Average Error: 39.7 → 30.6
Time: 26.1s
Precision: 64
$\frac{\frac{x}{n} - \frac{y}{m}}{\sqrt{\frac{{x}^{2}}{n} - \frac{{y}^{2}}{m}}}$
$\begin{array}{l} \mathbf{if}\;x \le 5.69003762507141083127778083459341345523 \cdot 10^{220}:\\ \;\;\;\;\frac{\frac{x}{n} - \frac{y}{m}}{\sqrt{{x}^{\left(\frac{2}{2}\right)} \cdot \frac{{x}^{\left(\frac{2}{2}\right)}}{n} - {y}^{\left(\frac{2}{2}\right)} \cdot \frac{{y}^{\left(\frac{2}{2}\right)}}{m}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{x}{n} - \frac{y}{m}}{\sqrt{\frac{{x}^{\left(\frac{2}{2}\right)}}{\sqrt{n}} + \sqrt{\frac{{y}^{2}}{m}}}}}{\sqrt{\frac{{x}^{\left(\frac{2}{2}\right)}}{\sqrt{n}} - \sqrt{\frac{{y}^{2}}{m}}}}\\ \end{array}$
\frac{\frac{x}{n} - \frac{y}{m}}{\sqrt{\frac{{x}^{2}}{n} - \frac{{y}^{2}}{m}}}
\begin{array}{l}
\mathbf{if}\;x \le 5.69003762507141083127778083459341345523 \cdot 10^{220}:\\
\;\;\;\;\frac{\frac{x}{n} - \frac{y}{m}}{\sqrt{{x}^{\left(\frac{2}{2}\right)} \cdot \frac{{x}^{\left(\frac{2}{2}\right)}}{n} - {y}^{\left(\frac{2}{2}\right)} \cdot \frac{{y}^{\left(\frac{2}{2}\right)}}{m}}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\frac{x}{n} - \frac{y}{m}}{\sqrt{\frac{{x}^{\left(\frac{2}{2}\right)}}{\sqrt{n}} + \sqrt{\frac{{y}^{2}}{m}}}}}{\sqrt{\frac{{x}^{\left(\frac{2}{2}\right)}}{\sqrt{n}} - \sqrt{\frac{{y}^{2}}{m}}}}\\

\end{array}
double f(double x, double n, double y, double m) {
double r479627 = x;
double r479628 = n;
double r479629 = r479627 / r479628;
double r479630 = y;
double r479631 = m;
double r479632 = r479630 / r479631;
double r479633 = r479629 - r479632;
double r479634 = 2.0;
double r479635 = pow(r479627, r479634);
double r479636 = r479635 / r479628;
double r479637 = pow(r479630, r479634);
double r479638 = r479637 / r479631;
double r479639 = r479636 - r479638;
double r479640 = sqrt(r479639);
double r479641 = r479633 / r479640;
return r479641;
}


double f(double x, double n, double y, double m) {
double r479642 = x;
double r479643 = 5.690037625071411e+220;
bool r479644 = r479642 <= r479643;
double r479645 = n;
double r479646 = r479642 / r479645;
double r479647 = y;
double r479648 = m;
double r479649 = r479647 / r479648;
double r479650 = r479646 - r479649;
double r479651 = 2.0;
double r479652 = 2.0;
double r479653 = r479651 / r479652;
double r479654 = pow(r479642, r479653);
double r479655 = r479654 / r479645;
double r479656 = r479654 * r479655;
double r479657 = pow(r479647, r479653);
double r479658 = r479657 / r479648;
double r479659 = r479657 * r479658;
double r479660 = r479656 - r479659;
double r479661 = sqrt(r479660);
double r479662 = r479650 / r479661;
double r479663 = sqrt(r479645);
double r479664 = r479654 / r479663;
double r479665 = pow(r479647, r479651);
double r479666 = r479665 / r479648;
double r479667 = sqrt(r479666);
double r479668 = r479664 + r479667;
double r479669 = sqrt(r479668);
double r479670 = r479650 / r479669;
double r479671 = r479664 - r479667;
double r479672 = sqrt(r479671);
double r479673 = r479670 / r479672;
double r479674 = r479644 ? r479662 : r479673;
return r479674;
}



