Average Error: 29.1 → 0.5
Time: 46.1s
Precision: 64
Internal Precision: 1344
$\frac{\frac{1}{\log x} - \frac{1}{\log \left(x + 300\right)}}{700}$
$\begin{array}{l} \mathbf{if}\;\left(\left(\frac{\frac{1}{700}}{\log x} - \frac{\frac{1}{700}}{\log x}\right) - \frac{\frac{\frac{\frac{900}{7}}{x}}{x}}{{\left(\log x\right)}^{3}}\right) - \left(\frac{\frac{450}{7}}{x \cdot x} - \frac{\frac{3}{7}}{x}\right) \cdot \frac{\frac{1}{\log x}}{\log x} \le -8.90664990904548 \cdot 10^{-210}:\\ \;\;\;\;\frac{\frac{1}{\log x}}{700} - \frac{\frac{1}{\log \left(x + 300\right)}}{700}\\ \mathbf{if}\;\left(\left(\frac{\frac{1}{700}}{\log x} - \frac{\frac{1}{700}}{\log x}\right) - \frac{\frac{\frac{\frac{900}{7}}{x}}{x}}{{\left(\log x\right)}^{3}}\right) - \left(\frac{\frac{450}{7}}{x \cdot x} - \frac{\frac{3}{7}}{x}\right) \cdot \frac{\frac{1}{\log x}}{\log x} \le 9.48947905936255 \cdot 10^{-12}:\\ \;\;\;\;\left(\left(\frac{\frac{\frac{\frac{270000}{7}}{x}}{x \cdot x}}{{\left(-\log x\right)}^{4}} - \frac{\frac{\frac{\frac{270000}{7}}{x}}{x \cdot x}}{\left(\log x \cdot \log x\right) \cdot \left(-\log x\right)}\right) + \left(\frac{\frac{\frac{900}{7}}{\log x \cdot \log x}}{\left(-\log x\right) \cdot \left(x \cdot x\right)} + \left(\frac{\frac{\frac{3}{7}}{x}}{\log x \cdot \log x} + \frac{\frac{\frac{\frac{90000}{7}}{x}}{x \cdot x}}{\log x \cdot \log x}\right)\right)\right) - \frac{\frac{\frac{450}{7}}{x \cdot x}}{\log x \cdot \log x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\log x}}{700} - \frac{\frac{1}{\log \left(x + 300\right)}}{700}\\ \end{array}$

# Try it out

Results

 In Out
Enter valid numbers for all inputs

# Derivation

1. Split input into 2 regimes
2. ## if (- (- (- (/ (/ 1 700) (log x)) (/ 1/700 (+ 0 (log x)))) (/ (/ (/ 900/7 x) x) (pow (+ 0 (log x)) 3))) (* (/ (/ 1 (+ 0 (log x))) (+ 0 (log x))) (- (/ 450/7 (* x x)) (/ 3/7 x)))) < -8.90664990904548e-210 or 9.48947905936255e-12 < (- (- (- (/ (/ 1 700) (log x)) (/ 1/700 (+ 0 (log x)))) (/ (/ (/ 900/7 x) x) (pow (+ 0 (log x)) 3))) (* (/ (/ 1 (+ 0 (log x))) (+ 0 (log x))) (- (/ 450/7 (* x x)) (/ 3/7 x))))

1. Initial program 0.4

$\frac{\frac{1}{\log x} - \frac{1}{\log \left(x + 300\right)}}{700}$
2. Using strategy rm
3. Applied div-sub0.4

$\leadsto \color{blue}{\frac{\frac{1}{\log x}}{700} - \frac{\frac{1}{\log \left(x + 300\right)}}{700}}$

## if -8.90664990904548e-210 < (- (- (- (/ (/ 1 700) (log x)) (/ 1/700 (+ 0 (log x)))) (/ (/ (/ 900/7 x) x) (pow (+ 0 (log x)) 3))) (* (/ (/ 1 (+ 0 (log x))) (+ 0 (log x))) (- (/ 450/7 (* x x)) (/ 3/7 x)))) < 9.48947905936255e-12

1. Initial program 59.3

$\frac{\frac{1}{\log x} - \frac{1}{\log \left(x + 300\right)}}{700}$
2. Taylor expanded around inf 1.5

