Average Error: 59.8 → 0.4
Time: 11.6s
Precision: 64
\[-0.0259999999999999988 \lt x \land x \lt 0.0259999999999999988\]
\[\frac{1}{x} - \frac{1}{\tan x}\]
↓
\[\mathsf{fma}\left(0.0222222222222222231, {x}^{3}, \mathsf{fma}\left(0.00211640211640211654, {x}^{5}, 0.333333333333333315 \cdot x\right)\right)\]
\frac{1}{x} - \frac{1}{\tan x}
↓
\mathsf{fma}\left(0.0222222222222222231, {x}^{3}, \mathsf{fma}\left(0.00211640211640211654, {x}^{5}, 0.333333333333333315 \cdot x\right)\right)
double f(double x) {
        double r106135 = 1.0;
        double r106136 = x;
        double r106137 = r106135 / r106136;
        double r106138 = tan(r106136);
        double r106139 = r106135 / r106138;
        double r106140 = r106137 - r106139;
        return r106140;
}
↓
double f(double x) {
        double r106141 = 0.022222222222222223;
        double r106142 = x;
        double r106143 = 3.0;
        double r106144 = pow(r106142, r106143);
        double r106145 = 0.0021164021164021165;
        double r106146 = 5.0;
        double r106147 = pow(r106142, r106146);
        double r106148 = 0.3333333333333333;
        double r106149 = r106148 * r106142;
        double r106150 = fma(r106145, r106147, r106149);
        double r106151 = fma(r106141, r106144, r106150);
        return r106151;
}

Error

Bits error versus x

Target

Original59.8
Target0.1
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;\left|x\right| \lt 0.0259999999999999988:\\ \;\;\;\;\frac{x}{3} \cdot \left(1 + \frac{x \cdot x}{15}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} - \frac{1}{\tan x}\\ \end{array}\]

Derivation

  1. Initial program 59.8

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

  2. Taylor expanded around 0 0.4

    \[\leadsto \color{blue}{0.0222222222222222231 \cdot {x}^{3} + \left(0.00211640211640211654 \cdot {x}^{5} + 0.333333333333333315 \cdot x\right)}\]

  3. Simplified0.4

    \[\leadsto \color{blue}{\mathsf{fma}\left(0.0222222222222222231, {x}^{3}, \mathsf{fma}\left(0.00211640211640211654, {x}^{5}, 0.333333333333333315 \cdot x\right)\right)}\]

  4. Final simplification0.4

    \[\leadsto \mathsf{fma}\left(0.0222222222222222231, {x}^{3}, \mathsf{fma}\left(0.00211640211640211654, {x}^{5}, 0.333333333333333315 \cdot x\right)\right)\]

Reproduce

herbie shell --seed 2020057 +o rules:numerics
(FPCore (x)
  :name "invcot (example 3.9)"
  :precision binary64
  :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))))