Average Error: 21.3 → 0.8
Time: 25.3s
Precision: 64
Internal Precision: 2368
$\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{1 + x \cdot x}}\right)}$
$\begin{array}{l} \mathbf{if}\;x \le -0.6974219668002979:\\ \;\;\;\;\left(\frac{0.00439453125}{0.25} - 0.078125\right) \cdot \frac{\frac{1}{\sqrt{0.25}}}{{x}^{5}} + \left(\frac{\frac{0.09375}{x \cdot x}}{\sqrt{0.25} \cdot x} - \frac{\sqrt{0.25}}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(\frac{x}{\left(x + \frac{\frac{1}{2}}{x}\right) - \frac{\frac{\frac{1}{8}}{x}}{x \cdot x}} + 1\right)}\\ \end{array}$

# Try it out

Results

 In Out
Enter valid numbers for all inputs

# Derivation

1. Split input into 2 regimes
2. ## if x < -0.6974219668002979

1. Initial program 60.6

$\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{1 + x \cdot x}}\right)}$
2. Taylor expanded around -inf 0.1

$\leadsto \color{blue}{\left(0.09375 \cdot \frac{1}{\sqrt{0.25} \cdot {x}^{3}} + 0.00439453125 \cdot \frac{1}{{\left(\sqrt{0.25}\right)}^{3} \cdot {x}^{5}}\right) - \left(0.078125 \cdot \frac{1}{\sqrt{0.25} \cdot {x}^{5}} + \frac{\sqrt{0.25}}{x}\right)}$
3. Simplified0.1

$\leadsto \color{blue}{\frac{\frac{1}{\sqrt{0.25}}}{{x}^{5}} \cdot \left(\frac{0.00439453125}{0.25} - 0.078125\right) + \left(\frac{\frac{0.09375}{x \cdot x}}{\sqrt{0.25} \cdot x} - \frac{\sqrt{0.25}}{x}\right)}$

## if -0.6974219668002979 < x

1. Initial program 8.3

$\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{1 + x \cdot x}}\right)}$
2. Taylor expanded around inf 1.0

$\leadsto \sqrt{0.5 \cdot \left(1 + \frac{x}{\color{blue}{\left(x + \frac{1}{2} \cdot \frac{1}{x}\right) - \frac{1}{8} \cdot \frac{1}{{x}^{3}}}}\right)}$
3. Simplified1.0

$\leadsto \sqrt{0.5 \cdot \left(1 + \frac{x}{\color{blue}{\left(x + \frac{\frac{1}{2}}{x}\right) - \frac{\frac{\frac{1}{8}}{x}}{x \cdot x}}}\right)}$
3. Recombined 2 regimes into one program.
4. Final simplification0.8

$\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.6974219668002979:\\ \;\;\;\;\left(\frac{0.00439453125}{0.25} - 0.078125\right) \cdot \frac{\frac{1}{\sqrt{0.25}}}{{x}^{5}} + \left(\frac{\frac{0.09375}{x \cdot x}}{\sqrt{0.25} \cdot x} - \frac{\sqrt{0.25}}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(\frac{x}{\left(x + \frac{\frac{1}{2}}{x}\right) - \frac{\frac{\frac{1}{8}}{x}}{x \cdot x}} + 1\right)}\\ \end{array}$

# Runtime

Time bar (total: 25.3s)Debug log

herbie shell --seed '#(2775764126 3555076145 3898259844 1891440260 2599947619 1948460636)'
(FPCore (x)
:name "sqrt( 0.5 * (1 + x/(sqrt(1 +x*x))))"
(sqrt (* 0.5 (+ 1 (/ x (sqrt (+ 1 (* x x))))))))