Average Error: 5.8 → 3.6
Time: 21.8s
Precision: 64
Internal Precision: 1344
$\frac{1}{{\left(1 + x\right)}^{n}}$
$\begin{array}{l} \mathbf{if}\;\frac{1}{\left(n \cdot x + 1\right) - \frac{1}{2} \cdot \left(n \cdot {x}^{2}\right)} \le 0.00021405454233365702:\\ \;\;\;\;{x}^{\left(-n\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\sqrt[3]{{\left(1 + x\right)}^{n}} \cdot \sqrt[3]{{\left(1 + x\right)}^{n}}\right) \cdot \sqrt[3]{{\left(1 + x\right)}^{n}}}\\ \end{array}$

# Try it out

Results

 In Out
Enter valid numbers for all inputs

# Derivation

1. Split input into 2 regimes
2. ## if (/ 1 (- (+ (* n x) 1) (* 1/2 (* n (pow x 2))))) < 0.00021405454233365702

1. Initial program 14.0

$\frac{1}{{\left(1 + x\right)}^{n}}$
2. Taylor expanded around inf 31.5

$\leadsto \frac{1}{\color{blue}{e^{-1 \cdot \left(n \cdot \log \left(\frac{1}{x}\right)\right)}}}$
3. Applied simplify8.1

$\leadsto \color{blue}{{x}^{\left(-n\right)}}$

## if 0.00021405454233365702 < (/ 1 (- (+ (* n x) 1) (* 1/2 (* n (pow x 2)))))

1. Initial program 1.0

$\frac{1}{{\left(1 + x\right)}^{n}}$
2. Using strategy rm
$\leadsto \frac{1}{\color{blue}{\left(\sqrt[3]{{\left(1 + x\right)}^{n}} \cdot \sqrt[3]{{\left(1 + x\right)}^{n}}\right) \cdot \sqrt[3]{{\left(1 + x\right)}^{n}}}}$
herbie shell --seed '#(2775764126 3555076145 3898259844 1891440260 2599947619 1948460636)'