :pre (and (< -0.0259999999999999988 x) (< x 0.0259999999999999988))

  :herbie-target
  (if (< (fabs x) 0.0259999999999999988) (* (/ x 3) (+ 1 (/ (* x x) 15))) (- (/ 1 x) (/ 1 (tan x))))

  (- (/ 1 x) (/ 1 (tan x))))