Average Error: 59.9 → 0.3
Time: 1.4m
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 r6667799 = 1.0;
        double r6667800 = x;
        double r6667801 = r6667799 / r6667800;
        double r6667802 = tan(r6667800);
        double r6667803 = r6667799 / r6667802;
        double r6667804 = r6667801 - r6667803;
        return r6667804;
}
↓
double f(double x) {
        double r6667805 = x;
        double r6667806 = 5.0;
        double r6667807 = pow(r6667805, r6667806);
        double r6667808 = 0.0021164021164021165;
        double r6667809 = 0.022222222222222223;
        double r6667810 = r6667809 * r6667805;
        double r6667811 = 0.3333333333333333;
        double r6667812 = fma(r6667805, r6667810, r6667811);
        double r6667813 = r6667812 * r6667805;
        double r6667814 = fma(r6667807, r6667808, r6667813);
        return r6667814;
}

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 2019121 +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))))