Average Error: 59.9 → 0.3
Time: 16.3s
Precision: 64
\[-0.0259999999999999988065102485279567190446 \lt x \land x \lt 0.0259999999999999988065102485279567190446\]
\[\frac{1}{x} - \frac{1}{\tan x}\]
↓
\[\mathsf{fma}\left(0.02222222222222222307030925492199457949027 \cdot \left(x \cdot x\right), x, \mathsf{fma}\left({x}^{5}, 0.002116402116402116544841005563171165704262, x \cdot 0.3333333333333333148296162562473909929395\right)\right)\]
\frac{1}{x} - \frac{1}{\tan x}
↓
\mathsf{fma}\left(0.02222222222222222307030925492199457949027 \cdot \left(x \cdot x\right), x, \mathsf{fma}\left({x}^{5}, 0.002116402116402116544841005563171165704262, x \cdot 0.3333333333333333148296162562473909929395\right)\right)
double f(double x) {
        double r2728334 = 1.0;
        double r2728335 = x;
        double r2728336 = r2728334 / r2728335;
        double r2728337 = tan(r2728335);
        double r2728338 = r2728334 / r2728337;
        double r2728339 = r2728336 - r2728338;
        return r2728339;
}
↓
double f(double x) {
        double r2728340 = 0.022222222222222223;
        double r2728341 = x;
        double r2728342 = r2728341 * r2728341;
        double r2728343 = r2728340 * r2728342;
        double r2728344 = 5.0;
        double r2728345 = pow(r2728341, r2728344);
        double r2728346 = 0.0021164021164021165;
        double r2728347 = 0.3333333333333333;
        double r2728348 = r2728341 * r2728347;
        double r2728349 = fma(r2728345, r2728346, r2728348);
        double r2728350 = fma(r2728343, r2728341, r2728349);
        return r2728350;
}

Error

Bits error versus x

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

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

  4. Final simplification0.3

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

Reproduce

herbie shell --seed 2019171 +o rules:numerics
(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))))