Average Error: 59.9 → 0.3
Time: 1.4m
Precision: 64
\[\frac{1}{x} - \frac{1}{\tan x}\]
↓
\[(\left({x}^{5}\right) \cdot \frac{2}{945} + \left((x \cdot \left(\frac{1}{45} \cdot x\right) + \frac{1}{3})_* \cdot x\right))_*\]
double f(double x) {
        double r9861421 = 1.0;
        double r9861422 = x;
        double r9861423 = r9861421 / r9861422;
        double r9861424 = tan(r9861422);
        double r9861425 = r9861421 / r9861424;
        double r9861426 = r9861423 - r9861425;
        return r9861426;
}
↓
double f(double x) {
        double r9861427 = x;
        double r9861428 = 5.0;
        double r9861429 = pow(r9861427, r9861428);
        double r9861430 = 0.0021164021164021165;
        double r9861431 = 0.022222222222222223;
        double r9861432 = r9861431 * r9861427;
        double r9861433 = 0.3333333333333333;
        double r9861434 = fma(r9861427, r9861432, r9861433);
        double r9861435 = r9861434 * r9861427;
        double r9861436 = fma(r9861429, r9861430, r9861435);
        return r9861436;
}


\frac{1}{x} - \frac{1}{\tan x}
↓
(\left({x}^{5}\right) \cdot \frac{2}{945} + \left((x \cdot \left(\frac{1}{45} \cdot x\right) + \frac{1}{3})_* \cdot x\right))_*

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

    \[\leadsto \color{blue}{(\left({x}^{5}\right) \cdot \frac{2}{945} + \left(x \cdot (x \cdot \left(\frac{1}{45} \cdot x\right) + \frac{1}{3})_*\right))_*}\]

  4. Final simplification0.3

    \[\leadsto (\left({x}^{5}\right) \cdot \frac{2}{945} + \left((x \cdot \left(\frac{1}{45} \cdot x\right) + \frac{1}{3})_* \cdot x\right))_*\]

Reproduce

herbie shell --seed 2019102 +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))))