\[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]

↓

\[\begin{array}{l} \mathbf{if}\;y.im \le -3.114802040969311027115783701676385953635 \cdot 10^{109}:\\ \;\;\;\;\frac{-x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \le 6.370255659610306169101339211646465545796 \cdot 10^{-234}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y.im, x.im, x.re \cdot y.re\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \le 4.694847582105375997704206679714705497211 \cdot 10^{-213}:\\ \;\;\;\;\frac{x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \le 1.048412145020906876185869327079410027246 \cdot 10^{133}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y.im, x.im, x.re \cdot y.re\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array}\]

\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}

↓

\begin{array}{l}
\mathbf{if}\;y.im \le -3.114802040969311027115783701676385953635 \cdot 10^{109}:\\
\;\;\;\;\frac{-x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\

\mathbf{elif}\;y.im \le 6.370255659610306169101339211646465545796 \cdot 10^{-234}:\\
\;\;\;\;\frac{\frac{\mathsf{fma}\left(y.im, x.im, x.re \cdot y.re\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\

\mathbf{elif}\;y.im \le 4.694847582105375997704206679714705497211 \cdot 10^{-213}:\\
\;\;\;\;\frac{x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\

\mathbf{elif}\;y.im \le 1.048412145020906876185869327079410027246 \cdot 10^{133}:\\
\;\;\;\;\frac{\frac{\mathsf{fma}\left(y.im, x.im, x.re \cdot y.re\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\

\end{array}

double f(double x_re, double x_im, double y_re, double y_im) {
        double r62091 = x_re;
        double r62092 = y_re;
        double r62093 = r62091 * r62092;
        double r62094 = x_im;
        double r62095 = y_im;
        double r62096 = r62094 * r62095;
        double r62097 = r62093 + r62096;
        double r62098 = r62092 * r62092;
        double r62099 = r62095 * r62095;
        double r62100 = r62098 + r62099;
        double r62101 = r62097 / r62100;
        return r62101;
}

↓

double f(double x_re, double x_im, double y_re, double y_im) {
        double r62102 = y_im;
        double r62103 = -3.114802040969311e+109;
        bool r62104 = r62102 <= r62103;
        double r62105 = x_im;
        double r62106 = -r62105;
        double r62107 = y_re;
        double r62108 = hypot(r62107, r62102);
        double r62109 = r62106 / r62108;
        double r62110 = 6.370255659610306e-234;
        bool r62111 = r62102 <= r62110;
        double r62112 = x_re;
        double r62113 = r62112 * r62107;
        double r62114 = fma(r62102, r62105, r62113);
        double r62115 = r62114 / r62108;
        double r62116 = r62115 / r62108;
        double r62117 = 4.694847582105376e-213;
        bool r62118 = r62102 <= r62117;
        double r62119 = r62112 / r62108;
        double r62120 = 1.0484121450209069e+133;
        bool r62121 = r62102 <= r62120;
        double r62122 = r62105 / r62108;
        double r62123 = r62121 ? r62116 : r62122;
        double r62124 = r62118 ? r62119 : r62123;
        double r62125 = r62111 ? r62116 : r62124;
        double r62126 = r62104 ? r62109 : r62125;
        return r62126;
}

