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

↓

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

double f(double x) {
        double r6335203 = 1.0;
        double r6335204 = x;
        double r6335205 = r6335203 / r6335204;
        double r6335206 = tan(r6335204);
        double r6335207 = r6335203 / r6335206;
        double r6335208 = r6335205 - r6335207;
        return r6335208;
}

↓

double f(double x) {
        double r6335209 = x;
        double r6335210 = 5.0;
        double r6335211 = pow(r6335209, r6335210);
        double r6335212 = 0.0021164021164021165;
        double r6335213 = 0.022222222222222223;
        double r6335214 = r6335213 * r6335209;
        double r6335215 = 0.3333333333333333;
        double r6335216 = fma(r6335209, r6335214, r6335215);
        double r6335217 = r6335216 * r6335216;
        double r6335218 = r6335216 * r6335217;
        double r6335219 = cbrt(r6335218);
        double r6335220 = r6335209 * r6335219;
        double r6335221 = fma(r6335211, r6335212, r6335220);
        return r6335221;
}

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

↓

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

Original	59.9
Target	0.1
Herbie	0.3

Derivation

Initial program 59.9
\[\frac{1}{x} - \frac{1}{\tan x}\]
Taylor expanded around 0 0.3
\[\leadsto \color{blue}{\frac{1}{3} \cdot x + \left(\frac{1}{45} \cdot {x}^{3} + \frac{2}{945} \cdot {x}^{5}\right)}\]
Simplified0.3
\[\leadsto \color{blue}{(\left({x}^{5}\right) \cdot \frac{2}{945} + \left(x \cdot (x \cdot \left(\frac{1}{45} \cdot x\right) + \frac{1}{3})_*\right))_*}\]
Using strategy rm
Applied add-cbrt-cube0.3
\[\leadsto (\left({x}^{5}\right) \cdot \frac{2}{945} + \left(x \cdot \color{blue}{\sqrt[3]{\left((x \cdot \left(\frac{1}{45} \cdot x\right) + \frac{1}{3})_* \cdot (x \cdot \left(\frac{1}{45} \cdot x\right) + \frac{1}{3})_*\right) \cdot (x \cdot \left(\frac{1}{45} \cdot x\right) + \frac{1}{3})_*}}\right))_*\]
Final simplification0.3
\[\leadsto (\left({x}^{5}\right) \cdot \frac{2}{945} + \left(x \cdot \sqrt[3]{(x \cdot \left(\frac{1}{45} \cdot x\right) + \frac{1}{3})_* \cdot \left((x \cdot \left(\frac{1}{45} \cdot x\right) + \frac{1}{3})_* \cdot (x \cdot \left(\frac{1}{45} \cdot x\right) + \frac{1}{3})_*\right)}\right))_*\]

Reproduce

herbie shell --seed 2019101 +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))))

Error

Target

Derivation

Reproduce