Average Error: 29.0 → 0.4
Time: 49.7s
Precision: 64
Internal Precision: 1344
$\frac{1}{700 \cdot \log x} - \frac{1}{700 \cdot \log \left(x + 500\right)}$
$\begin{array}{l} \mathbf{if}\;x \le 2264569.2881505126:\\ \;\;\;\;\frac{\frac{1}{700}}{\log x} - \frac{1}{700 \cdot \log \left(x + 500\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{\frac{\frac{2500}{7}}{\log x \cdot \log x}}{\left(-\log x\right) \cdot \left(x \cdot x\right)} + \left(\frac{\frac{\frac{5}{7}}{x}}{\log x \cdot \log x} + \frac{\frac{\frac{\frac{1250000}{21}}{x}}{x \cdot x}}{\log x \cdot \log x}\right)\right) + \left(\frac{\frac{\frac{\frac{1250000}{7}}{x}}{x \cdot x}}{{\left(-\log x\right)}^{4}} - \frac{\frac{\frac{\frac{1250000}{7}}{x}}{x \cdot x}}{\left(\log x \cdot \log x\right) \cdot \left(-\log x\right)}\right)\right) - \frac{\frac{\frac{1250}{7}}{x \cdot x}}{\log x \cdot \log x}\\ \end{array}$

# Try it out

Results

 In Out
Enter valid numbers for all inputs

# Derivation

1. Split input into 2 regimes
2. ## if x < 2264569.2881505126

1. Initial program 0.2

$\frac{1}{700 \cdot \log x} - \frac{1}{700 \cdot \log \left(x + 500\right)}$
2. Using strategy rm
3. Applied associate-/r*0.2

$\leadsto \color{blue}{\frac{\frac{1}{700}}{\log x}} - \frac{1}{700 \cdot \log \left(x + 500\right)}$

## if 2264569.2881505126 < x

1. Initial program 59.1

$\frac{1}{700 \cdot \log x} - \frac{1}{700 \cdot \log \left(x + 500\right)}$
2. Taylor expanded around inf 1.5

$\leadsto \color{blue}{\left(\frac{2500}{7} \cdot \frac{1}{{\left(\log \left(\frac{1}{x}\right)\right)}^{3} \cdot {x}^{2}} + \left(\frac{5}{7} \cdot \frac{1}{{\left(\log \left(\frac{1}{x}\right)\right)}^{2} \cdot x} + \left(\frac{1250000}{7} \cdot \frac{1}{{\left(\log \left(\frac{1}{x}\right)\right)}^{4} \cdot {x}^{3}} + \frac{1250000}{21} \cdot \frac{1}{{\left(\log \left(\frac{1}{x}\right)\right)}^{2} \cdot {x}^{3}}\right)\right)\right) - \left(\frac{1250}{7} \cdot \frac{1}{{\left(\log \left(\frac{1}{x}\right)\right)}^{2} \cdot {x}^{2}} + \frac{1250000}{7} \cdot \frac{1}{{\left(\log \left(\frac{1}{x}\right)\right)}^{3} \cdot {x}^{3}}\right)}$
3. Applied simplify0.6

$\leadsto \color{blue}{\left(\left(\frac{\frac{\frac{2500}{7}}{\log x \cdot \log x}}{\left(-\log x\right) \cdot \left(x \cdot x\right)} + \left(\frac{\frac{\frac{5}{7}}{x}}{\log x \cdot \log x} + \frac{\frac{\frac{\frac{1250000}{21}}{x}}{x \cdot x}}{\log x \cdot \log x}\right)\right) + \left(\frac{\frac{\frac{\frac{1250000}{7}}{x}}{x \cdot x}}{{\left(-\log x\right)}^{4}} - \frac{\frac{\frac{\frac{1250000}{7}}{x}}{x \cdot x}}{\left(\log x \cdot \log x\right) \cdot \left(-\log x\right)}\right)\right) - \frac{\frac{\frac{1250}{7}}{x \cdot x}}{\log x \cdot \log x}}$
3. Recombined 2 regimes into one program.

# Runtime

Time bar (total: 49.7s)Debug log

herbie shell --seed '#(2775764126 3555076145 3898259844 1891440260 2599947619 1948460636)'
(FPCore (x)
:name "1/(700 * log(x)) - 1/(700*log(x+500))"
(- (/ 1 (* 700 (log x))) (/ 1 (* 700 (log (+ x 500))))))