_divideComplex, imaginary part

Average Error: 26.3 → 13.1

Time: 4.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y.re \le -2.06095728035008259 \cdot 10^{150}:\\ \;\;\;\;{\left(\frac{-1 \cdot x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\right)}^{1}\\ \mathbf{elif}\;y.re \le 1.04482715632166882 \cdot 10^{105}:\\ \;\;\;\;{\left(\frac{\left(x.im \cdot y.re - x.re \cdot y.im\right) \cdot \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\right)}^{1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\right)}^{1}\\ \end{array}\]

C

double f(double x_re, double x_im, double y_re, double y_im) {
        double r49582 = x_im;
        double r49583 = y_re;
        double r49584 = r49582 * r49583;
        double r49585 = x_re;
        double r49586 = y_im;
        double r49587 = r49585 * r49586;
        double r49588 = r49584 - r49587;
        double r49589 = r49583 * r49583;
        double r49590 = r49586 * r49586;
        double r49591 = r49589 + r49590;
        double r49592 = r49588 / r49591;
        return r49592;
}

↓

double f(double x_re, double x_im, double y_re, double y_im) {
        double r49593 = y_re;
        double r49594 = -2.0609572803500826e+150;
        bool r49595 = r49593 <= r49594;
        double r49596 = -1.0;
        double r49597 = x_im;
        double r49598 = r49596 * r49597;
        double r49599 = y_im;
        double r49600 = hypot(r49593, r49599);
        double r49601 = r49598 / r49600;
        double r49602 = 1.0;
        double r49603 = pow(r49601, r49602);
        double r49604 = 1.0448271563216688e+105;
        bool r49605 = r49593 <= r49604;
        double r49606 = r49597 * r49593;
        double r49607 = x_re;
        double r49608 = r49607 * r49599;
        double r49609 = r49606 - r49608;
        double r49610 = r49602 / r49600;
        double r49611 = r49609 * r49610;
        double r49612 = r49611 / r49600;
        double r49613 = pow(r49612, r49602);
        double r49614 = r49597 / r49600;
        double r49615 = pow(r49614, r49602);
        double r49616 = r49605 ? r49613 : r49615;
        double r49617 = r49595 ? r49603 : r49616;
        return r49617;
}

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 < -2.0609572803500826e+150

   1. Initial program 44.3

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

   3. Applied add-sqr-sqrt 44.3

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

   4. Applied *-un-lft-identity 44.3

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

   5. Applied times-frac 44.3

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

   6. Simplified 44.3

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

   7. Simplified 27.8

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

   9. Applied pow1 27.8

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

   10. Applied pow1 27.8

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

   11. Applied pow-prod-down 27.8

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

   12. Simplified 27.8

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

   13. Taylor expanded around -inf 13.0

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

   if -2.0609572803500826e+150 < y.re < 1.0448271563216688e+105

   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. Using strategy rm

   3. Applied add-sqr-sqrt 19.2

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

   4. Applied *-un-lft-identity 19.2

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

   5. Applied times-frac 19.2

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

   6. Simplified 19.2

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

   7. Simplified 12.4

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

   9. Applied pow1 12.4

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

   10. Applied pow1 12.4

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

   11. Applied pow-prod-down 12.4

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

   12. Simplified 12.2

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

   14. Applied div-inv 12.3

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

   if 1.0448271563216688e+105 < y.re

   1. Initial program 39.7

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

   3. Applied add-sqr-sqrt 39.7

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

   4. Applied *-un-lft-identity 39.7

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

   5. Applied times-frac 39.8

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

   6. Simplified 39.8

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

   7. Simplified 27.1

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

   9. Applied pow1 27.1

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

   10. Applied pow1 27.1

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

   11. Applied pow-prod-down 27.1

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

   12. Simplified 27.1

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

   13. Taylor expanded around inf 16.2

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

4. Final simplification 13.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \le -2.06095728035008259 \cdot 10^{150}:\\ \;\;\;\;{\left(\frac{-1 \cdot x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\right)}^{1}\\ \mathbf{elif}\;y.re \le 1.04482715632166882 \cdot 10^{105}:\\ \;\;\;\;{\left(\frac{\left(x.im \cdot y.re - x.re \cdot y.im\right) \cdot \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\right)}^{1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\right)}^{1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020039 +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))))