_divideComplex, imaginary part

Average Error: 26.0 → 13.4

Time: 15.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y.re \le -9.293826114497670869139658346683561592978 \cdot 10^{210}:\\ \;\;\;\;\frac{-1}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot x.im\\ \mathbf{elif}\;y.re \le 1.15056779942975516580562280806905910555 \cdot 10^{182}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re - y.im \cdot x.re}{\mathsf{hypot}\left(y.im, y.re\right)}}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \frac{1}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \end{array}\]

C

double f(double x_re, double x_im, double y_re, double y_im) {
        double r2625755 = x_im;
        double r2625756 = y_re;
        double r2625757 = r2625755 * r2625756;
        double r2625758 = x_re;
        double r2625759 = y_im;
        double r2625760 = r2625758 * r2625759;
        double r2625761 = r2625757 - r2625760;
        double r2625762 = r2625756 * r2625756;
        double r2625763 = r2625759 * r2625759;
        double r2625764 = r2625762 + r2625763;
        double r2625765 = r2625761 / r2625764;
        return r2625765;
}

↓

double f(double x_re, double x_im, double y_re, double y_im) {
        double r2625766 = y_re;
        double r2625767 = -9.293826114497671e+210;
        bool r2625768 = r2625766 <= r2625767;
        double r2625769 = -1.0;
        double r2625770 = y_im;
        double r2625771 = hypot(r2625770, r2625766);
        double r2625772 = r2625769 / r2625771;
        double r2625773 = x_im;
        double r2625774 = r2625772 * r2625773;
        double r2625775 = 1.1505677994297552e+182;
        bool r2625776 = r2625766 <= r2625775;
        double r2625777 = r2625773 * r2625766;
        double r2625778 = x_re;
        double r2625779 = r2625770 * r2625778;
        double r2625780 = r2625777 - r2625779;
        double r2625781 = r2625780 / r2625771;
        double r2625782 = r2625781 / r2625771;
        double r2625783 = 1.0;
        double r2625784 = r2625783 / r2625771;
        double r2625785 = r2625773 * r2625784;
        double r2625786 = r2625776 ? r2625782 : r2625785;
        double r2625787 = r2625768 ? r2625774 : r2625786;
        return r2625787;
}

Error

Bits error versus x.re

Bits error versus x.im

Bits error versus y.re

Bits error versus y.im

Derivation

1. Split input into 3 regimes

2. if y.re < -9.293826114497671e+210

   1. Initial program 41.6

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

   2. Simplified 41.6

      \[\leadsto \color{blue}{\frac{x.im \cdot y.re - x.re \cdot y.im}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}}\]
   3. Using strategy rm

   4. Applied clear-num 41.6

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}{x.im \cdot y.re - x.re \cdot y.im}}}\]
   5. Using strategy rm

   6. Applied *-un-lft-identity 41.6

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}{\color{blue}{1 \cdot \left(x.im \cdot y.re - x.re \cdot y.im\right)}}}\]

   7. Applied add-sqr-sqrt 41.6

      \[\leadsto \frac{1}{\frac{\color{blue}{\sqrt{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)} \cdot \sqrt{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}}}{1 \cdot \left(x.im \cdot y.re - x.re \cdot y.im\right)}}\]

   8. Applied times-frac 41.6

      \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}}{1} \cdot \frac{\sqrt{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}}{x.im \cdot y.re - x.re \cdot y.im}}}\]

   9. Applied add-cube-cbrt 41.6

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\frac{\sqrt{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}}{1} \cdot \frac{\sqrt{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}}{x.im \cdot y.re - x.re \cdot y.im}}\]

   10. Applied times-frac 41.6

       \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\frac{\sqrt{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}}{1}} \cdot \frac{\sqrt[3]{1}}{\frac{\sqrt{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}}{x.im \cdot y.re - x.re \cdot y.im}}}\]

   11. Simplified 41.6

       \[\leadsto \color{blue}{\frac{1}{\mathsf{hypot}\left(y.im, y.re\right)}} \cdot \frac{\sqrt[3]{1}}{\frac{\sqrt{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}}{x.im \cdot y.re - x.re \cdot y.im}}\]

   12. Simplified 29.9

       \[\leadsto \frac{1}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot \color{blue}{\frac{y.re \cdot x.im - y.im \cdot x.re}{\mathsf{hypot}\left(y.im, y.re\right)}}\]

   13. Taylor expanded around -inf 10.2

       \[\leadsto \frac{1}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot \color{blue}{\left(-1 \cdot x.im\right)}\]

   14. Simplified 10.2

       \[\leadsto \frac{1}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot \color{blue}{\left(-x.im\right)}\]

   if -9.293826114497671e+210 < y.re < 1.1505677994297552e+182

   1. Initial program 22.3

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

   2. Simplified 22.3

      \[\leadsto \color{blue}{\frac{x.im \cdot y.re - x.re \cdot y.im}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}}\]
   3. Using strategy rm

