_divideComplex, imaginary part

Average Error: 25.6 → 12.8

Time: 3.9m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y.re \le -6.693395251828147 \cdot 10^{+97}:\\ \;\;\;\;\frac{-x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \le 1.321565434537417 \cdot 10^{+152}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re - x.re \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array}\]

C

double f(double x_re, double x_im, double y_re, double y_im) {
        double r12483162 = x_im;
        double r12483163 = y_re;
        double r12483164 = r12483162 * r12483163;
        double r12483165 = x_re;
        double r12483166 = y_im;
        double r12483167 = r12483165 * r12483166;
        double r12483168 = r12483164 - r12483167;
        double r12483169 = r12483163 * r12483163;
        double r12483170 = r12483166 * r12483166;
        double r12483171 = r12483169 + r12483170;
        double r12483172 = r12483168 / r12483171;
        return r12483172;
}

↓

double f(double x_re, double x_im, double y_re, double y_im) {
        double r12483173 = y_re;
        double r12483174 = -6.693395251828147e+97;
        bool r12483175 = r12483173 <= r12483174;
        double r12483176 = x_im;
        double r12483177 = -r12483176;
        double r12483178 = y_im;
        double r12483179 = hypot(r12483173, r12483178);
        double r12483180 = r12483177 / r12483179;
        double r12483181 = 1.321565434537417e+152;
        bool r12483182 = r12483173 <= r12483181;
        double r12483183 = r12483176 * r12483173;
        double r12483184 = x_re;
        double r12483185 = r12483184 * r12483178;
        double r12483186 = r12483183 - r12483185;
        double r12483187 = r12483186 / r12483179;
        double r12483188 = r12483187 / r12483179;
        double r12483189 = r12483176 / r12483179;
        double r12483190 = r12483182 ? r12483188 : r12483189;
        double r12483191 = r12483175 ? r12483180 : r12483190;
        return r12483191;
}

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 < -6.693395251828147e+97

   1. Initial program 37.9

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

   2. Simplified 37.9

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

   4. Applied add-sqr-sqrt 37.9

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

   5. Applied associate-/r* 37.8

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

   7. Applied *-un-lft-identity 37.8

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

   8. Applied sqrt-prod 37.8

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

   9. Applied *-un-lft-identity 37.8

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

   10. Applied *-un-lft-identity 37.8

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

   11. Applied times-frac 37.8

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

   12. Applied times-frac 37.8

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

   13. Simplified 37.8

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

   14. Simplified 25.5

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

   15. Taylor expanded around -inf 16.8

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

   16. Simplified 16.8

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

   if -6.693395251828147e+97 < y.re < 1.321565434537417e+152

   1. Initial program 18.8

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

   2. Simplified 18.8

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

   4. Applied add-sqr-sqrt 18.8

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

   5. Applied associate-/r* 18.7

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

   7. Applied *-un-lft-identity 18.7

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

   8. Applied sqrt-prod 18.7

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

   9. Applied *-un-lft-identity 18.7

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

   10. Applied *-un-lft-identity 18.7

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

   11. Applied times-frac 18.7

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

   12. Applied times-frac 18.7

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

   13. Simplified 18.7

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

   14. Simplified 11.6

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

   if 1.321565434537417e+152 < y.re

   1. Initial program 44.0

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

   2. Simplified 44.0

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

   4. Applied add-sqr-sqrt 44.0

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

   5. Applied associate-/r* 44.0

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

   7. Applied *-un-lft-identity 44.0

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

   8. Applied sqrt-prod 44.0

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

   9. Applied *-un-lft-identity 44.0

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

   10. Applied *-un-lft-identity 44.0

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

   11. Applied times-frac 44.0

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

   12. Applied times-frac 44.0

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

   13. Simplified 44.0

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

   14. Simplified 28.8

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

   15. Taylor expanded around inf 13.1

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

4. Final simplification 12.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \le -6.693395251828147 \cdot 10^{+97}:\\ \;\;\;\;\frac{-x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \le 1.321565434537417 \cdot 10^{+152}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re - x.re \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019107 +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))))