Average Error: 33.9 → 23.8
Time: 10.1s
Precision: 64
Internal Precision: 2368
$\sqrt{x \cdot x + y \cdot y} - x$
$\begin{array}{l} \mathbf{if}\;x \le -5.147403001023096 \cdot 10^{+84}:\\ \;\;\;\;\left(-x\right) - x\\ \mathbf{elif}\;x \le 6.294595312968477 \cdot 10^{+146}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y} - x\\ \mathbf{else}:\\ \;\;\;\;x - x\\ \end{array}$

# Try it out

Results

 In Out
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# Derivation

1. Split input into 3 regimes
2. ## if x < -5.147403001023096e+84

1. Initial program 45.3

$\sqrt{x \cdot x + y \cdot y} - x$
2. Taylor expanded around -inf 10.5

$\leadsto \color{blue}{-1 \cdot x} - x$
3. Simplified10.5

$\leadsto \color{blue}{\left(-x\right)} - x$

## if -5.147403001023096e+84 < x < 6.294595312968477e+146

1. Initial program 25.4

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

## if 6.294595312968477e+146 < x

1. Initial program 60.9

$\sqrt{x \cdot x + y \cdot y} - x$
2. Taylor expanded around inf 34.4

$\leadsto \color{blue}{x} - x$
3. Recombined 3 regimes into one program.
4. Final simplification23.8

$\leadsto \begin{array}{l} \mathbf{if}\;x \le -5.147403001023096 \cdot 10^{+84}:\\ \;\;\;\;\left(-x\right) - x\\ \mathbf{elif}\;x \le 6.294595312968477 \cdot 10^{+146}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y} - x\\ \mathbf{else}:\\ \;\;\;\;x - x\\ \end{array}$

# Runtime

Time bar (total: 10.1s)Debug log

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