Average Error: 59.8 → 0.4
Time: 32.9s
Precision: 64
\[-0.0259999999999999988065102485279567190446 \lt x \land x \lt 0.0259999999999999988065102485279567190446\]
\[\frac{1}{x} - \frac{1}{\tan x}\]
↓
\[0.02222222222222222307030925492199457949027 \cdot {x}^{3} + \left(\log \left(e^{0.002116402116402116544841005563171165704262 \cdot {x}^{5}}\right) + 0.3333333333333333148296162562473909929395 \cdot x\right)\]
\frac{1}{x} - \frac{1}{\tan x}
↓
0.02222222222222222307030925492199457949027 \cdot {x}^{3} + \left(\log \left(e^{0.002116402116402116544841005563171165704262 \cdot {x}^{5}}\right) + 0.3333333333333333148296162562473909929395 \cdot x\right)
double f(double x) {
        double r89271 = 1.0;
        double r89272 = x;
        double r89273 = r89271 / r89272;
        double r89274 = tan(r89272);
        double r89275 = r89271 / r89274;
        double r89276 = r89273 - r89275;
        return r89276;
}
↓
double f(double x) {
        double r89277 = 0.022222222222222223;
        double r89278 = x;
        double r89279 = 3.0;
        double r89280 = pow(r89278, r89279);
        double r89281 = r89277 * r89280;
        double r89282 = 0.0021164021164021165;
        double r89283 = 5.0;
        double r89284 = pow(r89278, r89283);
        double r89285 = r89282 * r89284;
        double r89286 = exp(r89285);
        double r89287 = log(r89286);
        double r89288 = 0.3333333333333333;
        double r89289 = r89288 * r89278;
        double r89290 = r89287 + r89289;
        double r89291 = r89281 + r89290;
        return r89291;
}

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 0.3

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

  4. Applied add-log-exp0.4

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

  5. Final simplification0.4

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

Reproduce

herbie shell --seed 2019304 
(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))))