Average Error: 14.5 → 7.4
Time: 13.5s
Precision: 64
Internal Precision: 576
$\sqrt{x \cdot \left(expt \cdot 2\right) + y \cdot \left(expt \cdot 2\right)}$
$\begin{array}{l} \mathbf{if}\;expt \le 8.6675727301626 \cdot 10^{-302}:\\ \;\;\;\;\sqrt{\left(expt \cdot 2\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{y + x} \cdot \sqrt{expt \cdot 2}\\ \end{array}$

# Try it out

Results

 In Out
Enter valid numbers for all inputs

# Derivation

1. Split input into 2 regimes
2. ## if expt < 8.6675727301626e-302

1. Initial program 14.2

$\sqrt{x \cdot \left(expt \cdot 2\right) + y \cdot \left(expt \cdot 2\right)}$
2. Initial simplification14.2

$\leadsto \sqrt{\left(x + y\right) \cdot \left(expt \cdot 2\right)}$

## if 8.6675727301626e-302 < expt

1. Initial program 14.8

$\sqrt{x \cdot \left(expt \cdot 2\right) + y \cdot \left(expt \cdot 2\right)}$
2. Using strategy rm
3. Applied distribute-rgt-out14.8

$\leadsto \sqrt{\color{blue}{\left(expt \cdot 2\right) \cdot \left(x + y\right)}}$
4. Applied sqrt-prod0.4

$\leadsto \color{blue}{\sqrt{expt \cdot 2} \cdot \sqrt{x + y}}$
3. Recombined 2 regimes into one program.
4. Final simplification7.4

$\leadsto \begin{array}{l} \mathbf{if}\;expt \le 8.6675727301626 \cdot 10^{-302}:\\ \;\;\;\;\sqrt{\left(expt \cdot 2\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{y + x} \cdot \sqrt{expt \cdot 2}\\ \end{array}$

# Runtime

Time bar (total: 13.5s)Debug log

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