Average Error: 39.4 → 11.3
Time: 16.6s
Precision: 64
$x \gt 0.0$
$\frac{\sqrt{x - 1}}{x + 1} - \frac{\sqrt{x + 1}}{x - 1}$
$\frac{-\left(\left(\frac{14}{{x}^{4}} + \frac{22}{{x}^{6}}\right) + \frac{6}{x \cdot x}\right)}{\frac{\sqrt{x - 1}}{x + 1} + \frac{\sqrt{x + 1}}{x - 1}}$
\frac{\sqrt{x - 1}}{x + 1} - \frac{\sqrt{x + 1}}{x - 1}
\frac{-\left(\left(\frac{14}{{x}^{4}} + \frac{22}{{x}^{6}}\right) + \frac{6}{x \cdot x}\right)}{\frac{\sqrt{x - 1}}{x + 1} + \frac{\sqrt{x + 1}}{x - 1}}
double f(double x) {
double r74312 = x;
double r74313 = 1.0;
double r74314 = r74312 - r74313;
double r74315 = sqrt(r74314);
double r74316 = r74312 + r74313;
double r74317 = r74315 / r74316;
double r74318 = sqrt(r74316);
double r74319 = r74318 / r74314;
double r74320 = r74317 - r74319;
return r74320;
}


double f(double x) {
double r74321 = 14.0;
double r74322 = x;
double r74323 = 4.0;
double r74324 = pow(r74322, r74323);
double r74325 = r74321 / r74324;
double r74326 = 22.0;
double r74327 = 6.0;
double r74328 = pow(r74322, r74327);
double r74329 = r74326 / r74328;
double r74330 = r74325 + r74329;
double r74331 = 6.0;
double r74332 = r74322 * r74322;
double r74333 = r74331 / r74332;
double r74334 = r74330 + r74333;
double r74335 = -r74334;
double r74336 = 1.0;
double r74337 = r74322 - r74336;
double r74338 = sqrt(r74337);
double r74339 = r74322 + r74336;
double r74340 = r74338 / r74339;
double r74341 = sqrt(r74339);
double r74342 = r74341 / r74337;
double r74343 = r74340 + r74342;
double r74344 = r74335 / r74343;
return r74344;
}



# Try it out

Results

 In Out
Enter valid numbers for all inputs

# Derivation

1. Initial program 39.4

$\frac{\sqrt{x - 1}}{x + 1} - \frac{\sqrt{x + 1}}{x - 1}$
2. Using strategy rm
3. Applied flip--39.4

$\leadsto \color{blue}{\frac{\frac{\sqrt{x - 1}}{x + 1} \cdot \frac{\sqrt{x - 1}}{x + 1} - \frac{\sqrt{x + 1}}{x - 1} \cdot \frac{\sqrt{x + 1}}{x - 1}}{\frac{\sqrt{x - 1}}{x + 1} + \frac{\sqrt{x + 1}}{x - 1}}}$
4. Simplified39.4

$\leadsto \frac{\color{blue}{\frac{x - 1}{\left(x + 1\right) \cdot \left(x + 1\right)} - \frac{x + 1}{\left(x - 1\right) \cdot \left(x - 1\right)}}}{\frac{\sqrt{x - 1}}{x + 1} + \frac{\sqrt{x + 1}}{x - 1}}$
5. Taylor expanded around inf 11.4

$\leadsto \frac{\color{blue}{-\left(6 \cdot \frac{1}{{x}^{2}} + \left(14 \cdot \frac{1}{{x}^{4}} + 22 \cdot \frac{1}{{x}^{6}}\right)\right)}}{\frac{\sqrt{x - 1}}{x + 1} + \frac{\sqrt{x + 1}}{x - 1}}$
6. Simplified11.3

$\leadsto \frac{\color{blue}{-\left(\left(\frac{14}{{x}^{4}} + \frac{22}{{x}^{6}}\right) + \frac{6}{x \cdot x}\right)}}{\frac{\sqrt{x - 1}}{x + 1} + \frac{\sqrt{x + 1}}{x - 1}}$
7. Final simplification11.3

$\leadsto \frac{-\left(\left(\frac{14}{{x}^{4}} + \frac{22}{{x}^{6}}\right) + \frac{6}{x \cdot x}\right)}{\frac{\sqrt{x - 1}}{x + 1} + \frac{\sqrt{x + 1}}{x - 1}}$

# Reproduce

herbie shell --seed 1
(FPCore (x)
:name "sqrt(x-1)/(x+1)-sqrt(x+1)/(x-1)"
:precision binary64
:pre (> x 0.0)
(- (/ (sqrt (- x 1)) (+ x 1)) (/ (sqrt (+ x 1)) (- x 1))))