Average Error: 59.9 → 0.4
Time: 15.9s
Precision: 64
\[-0.0259999999999999988065102485279567190446 \lt x \land x \lt 0.0259999999999999988065102485279567190446\]
\[\frac{1}{x} - \frac{1}{\tan x}\]
↓
\[\mathsf{fma}\left(0.002116402116402116544841005563171165704262, {x}^{5}, x \cdot \mathsf{fma}\left(0.02222222222222222307030925492199457949027, x \cdot x, 0.3333333333333333148296162562473909929395\right)\right)\]
\frac{1}{x} - \frac{1}{\tan x}
↓
\mathsf{fma}\left(0.002116402116402116544841005563171165704262, {x}^{5}, x \cdot \mathsf{fma}\left(0.02222222222222222307030925492199457949027, x \cdot x, 0.3333333333333333148296162562473909929395\right)\right)
double f(double x) {
        double r4418080 = 1.0;
        double r4418081 = x;
        double r4418082 = r4418080 / r4418081;
        double r4418083 = tan(r4418081);
        double r4418084 = r4418080 / r4418083;
        double r4418085 = r4418082 - r4418084;
        return r4418085;
}
↓
double f(double x) {
        double r4418086 = 0.0021164021164021165;
        double r4418087 = x;
        double r4418088 = 5.0;
        double r4418089 = pow(r4418087, r4418088);
        double r4418090 = 0.022222222222222223;
        double r4418091 = r4418087 * r4418087;
        double r4418092 = 0.3333333333333333;
        double r4418093 = fma(r4418090, r4418091, r4418092);
        double r4418094 = r4418087 * r4418093;
        double r4418095 = fma(r4418086, r4418089, r4418094);
        return r4418095;
}

Error

Bits error versus x

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

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

  3. Simplified0.4

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

  4. Final simplification0.4

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

Reproduce

herbie shell --seed 2019169 +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))))