Average Error: 60.0 → 0.3
Time: 28.7s
Precision: 64
\[-0.026 \lt x \land x \lt 0.026\]
\[\frac{1}{x} - \frac{1}{\tan x}\]
↓
\[\mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, \frac{1}{45}, \frac{1}{3}\right), \frac{2}{945} \cdot {x}^{5}\right)\]
\frac{1}{x} - \frac{1}{\tan x}
↓
\mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, \frac{1}{45}, \frac{1}{3}\right), \frac{2}{945} \cdot {x}^{5}\right)
double f(double x) {
        double r2641701 = 1.0;
        double r2641702 = x;
        double r2641703 = r2641701 / r2641702;
        double r2641704 = tan(r2641702);
        double r2641705 = r2641701 / r2641704;
        double r2641706 = r2641703 - r2641705;
        return r2641706;
}
↓
double f(double x) {
        double r2641707 = x;
        double r2641708 = r2641707 * r2641707;
        double r2641709 = 0.022222222222222223;
        double r2641710 = 0.3333333333333333;
        double r2641711 = fma(r2641708, r2641709, r2641710);
        double r2641712 = 0.0021164021164021165;
        double r2641713 = 5.0;
        double r2641714 = pow(r2641707, r2641713);
        double r2641715 = r2641712 * r2641714;
        double r2641716 = fma(r2641707, r2641711, r2641715);
        return r2641716;
}

Error

Bits error versus x

Target

Original60.0
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 60.0

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

  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

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

  4. Final simplification0.3

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

Reproduce

herbie shell --seed 2019164 +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))))