Try it out

Results

 In Out
Enter valid numbers for all inputs

Derivation

1. Split input into 2 regimes
2. if x < 5.690037625071411e+220

1. Initial program 37.9

$\frac{\frac{x}{n} - \frac{y}{m}}{\sqrt{\frac{{x}^{2}}{n} - \frac{{y}^{2}}{m}}}$
2. Using strategy rm
3. Applied *-un-lft-identity37.9

$\leadsto \frac{\frac{x}{n} - \frac{y}{m}}{\sqrt{\frac{{x}^{2}}{n} - \frac{{y}^{2}}{\color{blue}{1 \cdot m}}}}$
4. Applied sqr-pow37.9

$\leadsto \frac{\frac{x}{n} - \frac{y}{m}}{\sqrt{\frac{{x}^{2}}{n} - \frac{\color{blue}{{y}^{\left(\frac{2}{2}\right)} \cdot {y}^{\left(\frac{2}{2}\right)}}}{1 \cdot m}}}$
5. Applied times-frac33.7

$\leadsto \frac{\frac{x}{n} - \frac{y}{m}}{\sqrt{\frac{{x}^{2}}{n} - \color{blue}{\frac{{y}^{\left(\frac{2}{2}\right)}}{1} \cdot \frac{{y}^{\left(\frac{2}{2}\right)}}{m}}}}$
6. Simplified33.7

$\leadsto \frac{\frac{x}{n} - \frac{y}{m}}{\sqrt{\frac{{x}^{2}}{n} - \color{blue}{{y}^{\left(\frac{2}{2}\right)}} \cdot \frac{{y}^{\left(\frac{2}{2}\right)}}{m}}}$
7. Using strategy rm
8. Applied *-un-lft-identity33.7

$\leadsto \frac{\frac{x}{n} - \frac{y}{m}}{\sqrt{\frac{{x}^{2}}{\color{blue}{1 \cdot n}} - {y}^{\left(\frac{2}{2}\right)} \cdot \frac{{y}^{\left(\frac{2}{2}\right)}}{m}}}$
9. Applied sqr-pow33.7

$\leadsto \frac{\frac{x}{n} - \frac{y}{m}}{\sqrt{\frac{\color{blue}{{x}^{\left(\frac{2}{2}\right)} \cdot {x}^{\left(\frac{2}{2}\right)}}}{1 \cdot n} - {y}^{\left(\frac{2}{2}\right)} \cdot \frac{{y}^{\left(\frac{2}{2}\right)}}{m}}}$
10. Applied times-frac29.4

$\leadsto \frac{\frac{x}{n} - \frac{y}{m}}{\sqrt{\color{blue}{\frac{{x}^{\left(\frac{2}{2}\right)}}{1} \cdot \frac{{x}^{\left(\frac{2}{2}\right)}}{n}} - {y}^{\left(\frac{2}{2}\right)} \cdot \frac{{y}^{\left(\frac{2}{2}\right)}}{m}}}$
11. Simplified29.4

$\leadsto \frac{\frac{x}{n} - \frac{y}{m}}{\sqrt{\color{blue}{{x}^{\left(\frac{2}{2}\right)}} \cdot \frac{{x}^{\left(\frac{2}{2}\right)}}{n} - {y}^{\left(\frac{2}{2}\right)} \cdot \frac{{y}^{\left(\frac{2}{2}\right)}}{m}}}$

if 5.690037625071411e+220 < x

1. Initial program 63.0

$\frac{\frac{x}{n} - \frac{y}{m}}{\sqrt{\frac{{x}^{2}}{n} - \frac{{y}^{2}}{m}}}$
2. Using strategy rm

$\leadsto \frac{\frac{x}{n} - \frac{y}{m}}{\sqrt{\frac{{x}^{2}}{n} - \color{blue}{\sqrt{\frac{{y}^{2}}{m}} \cdot \sqrt{\frac{{y}^{2}}{m}}}}}$