Derivation

Split input into 4 regimes
if y.im < -3.114802040969311e+109
1. Initial program 40.1
  \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
2. Using strategy rm
3. Applied add-sqr-sqrt40.1
  \[\leadsto \frac{x.re \cdot y.re + x.im \cdot y.im}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}}\]
4. Applied *-un-lft-identity40.1
  \[\leadsto \frac{\color{blue}{1 \cdot \left(x.re \cdot y.re + x.im \cdot y.im\right)}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}\]
5. Applied times-frac40.1
  \[\leadsto \color{blue}{\frac{1}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}\]
6. Simplified40.1
  \[\leadsto \color{blue}{\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}} \cdot \frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\]
7. Simplified26.8
  \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\frac{\mathsf{fma}\left(y.re, x.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}\]
8. Using strategy rm
9. Applied *-un-lft-identity26.8
  \[\leadsto \frac{1}{\color{blue}{1 \cdot \mathsf{hypot}\left(y.re, y.im\right)}} \cdot \frac{\mathsf{fma}\left(y.re, x.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\]
10. Applied add-sqr-sqrt26.8
  \[\leadsto \frac{\color{blue}{\sqrt{1} \cdot \sqrt{1}}}{1 \cdot \mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(y.re, x.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\]
11. Applied times-frac26.8
  \[\leadsto \color{blue}{\left(\frac{\sqrt{1}}{1} \cdot \frac{\sqrt{1}}{\mathsf{hypot}\left(y.re, y.im\right)}\right)} \cdot \frac{\mathsf{fma}\left(y.re, x.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\]
12. Applied associate-*l*26.8
  \[\leadsto \color{blue}{\frac{\sqrt{1}}{1} \cdot \left(\frac{\sqrt{1}}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(y.re, x.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\right)}\]
13. Simplified26.8
  \[\leadsto \frac{\sqrt{1}}{1} \cdot \color{blue}{\frac{\frac{\mathsf{fma}\left(y.im, x.im, x.re \cdot y.re\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}}\]
14. Taylor expanded around -inf 16.1
  \[\leadsto \frac{\sqrt{1}}{1} \cdot \frac{\color{blue}{-1 \cdot x.im}}{\mathsf{hypot}\left(y.re, y.im\right)}\]
15. Simplified16.1
  \[\leadsto \frac{\sqrt{1}}{1} \cdot \frac{\color{blue}{-x.im}}{\mathsf{hypot}\left(y.re, y.im\right)}\]
if -3.114802040969311e+109 < y.im < 6.370255659610306e-234 or 4.694847582105376e-213 < y.im < 1.0484121450209069e+133
1. Initial program 18.3
  \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
2. Using strategy rm
3. Applied add-sqr-sqrt18.3
  \[\leadsto \frac{x.re \cdot y.re + x.im \cdot y.im}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}}\]
4. Applied *-un-lft-identity18.3
  \[\leadsto \frac{\color{blue}{1 \cdot \left(x.re \cdot y.re + x.im \cdot y.im\right)}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}\]
5. Applied times-frac18.3
  \[\leadsto \color{blue}{\frac{1}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}\]
6. Simplified18.3
  \[\leadsto \color{blue}{\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}} \cdot \frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\]
7. Simplified12.1
  \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\frac{\mathsf{fma}\left(y.re, x.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}\]
8. Using strategy rm
9. Applied *-un-lft-identity12.1
  \[\leadsto \frac{1}{\color{blue}{1 \cdot \mathsf{hypot}\left(y.re, y.im\right)}} \cdot \frac{\mathsf{fma}\left(y.re, x.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\]
10. Applied add-sqr-sqrt12.1
  \[\leadsto \frac{\color{blue}{\sqrt{1} \cdot \sqrt{1}}}{1 \cdot \mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(y.re, x.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\]
11. Applied times-frac12.1
  \[\leadsto \color{blue}{\left(\frac{\sqrt{1}}{1} \cdot \frac{\sqrt{1}}{\mathsf{hypot}\left(y.re, y.im\right)}\right)} \cdot \frac{\mathsf{fma}\left(y.re, x.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\]
12. Applied associate-*l*12.1
  \[\leadsto \color{blue}{\frac{\sqrt{1}}{1} \cdot \left(\frac{\sqrt{1}}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(y.re, x.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\right)}\]
13. Simplified11.9
  \[\leadsto \frac{\sqrt{1}}{1} \cdot \color{blue}{\frac{\frac{\mathsf{fma}\left(y.im, x.im, x.re \cdot y.re\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}}\]
if 6.370255659610306e-234 < y.im < 4.694847582105376e-213
1. Initial program 27.1
  \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
2. Using strategy rm
3. Applied add-sqr-sqrt27.1
  \[\leadsto \frac{x.re \cdot y.re + x.im \cdot y.im}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}}\]
4. Applied *-un-lft-identity27.1
  \[\leadsto \frac{\color{blue}{1 \cdot \left(x.re \cdot y.re + x.im \cdot y.im\right)}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}\]
