Average Error: 29.9 → 0.1
Time: 27.8s
Precision: 64
Internal Precision: 1344
$\sqrt{x \cdot x + x} - x$
$\begin{array}{l} \mathbf{if}\;x \le -1.3602604604102958 \cdot 10^{+154}:\\ \;\;\;\;\left(\frac{\frac{1}{8}}{x} - x\right) - \left(x + \frac{1}{2}\right)\\ \mathbf{if}\;x \le 164999.76010174042:\\ \;\;\;\;\sqrt{x \cdot x + x} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} - \frac{\frac{1}{8}}{x}\\ \end{array}$

# Try it out

Results

 In Out
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# Derivation

1. Split input into 3 regimes
2. ## if x < -1.3602604604102958e+154

1. Initial program 59.6

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

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

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

## if -1.3602604604102958e+154 < x < 164999.76010174042

1. Initial program 0.1

$\sqrt{x \cdot x + x} - x$

## if 164999.76010174042 < x

1. Initial program 60.3

$\sqrt{x \cdot x + x} - x$
2. Taylor expanded around inf 60.3

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

$\leadsto \color{blue}{\frac{1}{2} - \frac{\frac{1}{8}}{x}}$
3. Recombined 3 regimes into one program.

# Runtime

Time bar (total: 27.8s)Debug log

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