Average Error: 59.8 → 0.3
Time: 35.6s
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 r122593 = 1.0;
        double r122594 = x;
        double r122595 = r122593 / r122594;
        double r122596 = tan(r122594);
        double r122597 = r122593 / r122596;
        double r122598 = r122595 - r122597;
        return r122598;
}
↓
double f(double x) {
        double r122599 = 0.022222222222222223;
        double r122600 = x;
        double r122601 = 3.0;
        double r122602 = pow(r122600, r122601);
        double r122603 = 0.0021164021164021165;
        double r122604 = 5.0;
        double r122605 = pow(r122600, r122604);
        double r122606 = 0.3333333333333333;
        double r122607 = r122606 * r122600;
        double r122608 = fma(r122603, r122605, r122607);
        double r122609 = fma(r122599, r122602, r122608);
        return r122609;
}

Error

Bits error versus x

Target

Original59.8
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 59.8

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

  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. 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 2019235 +o rules:numerics
(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))))