Average Error: 39.6 → 27.3
Time: 31.7s
Precision: 64
$\frac{\frac{x}{n} - \frac{y}{m}}{\sqrt{\frac{{x}^{2}}{n} - \frac{{y}^{2}}{n}}}$
$\begin{array}{l} \mathbf{if}\;y \le -1.221347972685769510583718036205494926521 \cdot 10^{128}:\\ \;\;\;\;-\frac{\sqrt{-\sqrt[3]{n}}}{m} \cdot \sqrt[3]{n}\\ \mathbf{elif}\;y \le -1.865317888159859213866006310937897833894 \cdot 10^{-143}:\\ \;\;\;\;\frac{\frac{x}{n} - \frac{y}{m}}{\sqrt{{x}^{\left(\frac{2}{2}\right)} \cdot \frac{{x}^{\left(\frac{2}{2}\right)}}{n} - \frac{{y}^{2}}{n}}}\\ \mathbf{elif}\;y \le 1.053727101585239359603006732099792167747 \cdot 10^{-236}:\\ \;\;\;\;\frac{\frac{\frac{x}{n} - \frac{y}{m}}{\sqrt{\frac{\sqrt{{x}^{2}}}{\sqrt{n}} + \frac{{y}^{\left(\frac{2}{2}\right)}}{\sqrt{n}}}}}{\sqrt{\frac{\sqrt{{x}^{2}}}{\sqrt{n}} - \frac{{y}^{\left(\frac{2}{2}\right)}}{\sqrt{n}}}}\\ \mathbf{elif}\;y \le 4.411903899729097729728951692605696132411 \cdot 10^{52}:\\ \;\;\;\;\frac{\frac{x}{n} - \frac{y}{m}}{\sqrt{{x}^{\left(\frac{2}{2}\right)} \cdot \frac{{x}^{\left(\frac{2}{2}\right)}}{n} - \frac{{y}^{2}}{n}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{n} - \frac{y}{m}}{\sqrt{\frac{1}{\sqrt[3]{n} \cdot \sqrt[3]{n}}} \cdot \left(\sqrt{-\sqrt[3]{\frac{1}{n}}} \cdot y\right)}\\ \end{array}$
\frac{\frac{x}{n} - \frac{y}{m}}{\sqrt{\frac{{x}^{2}}{n} - \frac{{y}^{2}}{n}}}
\begin{array}{l}
\mathbf{if}\;y \le -1.221347972685769510583718036205494926521 \cdot 10^{128}:\\
\;\;\;\;-\frac{\sqrt{-\sqrt[3]{n}}}{m} \cdot \sqrt[3]{n}\\

\mathbf{elif}\;y \le -1.865317888159859213866006310937897833894 \cdot 10^{-143}:\\
\;\;\;\;\frac{\frac{x}{n} - \frac{y}{m}}{\sqrt{{x}^{\left(\frac{2}{2}\right)} \cdot \frac{{x}^{\left(\frac{2}{2}\right)}}{n} - \frac{{y}^{2}}{n}}}\\

\mathbf{elif}\;y \le 1.053727101585239359603006732099792167747 \cdot 10^{-236}:\\
\;\;\;\;\frac{\frac{\frac{x}{n} - \frac{y}{m}}{\sqrt{\frac{\sqrt{{x}^{2}}}{\sqrt{n}} + \frac{{y}^{\left(\frac{2}{2}\right)}}{\sqrt{n}}}}}{\sqrt{\frac{\sqrt{{x}^{2}}}{\sqrt{n}} - \frac{{y}^{\left(\frac{2}{2}\right)}}{\sqrt{n}}}}\\

\mathbf{elif}\;y \le 4.411903899729097729728951692605696132411 \cdot 10^{52}:\\
\;\;\;\;\frac{\frac{x}{n} - \frac{y}{m}}{\sqrt{{x}^{\left(\frac{2}{2}\right)} \cdot \frac{{x}^{\left(\frac{2}{2}\right)}}{n} - \frac{{y}^{2}}{n}}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{x}{n} - \frac{y}{m}}{\sqrt{\frac{1}{\sqrt[3]{n} \cdot \sqrt[3]{n}}} \cdot \left(\sqrt{-\sqrt[3]{\frac{1}{n}}} \cdot y\right)}\\

