Average Error: 29.2 → 19.2
Time: 27.1s
Precision: 64
Internal Precision: 1344
$\left(\left(e^{x} - 1\right) + \sqrt{y + 1}\right) - \sqrt{y}$
$\begin{array}{l} \mathbf{if}\;y \le 6698305061578466.0:\\ \;\;\;\;\frac{\left(\left(e^{x} - 1\right) + \sqrt{y + 1}\right) \cdot \left(\left(e^{x} - 1\right) + \sqrt{y + 1}\right) - \sqrt{y} \cdot \sqrt{y}}{\left(\left(e^{x} - 1\right) + \sqrt{y + 1}\right) + \sqrt{y}}\\ \mathbf{else}:\\ \;\;\;\;\left(e^{x} - 1\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\\ \end{array}$

# Try it out

Results

 In Out
Enter valid numbers for all inputs

# Derivation

1. Split input into 2 regimes
2. ## if y < 6698305061578466.0

1. Initial program 1.4

$\left(\left(e^{x} - 1\right) + \sqrt{y + 1}\right) - \sqrt{y}$
2. Using strategy rm
3. Applied flip--1.1

$\leadsto \color{blue}{\frac{\left(\left(e^{x} - 1\right) + \sqrt{y + 1}\right) \cdot \left(\left(e^{x} - 1\right) + \sqrt{y + 1}\right) - \sqrt{y} \cdot \sqrt{y}}{\left(\left(e^{x} - 1\right) + \sqrt{y + 1}\right) + \sqrt{y}}}$

## if 6698305061578466.0 < y

1. Initial program 60.6

$\left(\left(e^{x} - 1\right) + \sqrt{y + 1}\right) - \sqrt{y}$
2. Using strategy rm
3. Applied associate--l+39.6

$\leadsto \color{blue}{\left(e^{x} - 1\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)}$
3. Recombined 2 regimes into one program.
4. Final simplification19.2

$\leadsto \begin{array}{l} \mathbf{if}\;y \le 6698305061578466.0:\\ \;\;\;\;\frac{\left(\left(e^{x} - 1\right) + \sqrt{y + 1}\right) \cdot \left(\left(e^{x} - 1\right) + \sqrt{y + 1}\right) - \sqrt{y} \cdot \sqrt{y}}{\left(\left(e^{x} - 1\right) + \sqrt{y + 1}\right) + \sqrt{y}}\\ \mathbf{else}:\\ \;\;\;\;\left(e^{x} - 1\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\\ \end{array}$

# Runtime

Time bar (total: 27.1s)Debug log

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