Average Error: 59.9 → 0.3
Time: 25.6s
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 r1044671 = 1.0;
        double r1044672 = x;
        double r1044673 = r1044671 / r1044672;
        double r1044674 = tan(r1044672);
        double r1044675 = r1044671 / r1044674;
        double r1044676 = r1044673 - r1044675;
        return r1044676;
}
↓
double f(double x) {
        double r1044677 = x;
        double r1044678 = r1044677 * r1044677;
        double r1044679 = 0.022222222222222223;
        double r1044680 = r1044677 * r1044679;
        double r1044681 = 0.3333333333333333;
        double r1044682 = 0.0021164021164021165;
        double r1044683 = 5.0;
        double r1044684 = pow(r1044677, r1044683);
        double r1044685 = r1044682 * r1044684;
        double r1044686 = fma(r1044681, r1044677, r1044685);
        double r1044687 = fma(r1044678, r1044680, r1044686);
        return r1044687;
}

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))))