Average Error: 15.3 → 0.0
Time: 34.9s
Precision: 64
Internal Precision: 320
$\frac{x}{\sqrt{x \cdot x + 1}}$
$\begin{array}{l} \mathbf{if}\;x \le -1.3352415157792068 \cdot 10^{+154}:\\ \;\;\;\;\frac{x}{\frac{\frac{\frac{1}{8}}{x}}{x \cdot x} - \left(\frac{\frac{1}{2}}{x} + x\right)}\\ \mathbf{if}\;x \le 296.75299396492323:\\ \;\;\;\;\frac{x}{\sqrt{x \cdot x + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(x + \frac{\frac{1}{2}}{x}\right) - \frac{\frac{\frac{1}{8}}{x}}{x \cdot x}}\\ \end{array}$

# Try it out

Results

 In Out
Enter valid numbers for all inputs

# Derivation

1. Split input into 3 regimes
2. ## if x < -1.3352415157792068e+154

1. Initial program 62.0

$\frac{x}{\sqrt{x \cdot x + 1}}$
2. Taylor expanded around -inf 0

$\leadsto \frac{x}{\color{blue}{\frac{1}{8} \cdot \frac{1}{{x}^{3}} - \left(\frac{1}{2} \cdot \frac{1}{x} + x\right)}}$
3. Applied simplify0

$\leadsto \color{blue}{\frac{x}{\frac{\frac{\frac{1}{8}}{x}}{x \cdot x} - \left(\frac{\frac{1}{2}}{x} + x\right)}}$

## if -1.3352415157792068e+154 < x < 296.75299396492323

1. Initial program 0.0

$\frac{x}{\sqrt{x \cdot x + 1}}$

## if 296.75299396492323 < x

1. Initial program 30.0

$\frac{x}{\sqrt{x \cdot x + 1}}$
2. Taylor expanded around inf 0.0

$\leadsto \frac{x}{\color{blue}{\left(\frac{1}{2} \cdot \frac{1}{x} + x\right) - \frac{1}{8} \cdot \frac{1}{{x}^{3}}}}$
3. Applied simplify0.0

$\leadsto \color{blue}{\frac{x}{\left(x + \frac{\frac{1}{2}}{x}\right) - \frac{\frac{\frac{1}{8}}{x}}{x \cdot x}}}$
3. Recombined 3 regimes into one program.

# Runtime

Time bar (total: 34.9s)Debug log

herbie shell --seed '#(2775764126 3555076145 3898259844 1891440260 2599947619 1948460636)'
(FPCore (x)
:name "x/sqrt(x*x+1)"
(/ x (sqrt (+ (* x x) 1))))