Average Error: 59.9 → 0.3
Time: 29.3s
Precision: 64
\[-0.0259999999999999988065102485279567190446 \lt x \land x \lt 0.0259999999999999988065102485279567190446\]
\[\frac{1}{x} - \frac{1}{\tan x}\]
↓
\[\mathsf{fma}\left(0.02222222222222222307030925492199457949027, {x}^{3}, \mathsf{fma}\left(0.002116402116402116544841005563171165704262, {x}^{5}, 0.3333333333333333148296162562473909929395 \cdot x\right)\right)\]
\frac{1}{x} - \frac{1}{\tan x}
↓
\mathsf{fma}\left(0.02222222222222222307030925492199457949027, {x}^{3}, \mathsf{fma}\left(0.002116402116402116544841005563171165704262, {x}^{5}, 0.3333333333333333148296162562473909929395 \cdot x\right)\right)
double f(double x) {
        double r96814 = 1.0;
        double r96815 = x;
        double r96816 = r96814 / r96815;
        double r96817 = tan(r96815);
        double r96818 = r96814 / r96817;
        double r96819 = r96816 - r96818;
        return r96819;
}
↓
double f(double x) {
        double r96820 = 0.022222222222222223;
        double r96821 = x;
        double r96822 = 3.0;
        double r96823 = pow(r96821, r96822);
        double r96824 = 0.0021164021164021165;
        double r96825 = 5.0;
        double r96826 = pow(r96821, r96825);
        double r96827 = 0.3333333333333333;
        double r96828 = r96827 * r96821;
        double r96829 = fma(r96824, r96826, r96828);
        double r96830 = fma(r96820, r96823, r96829);
        return r96830;
}

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

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

  3. Simplified0.3

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

  4. Final simplification0.3

    \[\leadsto \mathsf{fma}\left(0.02222222222222222307030925492199457949027, {x}^{3}, \mathsf{fma}\left(0.002116402116402116544841005563171165704262, {x}^{5}, 0.3333333333333333148296162562473909929395 \cdot x\right)\right)\]

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(FPCore (x)
  :name "invcot (example 3.9)"
  :precision binary64
  :pre (and (< -0.026 x) (< x 0.026))

  :herbie-target
  (if (< (fabs x) 0.026) (* (/ x 3) (+ 1 (/ (* x x) 15))) (- (/ 1 x) (/ 1 (tan x))))

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