\end{array}
double f(double x, double n, double y, double m) {
double r447073 = x;
double r447074 = n;
double r447075 = r447073 / r447074;
double r447076 = y;
double r447077 = m;
double r447078 = r447076 / r447077;
double r447079 = r447075 - r447078;
double r447080 = 2.0;
double r447081 = pow(r447073, r447080);
double r447082 = r447081 / r447074;
double r447083 = pow(r447076, r447080);
double r447084 = r447083 / r447074;
double r447085 = r447082 - r447084;
double r447086 = sqrt(r447085);
double r447087 = r447079 / r447086;
return r447087;
}


double f(double x, double n, double y, double m) {
double r447088 = y;
double r447089 = -1.2213479726857695e+128;
bool r447090 = r447088 <= r447089;
double r447091 = n;
double r447092 = cbrt(r447091);
double r447093 = -r447092;
double r447094 = sqrt(r447093);
double r447095 = m;
double r447096 = r447094 / r447095;
double r447097 = r447096 * r447092;
double r447098 = -r447097;
double r447099 = -1.8653178881598592e-143;
bool r447100 = r447088 <= r447099;
double r447101 = x;
double r447102 = r447101 / r447091;
double r447103 = r447088 / r447095;
double r447104 = r447102 - r447103;
double r447105 = 2.0;
double r447106 = 2.0;
double r447107 = r447105 / r447106;
double r447108 = pow(r447101, r447107);
double r447109 = r447108 / r447091;
double r447110 = r447108 * r447109;
double r447111 = pow(r447088, r447105);
double r447112 = r447111 / r447091;
double r447113 = r447110 - r447112;
double r447114 = sqrt(r447113);
double r447115 = r447104 / r447114;
double r447116 = 1.0537271015852394e-236;
bool r447117 = r447088 <= r447116;
double r447118 = pow(r447101, r447105);
double r447119 = sqrt(r447118);
double r447120 = sqrt(r447091);
double r447121 = r447119 / r447120;
double r447122 = pow(r447088, r447107);
double r447123 = r447122 / r447120;
double r447124 = r447121 + r447123;
double r447125 = sqrt(r447124);
double r447126 = r447104 / r447125;
double r447127 = r447121 - r447123;
double r447128 = sqrt(r447127);
double r447129 = r447126 / r447128;
double r447130 = 4.4119038997290977e+52;
bool r447131 = r447088 <= r447130;
double r447132 = 1.0;
double r447133 = r447092 * r447092;
double r447134 = r447132 / r447133;
double r447135 = sqrt(r447134);
double r447136 = r447132 / r447091;
double r447137 = cbrt(r447136);
double r447138 = -r447137;
double r447139 = sqrt(r447138);
double r447140 = r447139 * r447088;
double r447141 = r447135 * r447140;
double r447142 = r447104 / r447141;
double r447143 = r447131 ? r447115 : r447142;
double r447144 = r447117 ? r447129 : r447143;
double r447145 = r447100 ? r447115 : r447144;
double r447146 = r447090 ? r447098 : r447145;
return r447146;
}



# Try it out

Results

 In Out
Enter valid numbers for all inputs

# Derivation

1. Split input into 4 regimes
2. ## if y < -1.2213479726857695e+128

1. Initial program 59.8

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

$\leadsto \frac{\frac{x}{n} - \frac{y}{m}}{\sqrt{\frac{{x}^{2}}{n} - \frac{{y}^{2}}{\color{blue}{\left(\sqrt[3]{n} \cdot \sqrt[3]{n}\right) \cdot \sqrt[3]{n}}}}}$
4. Applied *-un-lft-identity59.8

$\leadsto \frac{\frac{x}{n} - \frac{y}{m}}{\sqrt{\frac{{x}^{2}}{n} - \frac{{\color{blue}{\left(1 \cdot y\right)}}^{2}}{\left(\sqrt[3]{n} \cdot \sqrt[3]{n}\right) \cdot \sqrt[3]{n}}}}$
5. Applied unpow-prod-down59.8

