Average Error: 5.6 → 3.6
Time: 26.7s
Precision: 64
Internal Precision: 1344
${\left(1 + x\right)}^{n}$
$\begin{array}{l} \mathbf{if}\;\sqrt{\left(n \cdot x + 1\right) - \frac{1}{2} \cdot \left(n \cdot {x}^{2}\right)} \cdot \sqrt{\left(n \cdot x + 1\right) - \frac{1}{2} \cdot \left(n \cdot {x}^{2}\right)} \le 1066442948.7059948:\\ \;\;\;\;\left(n \cdot x + 1\right) - \frac{1}{2} \cdot \left(n \cdot {x}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{n}\\ \end{array}$

# Try it out

Results

 In Out
Enter valid numbers for all inputs

# Derivation

1. Split input into 2 regimes
2. ## if (* (sqrt (- (+ (* n x) 1) (* 1/2 (* n (pow x 2))))) (sqrt (- (+ (* n x) 1) (* 1/2 (* n (pow x 2)))))) < 1066442948.7059948

1. Initial program 0.9

${\left(1 + x\right)}^{n}$
2. Taylor expanded around 0 0.8

$\leadsto \color{blue}{\left(n \cdot x + 1\right) - \frac{1}{2} \cdot \left(n \cdot {x}^{2}\right)}$

## if 1066442948.7059948 < (* (sqrt (- (+ (* n x) 1) (* 1/2 (* n (pow x 2))))) (sqrt (- (+ (* n x) 1) (* 1/2 (* n (pow x 2))))))

1. Initial program 12.2

${\left(1 + x\right)}^{n}$
2. Taylor expanded around inf 26.9

$\leadsto \color{blue}{e^{-1 \cdot \left(n \cdot \log \left(\frac{1}{x}\right)\right)}}$
3. Applied simplify7.5

$\leadsto \color{blue}{{x}^{n}}$
3. Recombined 2 regimes into one program.

# Runtime

Time bar (total: 26.7s)Debug log

herbie shell --seed '#(2775764126 3555076145 3898259844 1891440260 2599947619 1948460636)'
(FPCore (x n)
:name "(1 + x)^n"
(pow (+ 1 x) n))