_divideComplex, imaginary part

Average Error: 26.3 → 12.9

Time: 10.5s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y.re \le -2.038119772413425795183034551315875124052 \cdot 10^{160}:\\ \;\;\;\;\frac{-x.im}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \sqrt{1}\\ \mathbf{elif}\;y.re \le 3.313512139340001910855338216147245066017 \cdot 10^{193}:\\ \;\;\;\;\sqrt{1} \cdot \frac{\frac{y.re \cdot x.im + \left(-y.im \cdot x.re\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1} \cdot \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array}\]

C

double f(double x_re, double x_im, double y_re, double y_im) {
        double r104192 = x_im;
        double r104193 = y_re;
        double r104194 = r104192 * r104193;
        double r104195 = x_re;
        double r104196 = y_im;
        double r104197 = r104195 * r104196;
        double r104198 = r104194 - r104197;
        double r104199 = r104193 * r104193;
        double r104200 = r104196 * r104196;
        double r104201 = r104199 + r104200;
        double r104202 = r104198 / r104201;
        return r104202;
}

↓

double f(double x_re, double x_im, double y_re, double y_im) {
        double r104203 = y_re;
        double r104204 = -2.0381197724134258e+160;
        bool r104205 = r104203 <= r104204;
        double r104206 = x_im;
        double r104207 = -r104206;
        double r104208 = y_im;
        double r104209 = hypot(r104203, r104208);
        double r104210 = r104207 / r104209;
        double r104211 = 1.0;
        double r104212 = sqrt(r104211);
        double r104213 = r104210 * r104212;
        double r104214 = 3.313512139340002e+193;
        bool r104215 = r104203 <= r104214;
        double r104216 = r104203 * r104206;
        double r104217 = x_re;
        double r104218 = r104208 * r104217;
        double r104219 = -r104218;
        double r104220 = r104216 + r104219;
        double r104221 = r104220 / r104209;
        double r104222 = r104221 / r104209;
        double r104223 = r104212 * r104222;
        double r104224 = r104206 / r104209;
        double r104225 = r104212 * r104224;
        double r104226 = r104215 ? r104223 : r104225;
        double r104227 = r104205 ? r104213 : r104226;
        return r104227;
}

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 < -2.0381197724134258e+160

   1. Initial program 45.8

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt 45.8

      \[\leadsto \frac{x.im \cdot y.re - x.re \cdot y.im}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}}\]

   4. Applied *-un-lft-identity 45.8

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

   5. Applied times-frac 45.8

      \[\leadsto \color{blue}{\frac{1}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}\]

   6. Simplified 45.8

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

   7. Simplified 30.6

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

   9. Applied *-un-lft-identity 30.6

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

   10. Applied add-sqr-sqrt 30.6

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

   11. Applied times-frac 30.6

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

   12. Applied associate-*l* 30.6

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

   13. Simplified 30.5

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

   14. Taylor expanded around -inf 12.6

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

   15. Simplified 12.6

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

   if -2.0381197724134258e+160 < y.re < 3.313512139340002e+193

   1. Initial program 20.9

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt 20.9

      \[\leadsto \frac{x.im \cdot y.re - x.re \cdot y.im}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}}\]

   4. Applied *-un-lft-identity 20.9

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

   5. Applied times-frac 21.0

      \[\leadsto \color{blue}{\frac{1}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}\]

   6. Simplified 21.0

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

   7. Simplified 13.1

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

   9. Applied *-un-lft-identity 13.1

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

   10. Applied add-sqr-sqrt 13.1

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

   11. Applied times-frac 13.1

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

   12. Applied associate-*l* 13.1

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

   13. Simplified 13.0

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

   15. Applied fma-udef 13.0

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

   16. Simplified 13.0

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

   if 3.313512139340002e+193 < y.re

   1. Initial program 43.2

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt 43.2

      \[\leadsto \frac{x.im \cdot y.re - x.re \cdot y.im}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}}\]

   4. Applied *-un-lft-identity 43.2

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

   5. Applied times-frac 43.2

      \[\leadsto \color{blue}{\frac{1}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}\]

   6. Simplified 43.2

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

   7. Simplified 30.1

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

   9. Applied *-un-lft-identity 30.1

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

   10. Applied add-sqr-sqrt 30.1

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

   11. Applied times-frac 30.1

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

   12. Applied associate-*l* 30.1

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

   13. Simplified 30.1

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

   14. Taylor expanded around inf 12.3

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

4. Final simplification 12.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \le -2.038119772413425795183034551315875124052 \cdot 10^{160}:\\ \;\;\;\;\frac{-x.im}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \sqrt{1}\\ \mathbf{elif}\;y.re \le 3.313512139340001910855338216147245066017 \cdot 10^{193}:\\ \;\;\;\;\sqrt{1} \cdot \frac{\frac{y.re \cdot x.im + \left(-y.im \cdot x.re\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1} \cdot \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019351 +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))))