Average Error: 29.8 → 0.2
Time: 14.9s
Precision: 64
Internal Precision: 1344
$\sqrt{x + 1} - \sqrt{x}$
$\frac{1}{\sqrt{1 + x} + \sqrt{x}}$

# Try it out

Results

 In Out
Enter valid numbers for all inputs

# Derivation

1. Initial program 29.8

$\sqrt{x + 1} - \sqrt{x}$
2. Using strategy rm
3. Applied flip--29.6

$\leadsto \color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}}$
4. Using strategy rm

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

$\leadsto \frac{\color{blue}{\log \left(e^{\sqrt{x + 1} \cdot \sqrt{x + 1}}\right)} - \log \left(e^{\sqrt{x} \cdot \sqrt{x}}\right)}{\sqrt{x + 1} + \sqrt{x}}$
7. Applied diff-log30.9

$\leadsto \frac{\color{blue}{\log \left(\frac{e^{\sqrt{x + 1} \cdot \sqrt{x + 1}}}{e^{\sqrt{x} \cdot \sqrt{x}}}\right)}}{\sqrt{x + 1} + \sqrt{x}}$
8. Simplified0.2

$\leadsto \frac{\log \color{blue}{e}}{\sqrt{x + 1} + \sqrt{x}}$
9. Final simplification0.2

$\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}}$

# Runtime

Time bar (total: 14.9s)Debug log

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