Average Error: 29.4 → 0.1
Time: 23.2s
Precision: 64
Internal Precision: 1344
$\log \left(x + 1\right) - \log x$
$\begin{array}{l} \mathbf{if}\;x \le 8446.004109786509:\\ \;\;\;\;\log \left(\frac{1 + x}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{\frac{\frac{1}{3}}{x}}{x} + 1\right) - \frac{\frac{1}{2}}{x}}{x}\\ \end{array}$

# Try it out

Results

 In Out
Enter valid numbers for all inputs

# Derivation

1. Split input into 2 regimes
2. ## if x < 8446.004109786509

1. Initial program 0.1

$\log \left(x + 1\right) - \log x$
2. Initial simplification0.1

$\leadsto \log \left(1 + x\right) - \log x$
3. Using strategy rm
4. Applied diff-log0.1

$\leadsto \color{blue}{\log \left(\frac{1 + x}{x}\right)}$

## if 8446.004109786509 < x

1. Initial program 59.6

$\log \left(x + 1\right) - \log x$
2. Initial simplification59.6

$\leadsto \log \left(1 + x\right) - \log x$
3. Taylor expanded around inf 0.0

$\leadsto \color{blue}{\left(\frac{1}{x} + \frac{1}{3} \cdot \frac{1}{{x}^{3}}\right) - \frac{1}{2} \cdot \frac{1}{{x}^{2}}}$
4. Simplified0.0

$\leadsto \color{blue}{\frac{\frac{\frac{\frac{1}{3}}{x}}{x} + \left(1 - \frac{\frac{1}{2}}{x}\right)}{x}}$
5. Using strategy rm
6. Applied associate-+r-0.0

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

$\leadsto \begin{array}{l} \mathbf{if}\;x \le 8446.004109786509:\\ \;\;\;\;\log \left(\frac{1 + x}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{\frac{\frac{1}{3}}{x}}{x} + 1\right) - \frac{\frac{1}{2}}{x}}{x}\\ \end{array}$

# Runtime

Time bar (total: 23.2s)Debug log

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