Average Error: 59.8 → 0.3
Time: 24.1s
Precision: 64
\[-0.026 \lt x \land x \lt 0.026\]
\[\frac{1}{x} - \frac{1}{\tan x}\]
↓
\[\mathsf{fma}\left({x}^{5}, \frac{2}{945}, \mathsf{fma}\left(\frac{1}{45}, x \cdot x, \frac{1}{3}\right) \cdot x\right)\]
\frac{1}{x} - \frac{1}{\tan x}
↓
\mathsf{fma}\left({x}^{5}, \frac{2}{945}, \mathsf{fma}\left(\frac{1}{45}, x \cdot x, \frac{1}{3}\right) \cdot x\right)
double f(double x) {
        double r2685339 = 1.0;
        double r2685340 = x;
        double r2685341 = r2685339 / r2685340;
        double r2685342 = tan(r2685340);
        double r2685343 = r2685339 / r2685342;
        double r2685344 = r2685341 - r2685343;
        return r2685344;
}
↓
double f(double x) {
        double r2685345 = x;
        double r2685346 = 5.0;
        double r2685347 = pow(r2685345, r2685346);
        double r2685348 = 0.0021164021164021165;
        double r2685349 = 0.022222222222222223;
        double r2685350 = r2685345 * r2685345;
        double r2685351 = 0.3333333333333333;
        double r2685352 = fma(r2685349, r2685350, r2685351);
        double r2685353 = r2685352 * r2685345;
        double r2685354 = fma(r2685347, r2685348, r2685353);
        return r2685354;
}

Error

Bits error versus x

Target

Original59.8
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 59.8

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

  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

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

  4. Final simplification0.3

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

Reproduce

herbie shell --seed 2019162 +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))))