   4. Applied clear-num 22.4

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}{x.im \cdot y.re - x.re \cdot y.im}}}\]
   5. Using strategy rm

   6. Applied *-un-lft-identity 22.4

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}{\color{blue}{1 \cdot \left(x.im \cdot y.re - x.re \cdot y.im\right)}}}\]

   7. Applied add-sqr-sqrt 22.4

      \[\leadsto \frac{1}{\frac{\color{blue}{\sqrt{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)} \cdot \sqrt{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}}}{1 \cdot \left(x.im \cdot y.re - x.re \cdot y.im\right)}}\]

   8. Applied times-frac 22.4

      \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}}{1} \cdot \frac{\sqrt{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}}{x.im \cdot y.re - x.re \cdot y.im}}}\]

   9. Applied add-cube-cbrt 22.4

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\frac{\sqrt{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}}{1} \cdot \frac{\sqrt{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}}{x.im \cdot y.re - x.re \cdot y.im}}\]

   10. Applied times-frac 22.3

       \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\frac{\sqrt{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}}{1}} \cdot \frac{\sqrt[3]{1}}{\frac{\sqrt{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}}{x.im \cdot y.re - x.re \cdot y.im}}}\]

   11. Simplified 22.3

       \[\leadsto \color{blue}{\frac{1}{\mathsf{hypot}\left(y.im, y.re\right)}} \cdot \frac{\sqrt[3]{1}}{\frac{\sqrt{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}}{x.im \cdot y.re - x.re \cdot y.im}}\]

   12. Simplified 14.0

       \[\leadsto \frac{1}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot \color{blue}{\frac{y.re \cdot x.im - y.im \cdot x.re}{\mathsf{hypot}\left(y.im, y.re\right)}}\]
   13. Using strategy rm

   14. Applied associate-*r/ 14.0

       \[\leadsto \color{blue}{\frac{\frac{1}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot \left(y.re \cdot x.im - y.im \cdot x.re\right)}{\mathsf{hypot}\left(y.im, y.re\right)}}\]

   15. Simplified 13.9

       \[\leadsto \frac{\color{blue}{\frac{y.re \cdot x.im - y.im \cdot x.re}{\mathsf{hypot}\left(y.im, y.re\right)}}}{\mathsf{hypot}\left(y.im, y.re\right)}\]

   if 1.1505677994297552e+182 < y.re

   1. Initial program 42.1

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

   2. Simplified 42.1

      \[\leadsto \color{blue}{\frac{x.im \cdot y.re - x.re \cdot y.im}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}}\]
   3. Using strategy rm

   4. Applied clear-num 42.1

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}{x.im \cdot y.re - x.re \cdot y.im}}}\]
   5. Using strategy rm

   6. Applied *-un-lft-identity 42.1

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}{\color{blue}{1 \cdot \left(x.im \cdot y.re - x.re \cdot y.im\right)}}}\]

   7. Applied add-sqr-sqrt 42.1

      \[\leadsto \frac{1}{\frac{\color{blue}{\sqrt{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)} \cdot \sqrt{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}}}{1 \cdot \left(x.im \cdot y.re - x.re \cdot y.im\right)}}\]

   8. Applied times-frac 42.1

      \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}}{1} \cdot \frac{\sqrt{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}}{x.im \cdot y.re - x.re \cdot y.im}}}\]

   9. Applied add-cube-cbrt 42.1

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\frac{\sqrt{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}}{1} \cdot \frac{\sqrt{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}}{x.im \cdot y.re - x.re \cdot y.im}}\]

   10. Applied times-frac 42.1

       \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\frac{\sqrt{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}}{1}} \cdot \frac{\sqrt[3]{1}}{\frac{\sqrt{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}}{x.im \cdot y.re - x.re \cdot y.im}}}\]

   11. Simplified 42.1

       \[\leadsto \color{blue}{\frac{1}{\mathsf{hypot}\left(y.im, y.re\right)}} \cdot \frac{\sqrt[3]{1}}{\frac{\sqrt{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}}{x.im \cdot y.re - x.re \cdot y.im}}\]

   12. Simplified 29.5

       \[\leadsto \frac{1}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot \color{blue}{\frac{y.re \cdot x.im - y.im \cdot x.re}{\mathsf{hypot}\left(y.im, y.re\right)}}\]

   13. Taylor expanded around inf 12.1

       \[\leadsto \frac{1}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot \color{blue}{x.im}\]
3. Recombined 3 regimes into one program.

4. Final simplification 13.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \le -9.293826114497670869139658346683561592978 \cdot 10^{210}:\\ \;\;\;\;\frac{-1}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot x.im\\ \mathbf{elif}\;y.re \le 1.15056779942975516580562280806905910555 \cdot 10^{182}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re - y.im \cdot x.re}{\mathsf{hypot}\left(y.im, y.re\right)}}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \frac{1}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019192 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_divideComplex, imaginary part"
  (/ (- (* x.im y.re) (* x.re y.im)) (+ (* y.re y.re) (* y.im y.im))))