$\leadsto \color{blue}{\left(\frac{900}{7} \cdot \frac{1}{{\left(\log \left(\frac{1}{x}\right)\right)}^{3} \cdot {x}^{2}} + \left(\frac{3}{7} \cdot \frac{1}{{\left(\log \left(\frac{1}{x}\right)\right)}^{2} \cdot x} + \left(\frac{270000}{7} \cdot \frac{1}{{\left(\log \left(\frac{1}{x}\right)\right)}^{4} \cdot {x}^{3}} + \frac{90000}{7} \cdot \frac{1}{{\left(\log \left(\frac{1}{x}\right)\right)}^{2} \cdot {x}^{3}}\right)\right)\right) - \left(\frac{450}{7} \cdot \frac{1}{{\left(\log \left(\frac{1}{x}\right)\right)}^{2} \cdot {x}^{2}} + \frac{270000}{7} \cdot \frac{1}{{\left(\log \left(\frac{1}{x}\right)\right)}^{3} \cdot {x}^{3}}\right)}$
3. Applied simplify0.6

$\leadsto \color{blue}{\left(\left(\frac{\frac{\frac{900}{7}}{\log x \cdot \log x}}{\left(-\log x\right) \cdot \left(x \cdot x\right)} + \left(\frac{\frac{\frac{3}{7}}{x}}{\log x \cdot \log x} + \frac{\frac{\frac{\frac{90000}{7}}{x}}{x \cdot x}}{\log x \cdot \log x}\right)\right) + \left(\frac{\frac{\frac{\frac{270000}{7}}{x}}{x \cdot x}}{{\left(-\log x\right)}^{4}} - \frac{\frac{\frac{\frac{270000}{7}}{x}}{x \cdot x}}{\left(\log x \cdot \log x\right) \cdot \left(-\log x\right)}\right)\right) - \frac{\frac{\frac{450}{7}}{x \cdot x}}{\log x \cdot \log x}}$
3. Recombined 2 regimes into one program.
4. Applied simplify0.5

$\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\left(\left(\frac{\frac{1}{700}}{\log x} - \frac{\frac{1}{700}}{\log x}\right) - \frac{\frac{\frac{\frac{900}{7}}{x}}{x}}{{\left(\log x\right)}^{3}}\right) - \left(\frac{\frac{450}{7}}{x \cdot x} - \frac{\frac{3}{7}}{x}\right) \cdot \frac{\frac{1}{\log x}}{\log x} \le -8.90664990904548 \cdot 10^{-210}:\\ \;\;\;\;\frac{\frac{1}{\log x}}{700} - \frac{\frac{1}{\log \left(x + 300\right)}}{700}\\ \mathbf{if}\;\left(\left(\frac{\frac{1}{700}}{\log x} - \frac{\frac{1}{700}}{\log x}\right) - \frac{\frac{\frac{\frac{900}{7}}{x}}{x}}{{\left(\log x\right)}^{3}}\right) - \left(\frac{\frac{450}{7}}{x \cdot x} - \frac{\frac{3}{7}}{x}\right) \cdot \frac{\frac{1}{\log x}}{\log x} \le 9.48947905936255 \cdot 10^{-12}:\\ \;\;\;\;\left(\left(\frac{\frac{\frac{\frac{270000}{7}}{x}}{x \cdot x}}{{\left(-\log x\right)}^{4}} - \frac{\frac{\frac{\frac{270000}{7}}{x}}{x \cdot x}}{\left(\log x \cdot \log x\right) \cdot \left(-\log x\right)}\right) + \left(\frac{\frac{\frac{900}{7}}{\log x \cdot \log x}}{\left(-\log x\right) \cdot \left(x \cdot x\right)} + \left(\frac{\frac{\frac{3}{7}}{x}}{\log x \cdot \log x} + \frac{\frac{\frac{\frac{90000}{7}}{x}}{x \cdot x}}{\log x \cdot \log x}\right)\right)\right) - \frac{\frac{\frac{450}{7}}{x \cdot x}}{\log x \cdot \log x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\log x}}{700} - \frac{\frac{1}{\log \left(x + 300\right)}}{700}\\ \end{array}}$

# Runtime

Time bar (total: 46.1s)Debug log

herbie shell --seed '#(2775764126 3555076145 3898259844 1891440260 2599947619 1948460636)'
(FPCore (x)
:name "(1/(log(x)) - 1/(log(x+300)))/700"
(/ (- (/ 1 (log x)) (/ 1 (log (+ x 300)))) 700))