5. Applied times-frac27.1
  \[\leadsto \color{blue}{\frac{1}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}\]
6. Simplified27.1
  \[\leadsto \color{blue}{\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}} \cdot \frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\]
7. Simplified13.8
  \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\frac{\mathsf{fma}\left(y.re, x.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}\]
8. Using strategy rm
9. Applied *-un-lft-identity13.8
  \[\leadsto \frac{1}{\color{blue}{1 \cdot \mathsf{hypot}\left(y.re, y.im\right)}} \cdot \frac{\mathsf{fma}\left(y.re, x.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\]
10. Applied add-sqr-sqrt13.8
  \[\leadsto \frac{\color{blue}{\sqrt{1} \cdot \sqrt{1}}}{1 \cdot \mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(y.re, x.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\]
11. Applied times-frac13.8
  \[\leadsto \color{blue}{\left(\frac{\sqrt{1}}{1} \cdot \frac{\sqrt{1}}{\mathsf{hypot}\left(y.re, y.im\right)}\right)} \cdot \frac{\mathsf{fma}\left(y.re, x.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\]
12. Applied associate-*l*13.8
  \[\leadsto \color{blue}{\frac{\sqrt{1}}{1} \cdot \left(\frac{\sqrt{1}}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(y.re, x.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\right)}\]
13. Simplified13.6
  \[\leadsto \frac{\sqrt{1}}{1} \cdot \color{blue}{\frac{\frac{\mathsf{fma}\left(y.im, x.im, x.re \cdot y.re\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}}\]
14. Taylor expanded around 0 36.1
  \[\leadsto \frac{\sqrt{1}}{1} \cdot \frac{\color{blue}{x.re}}{\mathsf{hypot}\left(y.re, y.im\right)}\]
if 1.0484121450209069e+133 < y.im
1. Initial program 42.6
  \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
2. Using strategy rm
3. Applied add-sqr-sqrt42.6
  \[\leadsto \frac{x.re \cdot y.re + x.im \cdot y.im}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}}\]
4. Applied *-un-lft-identity42.6
  \[\leadsto \frac{\color{blue}{1 \cdot \left(x.re \cdot y.re + x.im \cdot y.im\right)}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}\]
5. Applied times-frac42.6
  \[\leadsto \color{blue}{\frac{1}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}\]
6. Simplified42.6
  \[\leadsto \color{blue}{\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}} \cdot \frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\]
7. Simplified27.2
  \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\frac{\mathsf{fma}\left(y.re, x.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}\]
8. Using strategy rm
9. Applied *-un-lft-identity27.2
  \[\leadsto \frac{1}{\color{blue}{1 \cdot \mathsf{hypot}\left(y.re, y.im\right)}} \cdot \frac{\mathsf{fma}\left(y.re, x.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\]
10. Applied add-sqr-sqrt27.2
  \[\leadsto \frac{\color{blue}{\sqrt{1} \cdot \sqrt{1}}}{1 \cdot \mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(y.re, x.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\]
11. Applied times-frac27.2
  \[\leadsto \color{blue}{\left(\frac{\sqrt{1}}{1} \cdot \frac{\sqrt{1}}{\mathsf{hypot}\left(y.re, y.im\right)}\right)} \cdot \frac{\mathsf{fma}\left(y.re, x.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\]
12. Applied associate-*l*27.2
  \[\leadsto \color{blue}{\frac{\sqrt{1}}{1} \cdot \left(\frac{\sqrt{1}}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(y.re, x.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\right)}\]
13. Simplified27.1
  \[\leadsto \frac{\sqrt{1}}{1} \cdot \color{blue}{\frac{\frac{\mathsf{fma}\left(y.im, x.im, x.re \cdot y.re\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}}\]
14. Taylor expanded around inf 14.0
  \[\leadsto \frac{\sqrt{1}}{1} \cdot \frac{\color{blue}{x.im}}{\mathsf{hypot}\left(y.re, y.im\right)}\]
Recombined 4 regimes into one program.
Final simplification13.4
\[\leadsto \begin{array}{l} \mathbf{if}\;y.im \le -3.114802040969311027115783701676385953635 \cdot 10^{109}:\\ \;\;\;\;\frac{-x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \le 6.370255659610306169101339211646465545796 \cdot 10^{-234}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y.im, x.im, x.re \cdot y.re\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \le 4.694847582105375997704206679714705497211 \cdot 10^{-213}:\\ \;\;\;\;\frac{x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \le 1.048412145020906876185869327079410027246 \cdot 10^{133}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y.im, x.im, x.re \cdot y.re\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array}\]

Error

Derivation

`if y.im < -3.114802040969311e+109`

`if -3.114802040969311e+109 < y.im < 6.370255659610306e-234 or 4.694847582105376e-213 < y.im < 1.0484121450209069e+133`

`if 6.370255659610306e-234 < y.im < 4.694847582105376e-213`

`if 1.0484121450209069e+133 < y.im`

Reproduce