Average Error: 59.9 → 0.3
Time: 1.5m
Precision: 64
\[-0.026 \lt x \land x \lt 0.026\]
\[\frac{1}{x} - \frac{1}{\tan x}\]
↓
\[\mathsf{fma}\left(\left({x}^{5}\right), \frac{2}{945}, \left(\mathsf{fma}\left(x, \left(\frac{1}{45} \cdot x\right), \frac{1}{3}\right) \cdot x\right)\right)\]
\frac{1}{x} - \frac{1}{\tan x}
↓
\mathsf{fma}\left(\left({x}^{5}\right), \frac{2}{945}, \left(\mathsf{fma}\left(x, \left(\frac{1}{45} \cdot x\right), \frac{1}{3}\right) \cdot x\right)\right)
double f(double x) {
        double r7669244 = 1.0;
        double r7669245 = x;
        double r7669246 = r7669244 / r7669245;
        double r7669247 = tan(r7669245);
        double r7669248 = r7669244 / r7669247;
        double r7669249 = r7669246 - r7669248;
        return r7669249;
}
↓
double f(double x) {
        double r7669250 = x;
        double r7669251 = 5.0;
        double r7669252 = pow(r7669250, r7669251);
        double r7669253 = 0.0021164021164021165;
        double r7669254 = 0.022222222222222223;
        double r7669255 = r7669254 * r7669250;
        double r7669256 = 0.3333333333333333;
        double r7669257 = fma(r7669250, r7669255, r7669256);
        double r7669258 = r7669257 * r7669250;
        double r7669259 = fma(r7669252, r7669253, r7669258);
        return r7669259;
}

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

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

  4. Final simplification0.3

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

Reproduce

herbie shell --seed 2019120 +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))))