$\leadsto \frac{\frac{x}{n} - \frac{y}{m}}{\sqrt{\frac{{x}^{2}}{n} - \frac{\color{blue}{{1}^{2} \cdot {y}^{2}}}{\left(\sqrt[3]{n} \cdot \sqrt[3]{n}\right) \cdot \sqrt[3]{n}}}}$
6. Applied times-frac59.8

$\leadsto \frac{\frac{x}{n} - \frac{y}{m}}{\sqrt{\frac{{x}^{2}}{n} - \color{blue}{\frac{{1}^{2}}{\sqrt[3]{n} \cdot \sqrt[3]{n}} \cdot \frac{{y}^{2}}{\sqrt[3]{n}}}}}$

$\leadsto \frac{\frac{x}{n} - \frac{y}{m}}{\sqrt{\frac{{x}^{2}}{\color{blue}{\left(\sqrt[3]{n} \cdot \sqrt[3]{n}\right) \cdot \sqrt[3]{n}}} - \frac{{1}^{2}}{\sqrt[3]{n} \cdot \sqrt[3]{n}} \cdot \frac{{y}^{2}}{\sqrt[3]{n}}}}$
8. Applied *-un-lft-identity59.8

$\leadsto \frac{\frac{x}{n} - \frac{y}{m}}{\sqrt{\frac{{\color{blue}{\left(1 \cdot x\right)}}^{2}}{\left(\sqrt[3]{n} \cdot \sqrt[3]{n}\right) \cdot \sqrt[3]{n}} - \frac{{1}^{2}}{\sqrt[3]{n} \cdot \sqrt[3]{n}} \cdot \frac{{y}^{2}}{\sqrt[3]{n}}}}$
9. Applied unpow-prod-down59.8

$\leadsto \frac{\frac{x}{n} - \frac{y}{m}}{\sqrt{\frac{\color{blue}{{1}^{2} \cdot {x}^{2}}}{\left(\sqrt[3]{n} \cdot \sqrt[3]{n}\right) \cdot \sqrt[3]{n}} - \frac{{1}^{2}}{\sqrt[3]{n} \cdot \sqrt[3]{n}} \cdot \frac{{y}^{2}}{\sqrt[3]{n}}}}$
10. Applied times-frac59.8

$\leadsto \frac{\frac{x}{n} - \frac{y}{m}}{\sqrt{\color{blue}{\frac{{1}^{2}}{\sqrt[3]{n} \cdot \sqrt[3]{n}} \cdot \frac{{x}^{2}}{\sqrt[3]{n}}} - \frac{{1}^{2}}{\sqrt[3]{n} \cdot \sqrt[3]{n}} \cdot \frac{{y}^{2}}{\sqrt[3]{n}}}}$
11. Applied distribute-lft-out--59.7

$\leadsto \frac{\frac{x}{n} - \frac{y}{m}}{\sqrt{\color{blue}{\frac{{1}^{2}}{\sqrt[3]{n} \cdot \sqrt[3]{n}} \cdot \left(\frac{{x}^{2}}{\sqrt[3]{n}} - \frac{{y}^{2}}{\sqrt[3]{n}}\right)}}}$
12. Applied sqrt-prod59.2

$\leadsto \frac{\frac{x}{n} - \frac{y}{m}}{\color{blue}{\sqrt{\frac{{1}^{2}}{\sqrt[3]{n} \cdot \sqrt[3]{n}}} \cdot \sqrt{\frac{{x}^{2}}{\sqrt[3]{n}} - \frac{{y}^{2}}{\sqrt[3]{n}}}}}$
13. Simplified59.2

$\leadsto \frac{\frac{x}{n} - \frac{y}{m}}{\color{blue}{\sqrt{\frac{1}{\sqrt[3]{n} \cdot \sqrt[3]{n}}}} \cdot \sqrt{\frac{{x}^{2}}{\sqrt[3]{n}} - \frac{{y}^{2}}{\sqrt[3]{n}}}}$
14. Taylor expanded around 0 64.0

