Average Error: 60.0 → 0.3
Time: 33.2s
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 r4605663 = 1.0;
        double r4605664 = x;
        double r4605665 = r4605663 / r4605664;
        double r4605666 = tan(r4605664);
        double r4605667 = r4605663 / r4605666;
        double r4605668 = r4605665 - r4605667;
        return r4605668;
}
↓
double f(double x) {
        double r4605669 = x;
        double r4605670 = r4605669 * r4605669;
        double r4605671 = 0.022222222222222223;
        double r4605672 = 0.3333333333333333;
        double r4605673 = fma(r4605670, r4605671, r4605672);
        double r4605674 = 0.0021164021164021165;
        double r4605675 = 5.0;
        double r4605676 = pow(r4605669, r4605675);
        double r4605677 = r4605674 * r4605676;
        double r4605678 = fma(r4605669, r4605673, r4605677);
        return r4605678;
}

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 2019163 +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))))