_divideComplex, imaginary part

Average Error: 25.7 → 12.5

Time: 11.9s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y.re \le -5.9876509466672635 \cdot 10^{162}:\\ \;\;\;\;\frac{-x.im}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{elif}\;y.re \le 2.5019882417216274 \cdot 10^{136}:\\ \;\;\;\;\frac{\frac{1}{\frac{\mathsf{hypot}\left(y.im, y.re\right)}{\mathsf{fma}\left(x.im, y.re, -y.im \cdot x.re\right)}}}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \end{array}\]

C

double f(double x_re, double x_im, double y_re, double y_im) {
        double r64972 = x_im;
        double r64973 = y_re;
        double r64974 = r64972 * r64973;
        double r64975 = x_re;
        double r64976 = y_im;
        double r64977 = r64975 * r64976;
        double r64978 = r64974 - r64977;
        double r64979 = r64973 * r64973;
        double r64980 = r64976 * r64976;
        double r64981 = r64979 + r64980;
        double r64982 = r64978 / r64981;
        return r64982;
}

↓

double f(double x_re, double x_im, double y_re, double y_im) {
        double r64983 = y_re;
        double r64984 = -5.9876509466672635e+162;
        bool r64985 = r64983 <= r64984;
        double r64986 = x_im;
        double r64987 = -r64986;
        double r64988 = y_im;
        double r64989 = hypot(r64988, r64983);
        double r64990 = r64987 / r64989;
        double r64991 = 2.5019882417216274e+136;
        bool r64992 = r64983 <= r64991;
        double r64993 = 1.0;
        double r64994 = x_re;
        double r64995 = r64988 * r64994;
        double r64996 = -r64995;
        double r64997 = fma(r64986, r64983, r64996);
        double r64998 = r64989 / r64997;
        double r64999 = r64993 / r64998;
        double r65000 = r64999 / r64989;
        double r65001 = r64986 / r64989;
        double r65002 = r64992 ? r65000 : r65001;
        double r65003 = r64985 ? r64990 : r65002;
        return r65003;
}

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 < -5.9876509466672635e+162

   1. Initial program 43.8

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

   2. Simplified 43.8

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

   4. Applied add-sqr-sqrt 43.8

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

   5. Applied *-un-lft-identity 43.8

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

   6. Applied times-frac 43.8

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

   7. Simplified 43.8

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

   8. Simplified 29.5

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

   10. Applied *-un-lft-identity 29.5

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

   11. Applied *-un-lft-identity 29.5

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

   12. Applied times-frac 29.5

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

   13. Applied associate-*l* 29.5

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

   14. Simplified 29.5

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

   15. Taylor expanded around -inf 12.6

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

   16. Simplified 12.6

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

   if -5.9876509466672635e+162 < y.re < 2.5019882417216274e+136

   1. Initial program 19.2

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

   2. Simplified 19.2

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

   4. Applied add-sqr-sqrt 19.2

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

   5. Applied *-un-lft-identity 19.2

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

   6. Applied times-frac 19.2

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

   7. Simplified 19.2

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

   8. Simplified 12.1

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

   10. Applied *-un-lft-identity 12.1

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

   11. Applied *-un-lft-identity 12.1

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

   12. Applied times-frac 12.1

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

   13. Applied associate-*l* 12.1

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

   14. Simplified 12.0

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

   16. Applied clear-num 12.1

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

   if 2.5019882417216274e+136 < y.re

   1. Initial program 42.2

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

   2. Simplified 42.2

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

   4. Applied add-sqr-sqrt 42.2

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

   5. Applied *-un-lft-identity 42.2

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

   6. Applied times-frac 42.2

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

   7. Simplified 42.2

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

   8. Simplified 27.3

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

   10. Applied *-un-lft-identity 27.3

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

   11. Applied *-un-lft-identity 27.3

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

   12. Applied times-frac 27.3

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

   13. Applied associate-*l* 27.3

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

   14. Simplified 27.3

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

   15. Taylor expanded around inf 14.6

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

4. Final simplification 12.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \le -5.9876509466672635 \cdot 10^{162}:\\ \;\;\;\;\frac{-x.im}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{elif}\;y.re \le 2.5019882417216274 \cdot 10^{136}:\\ \;\;\;\;\frac{\frac{1}{\frac{\mathsf{hypot}\left(y.im, y.re\right)}{\mathsf{fma}\left(x.im, y.re, -y.im \cdot x.re\right)}}}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_divideComplex, imaginary part"
  :precision binary64
  (/ (- (* x.im y.re) (* x.re y.im)) (+ (* y.re y.re) (* y.im y.im))))