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;
}

Error

Bits error versus x

Bits error versus n

Bits error versus y

Bits error versus m

Try it out

Your Program's Arguments

Results

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
    3. Applied add-cube-cbrt59.8

      \[\leadsto \frac{\frac{x}{n} - \frac{y}{m}}{\sqrt{\frac{{x}^{2}}{n} - \frac{{y}^{2}}{\color{blue}{\left(\sqrt[3]{n} \cdot \sqrt[3]{n}\right) \cdot \sqrt[3]{n}}}}}\]
    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

      \[\leadsto \frac{\frac{x}{n} - \frac{y}{m}}{\sqrt{\frac{{x}^{2}}{n} - \color{blue}{\frac{{1}^{2}}{\sqrt[3]{n} \cdot \sqrt[3]{n}} \cdot \frac{{y}^{2}}{\sqrt[3]{n}}}}}\]
    7. Applied add-cube-cbrt59.8

      \[\leadsto \frac{\frac{x}{n} - \frac{y}{m}}{\sqrt{\frac{{x}^{2}}{\color{blue}{\left(\sqrt[3]{n} \cdot \sqrt[3]{n}\right) \cdot \sqrt[3]{n}}} - \frac{{1}^{2}}{\sqrt[3]{n} \cdot \sqrt[3]{n}} \cdot \frac{{y}^{2}}{\sqrt[3]{n}}}}\]
    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
    3. Applied add-sqr-sqrt38.6

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

      \[\leadsto \frac{\frac{x}{n} - \frac{y}{m}}{\sqrt{\frac{{x}^{2}}{n} - \color{blue}{\frac{{y}^{\left(\frac{2}{2}\right)}}{\sqrt{n}} \cdot \frac{{y}^{\left(\frac{2}{2}\right)}}{\sqrt{n}}}}}\]
    6. Applied add-sqr-sqrt38.7

      \[\leadsto \frac{\frac{x}{n} - \frac{y}{m}}{\sqrt{\frac{{x}^{2}}{\color{blue}{\sqrt{n} \cdot \sqrt{n}}} - \frac{{y}^{\left(\frac{2}{2}\right)}}{\sqrt{n}} \cdot \frac{{y}^{\left(\frac{2}{2}\right)}}{\sqrt{n}}}}\]
    7. Applied add-sqr-sqrt38.7

      \[\leadsto \frac{\frac{x}{n} - \frac{y}{m}}{\sqrt{\frac{\color{blue}{\sqrt{{x}^{2}} \cdot \sqrt{{x}^{2}}}}{\sqrt{n} \cdot \sqrt{n}} - \frac{{y}^{\left(\frac{2}{2}\right)}}{\sqrt{n}} \cdot \frac{{y}^{\left(\frac{2}{2}\right)}}{\sqrt{n}}}}\]
    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
    3. Applied add-cube-cbrt52.2

      \[\leadsto \frac{\frac{x}{n} - \frac{y}{m}}{\sqrt{\frac{{x}^{2}}{n} - \frac{{y}^{2}}{\color{blue}{\left(\sqrt[3]{n} \cdot \sqrt[3]{n}\right) \cdot \sqrt[3]{n}}}}}\]
    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

      \[\leadsto \frac{\frac{x}{n} - \frac{y}{m}}{\sqrt{\frac{{x}^{2}}{n} - \color{blue}{\frac{{1}^{2}}{\sqrt[3]{n} \cdot \sqrt[3]{n}} \cdot \frac{{y}^{2}}{\sqrt[3]{n}}}}}\]
    7. Applied add-cube-cbrt52.2

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