$\leadsto \frac{\frac{x}{n} - \frac{y}{m}}{\sqrt{\frac{{x}^{2}}{\color{blue}{\sqrt{n} \cdot \sqrt{n}}} - \sqrt{\frac{{y}^{2}}{m}} \cdot \sqrt{\frac{{y}^{2}}{m}}}}$
5. Applied sqr-pow63.4

$\leadsto \frac{\frac{x}{n} - \frac{y}{m}}{\sqrt{\frac{\color{blue}{{x}^{\left(\frac{2}{2}\right)} \cdot {x}^{\left(\frac{2}{2}\right)}}}{\sqrt{n} \cdot \sqrt{n}} - \sqrt{\frac{{y}^{2}}{m}} \cdot \sqrt{\frac{{y}^{2}}{m}}}}$
6. Applied times-frac59.1

$\leadsto \frac{\frac{x}{n} - \frac{y}{m}}{\sqrt{\color{blue}{\frac{{x}^{\left(\frac{2}{2}\right)}}{\sqrt{n}} \cdot \frac{{x}^{\left(\frac{2}{2}\right)}}{\sqrt{n}}} - \sqrt{\frac{{y}^{2}}{m}} \cdot \sqrt{\frac{{y}^{2}}{m}}}}$
7. Applied difference-of-squares59.1

$\leadsto \frac{\frac{x}{n} - \frac{y}{m}}{\sqrt{\color{blue}{\left(\frac{{x}^{\left(\frac{2}{2}\right)}}{\sqrt{n}} + \sqrt{\frac{{y}^{2}}{m}}\right) \cdot \left(\frac{{x}^{\left(\frac{2}{2}\right)}}{\sqrt{n}} - \sqrt{\frac{{y}^{2}}{m}}\right)}}}$
8. Applied sqrt-prod46.8

$\leadsto \frac{\frac{x}{n} - \frac{y}{m}}{\color{blue}{\sqrt{\frac{{x}^{\left(\frac{2}{2}\right)}}{\sqrt{n}} + \sqrt{\frac{{y}^{2}}{m}}} \cdot \sqrt{\frac{{x}^{\left(\frac{2}{2}\right)}}{\sqrt{n}} - \sqrt{\frac{{y}^{2}}{m}}}}}$
9. Applied associate-/r*46.7

$\leadsto \color{blue}{\frac{\frac{\frac{x}{n} - \frac{y}{m}}{\sqrt{\frac{{x}^{\left(\frac{2}{2}\right)}}{\sqrt{n}} + \sqrt{\frac{{y}^{2}}{m}}}}}{\sqrt{\frac{{x}^{\left(\frac{2}{2}\right)}}{\sqrt{n}} - \sqrt{\frac{{y}^{2}}{m}}}}}$
3. Recombined 2 regimes into one program.
4. Final simplification30.6

$\leadsto \begin{array}{l} \mathbf{if}\;x \le 5.69003762507141083127778083459341345523 \cdot 10^{220}:\\ \;\;\;\;\frac{\frac{x}{n} - \frac{y}{m}}{\sqrt{{x}^{\left(\frac{2}{2}\right)} \cdot \frac{{x}^{\left(\frac{2}{2}\right)}}{n} - {y}^{\left(\frac{2}{2}\right)} \cdot \frac{{y}^{\left(\frac{2}{2}\right)}}{m}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{x}{n} - \frac{y}{m}}{\sqrt{\frac{{x}^{\left(\frac{2}{2}\right)}}{\sqrt{n}} + \sqrt{\frac{{y}^{2}}{m}}}}}{\sqrt{\frac{{x}^{\left(\frac{2}{2}\right)}}{\sqrt{n}} - \sqrt{\frac{{y}^{2}}{m}}}}\\ \end{array}$

Reproduce

herbie shell --seed 1
(FPCore (x n y m)
:name "(x/n - y/m)/sqrt(x^2/n - y^2/m)"
:precision binary64
(/ (- (/ x n) (/ y m)) (sqrt (- (/ (pow x 2) n) (/ (pow y 2) m)))))