$\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{-1 \cdot {n}^{\frac{1}{3}}}}{m} \cdot {n}^{\frac{1}{3}}\right)}$
15. Simplified20.6

$\leadsto \color{blue}{-\frac{\sqrt{-\sqrt[3]{n}}}{m} \cdot \sqrt[3]{n}}$

## if -1.2213479726857695e+128 < y < -1.8653178881598592e-143 or 1.0537271015852394e-236 < y < 4.4119038997290977e+52

1. Initial program 28.5

$\frac{\frac{x}{n} - \frac{y}{m}}{\sqrt{\frac{{x}^{2}}{n} - \frac{{y}^{2}}{n}}}$
2. Using strategy rm
3. Applied *-un-lft-identity28.5

$\leadsto \frac{\frac{x}{n} - \frac{y}{m}}{\sqrt{\frac{{x}^{2}}{\color{blue}{1 \cdot n}} - \frac{{y}^{2}}{n}}}$
4. Applied sqr-pow28.5

$\leadsto \frac{\frac{x}{n} - \frac{y}{m}}{\sqrt{\frac{\color{blue}{{x}^{\left(\frac{2}{2}\right)} \cdot {x}^{\left(\frac{2}{2}\right)}}}{1 \cdot n} - \frac{{y}^{2}}{n}}}$
5. Applied times-frac24.0

$\leadsto \frac{\frac{x}{n} - \frac{y}{m}}{\sqrt{\color{blue}{\frac{{x}^{\left(\frac{2}{2}\right)}}{1} \cdot \frac{{x}^{\left(\frac{2}{2}\right)}}{n}} - \frac{{y}^{2}}{n}}}$
6. Simplified24.0

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

## if -1.8653178881598592e-143 < y < 1.0537271015852394e-236

1. Initial program 37.9

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

$\leadsto \frac{\frac{x}{n} - \frac{y}{m}}{\sqrt{\frac{{x}^{2}}{n} - \frac{{y}^{2}}{\color{blue}{\sqrt{n} \cdot \sqrt{n}}}}}$
4. Applied sqr-pow38.6

$\leadsto \frac{\frac{x}{n} - \frac{y}{m}}{\sqrt{\frac{{x}^{2}}{n} - \frac{\color{blue}{{y}^{\left(\frac{2}{2}\right)} \cdot {y}^{\left(\frac{2}{2}\right)}}}{\sqrt{n} \cdot \sqrt{n}}}}$
5. Applied times-frac38.6

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

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

$\leadsto \frac{\frac{x}{n} - \frac{y}{m}}{\sqrt{\frac{\color{blue}{\sqrt{{x}^{2}} \cdot \sqrt{{x}^{2}}}}{\sqrt{n} \cdot \sqrt{n}} - \frac{{y}^{\left(\frac{2}{2}\right)}}{\sqrt{n}} \cdot \frac{{y}^{\left(\frac{2}{2}\right)}}{\sqrt{n}}}}$
8. Applied times-frac38.7

$\leadsto \frac{\frac{x}{n} - \frac{y}{m}}{\sqrt{\color{blue}{\frac{\sqrt{{x}^{2}}}{\sqrt{n}} \cdot \frac{\sqrt{{x}^{2}}}{\sqrt{n}}} - \frac{{y}^{\left(\frac{2}{2}\right)}}{\sqrt{n}} \cdot \frac{{y}^{\left(\frac{2}{2}\right)}}{\sqrt{n}}}}$
9. Applied difference-of-squares38.7

$\leadsto \frac{\frac{x}{n} - \frac{y}{m}}{\sqrt{\color{blue}{\left(\frac{\sqrt{{x}^{2}}}{\sqrt{n}} + \frac{{y}^{\left(\frac{2}{2}\right)}}{\sqrt{n}}\right) \cdot \left(\frac{\sqrt{{x}^{2}}}{\sqrt{n}} - \frac{{y}^{\left(\frac{2}{2}\right)}}{\sqrt{n}}\right)}}}$
10. Applied sqrt-prod33.4

