Average Error: 59.9 → 0.3
Time: 29.2s
Precision: 64
\[-0.026 \lt x \land x \lt 0.026\]
\[\frac{1}{x} - \frac{1}{\tan x}\]
↓
\[\mathsf{fma}\left(x \cdot x, x \cdot \frac{1}{45}, \mathsf{fma}\left(\frac{1}{3}, x, \frac{2}{945} \cdot {x}^{5}\right)\right)\]
\frac{1}{x} - \frac{1}{\tan x}
↓
\mathsf{fma}\left(x \cdot x, x \cdot \frac{1}{45}, \mathsf{fma}\left(\frac{1}{3}, x, \frac{2}{945} \cdot {x}^{5}\right)\right)
double f(double x) {
        double r1977655 = 1.0;
        double r1977656 = x;
        double r1977657 = r1977655 / r1977656;
        double r1977658 = tan(r1977656);
        double r1977659 = r1977655 / r1977658;
        double r1977660 = r1977657 - r1977659;
        return r1977660;
}
↓
double f(double x) {
        double r1977661 = x;
        double r1977662 = r1977661 * r1977661;
        double r1977663 = 0.022222222222222223;
        double r1977664 = r1977661 * r1977663;
        double r1977665 = 0.3333333333333333;
        double r1977666 = 0.0021164021164021165;
        double r1977667 = 5.0;
        double r1977668 = pow(r1977661, r1977667);
        double r1977669 = r1977666 * r1977668;
        double r1977670 = fma(r1977665, r1977661, r1977669);
        double r1977671 = fma(r1977662, r1977664, r1977670);
        return r1977671;
}

Error

Bits error versus x

Target

Original59.9
Target0.1
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;\left|x\right| \lt 0.026:\\ \;\;\;\;\frac{x}{3} \cdot \left(1 + \frac{x \cdot x}{15}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} - \frac{1}{\tan x}\\ \end{array}\]

Derivation

  1. Initial program 59.9

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

  2. Taylor expanded around 0 0.3

    \[\leadsto \color{blue}{\frac{1}{3} \cdot x + \left(\frac{1}{45} \cdot {x}^{3} + \frac{2}{945} \cdot {x}^{5}\right)}\]

  3. Simplified0.3

    \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, x \cdot \frac{1}{45}, \mathsf{fma}\left(\frac{1}{3}, x, {x}^{5} \cdot \frac{2}{945}\right)\right)}\]

  4. Final simplification0.3

    \[\leadsto \mathsf{fma}\left(x \cdot x, x \cdot \frac{1}{45}, \mathsf{fma}\left(\frac{1}{3}, x, \frac{2}{945} \cdot {x}^{5}\right)\right)\]

Reproduce

herbie shell --seed 2019153 +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) (+ 1 (/ (* x x) 15))) (- (/ 1 x) (/ 1 (tan x))))

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