Average Error: 39.1 → 0.1
Time: 7.1s
Precision: 64
Internal Precision: 1344
$\sqrt{x + 4} - 2$
$\begin{array}{l} \mathbf{if}\;x \le 0.0007696105650111537:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\frac{1}{512} \cdot x - \frac{1}{64}\right) + \frac{1}{4} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 4} - 2\\ \end{array}$

# Try it out

Results

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# Derivation

1. Split input into 2 regimes
2. ## if x < 0.0007696105650111537

1. Initial program 58.9

$\sqrt{x + 4} - 2$
2. Taylor expanded around 0 0.2

$\leadsto \color{blue}{\left(\frac{1}{512} \cdot {x}^{3} + \frac{1}{4} \cdot x\right) - \frac{1}{64} \cdot {x}^{2}}$
3. Simplified0.2

$\leadsto \color{blue}{\frac{1}{4} \cdot x + \left(\frac{1}{512} \cdot x - \frac{1}{64}\right) \cdot \left(x \cdot x\right)}$

## if 0.0007696105650111537 < x

1. Initial program 0.1

$\sqrt{x + 4} - 2$
3. Recombined 2 regimes into one program.
4. Final simplification0.1

$\leadsto \begin{array}{l} \mathbf{if}\;x \le 0.0007696105650111537:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\frac{1}{512} \cdot x - \frac{1}{64}\right) + \frac{1}{4} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 4} - 2\\ \end{array}$

# Runtime

Time bar (total: 7.1s)Debug log

herbie shell --seed '#(2775764126 3555076145 3898259844 1891440260 2599947619 1948460636)'
(FPCore (x)
:name "sqrt(x+4)-2"
(- (sqrt (+ x 4)) 2))