$\leadsto \frac{\frac{x}{n} - \frac{y}{m}}{\color{blue}{\sqrt{\frac{\sqrt{{x}^{2}}}{\sqrt{n}} + \frac{{y}^{\left(\frac{2}{2}\right)}}{\sqrt{n}}} \cdot \sqrt{\frac{\sqrt{{x}^{2}}}{\sqrt{n}} - \frac{{y}^{\left(\frac{2}{2}\right)}}{\sqrt{n}}}}}$
11. Applied associate-/r*33.4

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

## if 4.4119038997290977e+52 < y

1. Initial program 52.1

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

$\leadsto \frac{\frac{x}{n} - \frac{y}{m}}{\sqrt{\frac{{x}^{2}}{n} - \frac{{y}^{2}}{\color{blue}{\left(\sqrt[3]{n} \cdot \sqrt[3]{n}\right) \cdot \sqrt[3]{n}}}}}$
4. Applied *-un-lft-identity52.2

$\leadsto \frac{\frac{x}{n} - \frac{y}{m}}{\sqrt{\frac{{x}^{2}}{n} - \frac{{\color{blue}{\left(1 \cdot y\right)}}^{2}}{\left(\sqrt[3]{n} \cdot \sqrt[3]{n}\right) \cdot \sqrt[3]{n}}}}$
5. Applied unpow-prod-down52.2

$\leadsto \frac{\frac{x}{n} - \frac{y}{m}}{\sqrt{\frac{{x}^{2}}{n} - \frac{\color{blue}{{1}^{2} \cdot {y}^{2}}}{\left(\sqrt[3]{n} \cdot \sqrt[3]{n}\right) \cdot \sqrt[3]{n}}}}$
6. Applied times-frac52.2

$\leadsto \frac{\frac{x}{n} - \frac{y}{m}}{\sqrt{\frac{{x}^{2}}{n} - \color{blue}{\frac{{1}^{2}}{\sqrt[3]{n} \cdot \sqrt[3]{n}} \cdot \frac{{y}^{2}}{\sqrt[3]{n}}}}}$

$\leadsto \frac{\frac{x}{n} - \frac{y}{m}}{\sqrt{\frac{{x}^{2}}{\color{blue}{\left(\sqrt[3]{n} \cdot \sqrt[3]{n}\right) \cdot \sqrt[3]{n}}} - \frac{{1}^{2}}{\sqrt[3]{n} \cdot \sqrt[3]{n}} \cdot \frac{{y}^{2}}{\sqrt[3]{n}}}}$
8. Applied *-un-lft-identity52.2

$\leadsto \frac{\frac{x}{n} - \frac{y}{m}}{\sqrt{\frac{{\color{blue}{\left(1 \cdot x\right)}}^{2}}{\left(\sqrt[3]{n} \cdot \sqrt[3]{n}\right) \cdot \sqrt[3]{n}} - \frac{{1}^{2}}{\sqrt[3]{n} \cdot \sqrt[3]{n}} \cdot \frac{{y}^{2}}{\sqrt[3]{n}}}}$
9. Applied unpow-prod-down52.2

$\leadsto \frac{\frac{x}{n} - \frac{y}{m}}{\sqrt{\frac{\color{blue}{{1}^{2} \cdot {x}^{2}}}{\left(\sqrt[3]{n} \cdot \sqrt[3]{n}\right) \cdot \sqrt[3]{n}} - \frac{{1}^{2}}{\sqrt[3]{n} \cdot \sqrt[3]{n}} \cdot \frac{{y}^{2}}{\sqrt[3]{n}}}}$
10. Applied times-frac52.2

$\leadsto \frac{\frac{x}{n} - \frac{y}{m}}{\sqrt{\color{blue}{\frac{{1}^{2}}{\sqrt[3]{n} \cdot \sqrt[3]{n}} \cdot \frac{{x}^{2}}{\sqrt[3]{n}}} - \frac{{1}^{2}}{\sqrt[3]{n} \cdot \sqrt[3]{n}} \cdot \frac{{y}^{2}}{\sqrt[3]{n}}}}$
11. Applied distribute-lft-out--52.1

