Average Error: 59.9 → 0.3
Time: 29.5s
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 r76719 = 1.0;
        double r76720 = x;
        double r76721 = r76719 / r76720;
        double r76722 = tan(r76720);
        double r76723 = r76719 / r76722;
        double r76724 = r76721 - r76723;
        return r76724;
}
↓
double f(double x) {
        double r76725 = 0.022222222222222223;
        double r76726 = x;
        double r76727 = 3.0;
        double r76728 = pow(r76726, r76727);
        double r76729 = 0.0021164021164021165;
        double r76730 = 5.0;
        double r76731 = pow(r76726, r76730);
        double r76732 = 0.3333333333333333;
        double r76733 = r76732 * r76726;
        double r76734 = fma(r76729, r76731, r76733);
        double r76735 = fma(r76725, r76728, r76734);
        return r76735;
}

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))))