Average Error: 59.9 → 0.4
Time: 1.3m
Precision: 64
\[-0.026 \lt x \land x \lt 0.026\]
\[\frac{1}{x} - \frac{1}{\tan x}\]
↓
\[(\left({x}^{5}\right) \cdot \frac{2}{945} + \left((x \cdot \left(\frac{1}{45} \cdot x\right) + \frac{1}{3})_* \cdot x\right))_*\]
\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 r6456061 = 1.0;
        double r6456062 = x;
        double r6456063 = r6456061 / r6456062;
        double r6456064 = tan(r6456062);
        double r6456065 = r6456061 / r6456064;
        double r6456066 = r6456063 - r6456065;
        return r6456066;
}
↓
double f(double x) {
        double r6456067 = x;
        double r6456068 = 5.0;
        double r6456069 = pow(r6456067, r6456068);
        double r6456070 = 0.0021164021164021165;
        double r6456071 = 0.022222222222222223;
        double r6456072 = r6456071 * r6456067;
        double r6456073 = 0.3333333333333333;
        double r6456074 = fma(r6456067, r6456072, r6456073);
        double r6456075 = r6456074 * r6456067;
        double r6456076 = fma(r6456069, r6456070, r6456075);
        return r6456076;
}

Error

Bits error versus x

Target

Original59.9
Target0.1
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;\left|x\right| \lt 0.026:\\ \;\;\;\;\frac{x}{3} \cdot \left(1 + \frac{x \cdot x}{15}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} - \frac{1}{\tan x}\\ \end{array}\]

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. Simplified0.4

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

  4. Final simplification0.4

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

Reproduce

herbie shell --seed 2019119 +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))))