$\leadsto \frac{\frac{x}{n} - \frac{y}{m}}{\sqrt{\color{blue}{\frac{{1}^{2}}{\sqrt[3]{n} \cdot \sqrt[3]{n}} \cdot \left(\frac{{x}^{2}}{\sqrt[3]{n}} - \frac{{y}^{2}}{\sqrt[3]{n}}\right)}}}$
12. Applied sqrt-prod49.3

$\leadsto \frac{\frac{x}{n} - \frac{y}{m}}{\color{blue}{\sqrt{\frac{{1}^{2}}{\sqrt[3]{n} \cdot \sqrt[3]{n}}} \cdot \sqrt{\frac{{x}^{2}}{\sqrt[3]{n}} - \frac{{y}^{2}}{\sqrt[3]{n}}}}}$
13. Simplified49.3

$\leadsto \frac{\frac{x}{n} - \frac{y}{m}}{\color{blue}{\sqrt{\frac{1}{\sqrt[3]{n} \cdot \sqrt[3]{n}}}} \cdot \sqrt{\frac{{x}^{2}}{\sqrt[3]{n}} - \frac{{y}^{2}}{\sqrt[3]{n}}}}$
14. Taylor expanded around 0 64.0

$\leadsto \frac{\frac{x}{n} - \frac{y}{m}}{\sqrt{\frac{1}{\sqrt[3]{n} \cdot \sqrt[3]{n}}} \cdot \color{blue}{\left(y \cdot \sqrt{-{\left(\frac{1}{n}\right)}^{\frac{1}{3}}}\right)}}$
15. Simplified34.0

$\leadsto \frac{\frac{x}{n} - \frac{y}{m}}{\sqrt{\frac{1}{\sqrt[3]{n} \cdot \sqrt[3]{n}}} \cdot \color{blue}{\left(\sqrt{-\sqrt[3]{\frac{1}{n}}} \cdot y\right)}}$
3. Recombined 4 regimes into one program.
4. Final simplification27.3

$\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.221347972685769510583718036205494926521 \cdot 10^{128}:\\ \;\;\;\;-\frac{\sqrt{-\sqrt[3]{n}}}{m} \cdot \sqrt[3]{n}\\ \mathbf{elif}\;y \le -1.865317888159859213866006310937897833894 \cdot 10^{-143}:\\ \;\;\;\;\frac{\frac{x}{n} - \frac{y}{m}}{\sqrt{{x}^{\left(\frac{2}{2}\right)} \cdot \frac{{x}^{\left(\frac{2}{2}\right)}}{n} - \frac{{y}^{2}}{n}}}\\ \mathbf{elif}\;y \le 1.053727101585239359603006732099792167747 \cdot 10^{-236}:\\ \;\;\;\;\frac{\frac{\frac{x}{n} - \frac{y}{m}}{\sqrt{\frac{\sqrt{{x}^{2}}}{\sqrt{n}} + \frac{{y}^{\left(\frac{2}{2}\right)}}{\sqrt{n}}}}}{\sqrt{\frac{\sqrt{{x}^{2}}}{\sqrt{n}} - \frac{{y}^{\left(\frac{2}{2}\right)}}{\sqrt{n}}}}\\ \mathbf{elif}\;y \le 4.411903899729097729728951692605696132411 \cdot 10^{52}:\\ \;\;\;\;\frac{\frac{x}{n} - \frac{y}{m}}{\sqrt{{x}^{\left(\frac{2}{2}\right)} \cdot \frac{{x}^{\left(\frac{2}{2}\right)}}{n} - \frac{{y}^{2}}{n}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{n} - \frac{y}{m}}{\sqrt{\frac{1}{\sqrt[3]{n} \cdot \sqrt[3]{n}}} \cdot \left(\sqrt{-\sqrt[3]{\frac{1}{n}}} \cdot y\right)}\\ \end{array}$

# Reproduce

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