Complex division, imag part

Average Error: 26.5 → 14.9

Time: 5.1s

Precision: 64

Math

\[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]

↓

\[\begin{array}{l} \mathbf{if}\;d \le -5.0306813936220228 \cdot 10^{154}:\\ \;\;\;\;{\left(\frac{\frac{b \cdot c - a \cdot d}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)}\right)}^{1}\\ \mathbf{elif}\;d \le -7.28046239192008739 \cdot 10^{83}:\\ \;\;\;\;\frac{b}{\frac{\mathsf{fma}\left(c, c, d \cdot d\right)}{c}} - \frac{a}{\frac{\mathsf{fma}\left(c, c, d \cdot d\right)}{d}}\\ \mathbf{elif}\;d \le 3.8050518390474684 \cdot 10^{-78}:\\ \;\;\;\;{\left(\frac{\frac{b \cdot c - a \cdot d}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)}\right)}^{1}\\ \mathbf{elif}\;d \le 5.27378373520958845 \cdot 10^{-64}:\\ \;\;\;\;{\left(\frac{b}{\mathsf{hypot}\left(c, d\right)}\right)}^{1}\\ \mathbf{elif}\;d \le 1.9537314790331482 \cdot 10^{152}:\\ \;\;\;\;{\left(\frac{\frac{b \cdot c - a \cdot d}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)}\right)}^{1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{-1 \cdot a}{\mathsf{hypot}\left(c, d\right)}\right)}^{1}\\ \end{array}\]

C

double f(double a, double b, double c, double d) {
        double r110874 = b;
        double r110875 = c;
        double r110876 = r110874 * r110875;
        double r110877 = a;
        double r110878 = d;
        double r110879 = r110877 * r110878;
        double r110880 = r110876 - r110879;
        double r110881 = r110875 * r110875;
        double r110882 = r110878 * r110878;
        double r110883 = r110881 + r110882;
        double r110884 = r110880 / r110883;
        return r110884;
}

↓

double f(double a, double b, double c, double d) {
        double r110885 = d;
        double r110886 = -5.030681393622023e+154;
        bool r110887 = r110885 <= r110886;
        double r110888 = b;
        double r110889 = c;
        double r110890 = r110888 * r110889;
        double r110891 = a;
        double r110892 = r110891 * r110885;
        double r110893 = r110890 - r110892;
        double r110894 = hypot(r110889, r110885);
        double r110895 = r110893 / r110894;
        double r110896 = r110895 / r110894;
        double r110897 = 1.0;
        double r110898 = pow(r110896, r110897);
        double r110899 = -7.280462391920087e+83;
        bool r110900 = r110885 <= r110899;
        double r110901 = r110885 * r110885;
        double r110902 = fma(r110889, r110889, r110901);
        double r110903 = r110902 / r110889;
        double r110904 = r110888 / r110903;
        double r110905 = r110902 / r110885;
        double r110906 = r110891 / r110905;
        double r110907 = r110904 - r110906;
        double r110908 = 3.8050518390474684e-78;
        bool r110909 = r110885 <= r110908;
        double r110910 = 5.2737837352095884e-64;
        bool r110911 = r110885 <= r110910;
        double r110912 = r110888 / r110894;
        double r110913 = pow(r110912, r110897);
        double r110914 = 1.9537314790331482e+152;
        bool r110915 = r110885 <= r110914;
        double r110916 = -1.0;
        double r110917 = r110916 * r110891;
        double r110918 = r110917 / r110894;
        double r110919 = pow(r110918, r110897);
        double r110920 = r110915 ? r110898 : r110919;
        double r110921 = r110911 ? r110913 : r110920;
        double r110922 = r110909 ? r110898 : r110921;
        double r110923 = r110900 ? r110907 : r110922;
        double r110924 = r110887 ? r110898 : r110923;
        return r110924;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original	26.5
Target	0.4
Herbie	14.9

\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-a\right) + b \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

1. Split input into 4 regimes

2. if d < -5.030681393622023e+154 or -7.280462391920087e+83 < d < 3.8050518390474684e-78 or 5.2737837352095884e-64 < d < 1.9537314790331482e+152

   1. Initial program 23.7

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt 23.7

      \[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]

   4. Applied *-un-lft-identity 23.7

      \[\leadsto \frac{\color{blue}{1 \cdot \left(b \cdot c - a \cdot d\right)}}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}\]

   5. Applied times-frac 23.7

      \[\leadsto \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}}\]

   6. Simplified 23.7

      \[\leadsto \color{blue}{\frac{1}{\mathsf{hypot}\left(c, d\right) \cdot 1}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}\]

   7. Simplified 14.9

      \[\leadsto \frac{1}{\mathsf{hypot}\left(c, d\right) \cdot 1} \cdot \color{blue}{\frac{b \cdot c - a \cdot d}{\mathsf{hypot}\left(c, d\right)}}\]
   8. Using strategy rm

   9. Applied pow1 14.9

      \[\leadsto \frac{1}{\mathsf{hypot}\left(c, d\right) \cdot 1} \cdot \color{blue}{{\left(\frac{b \cdot c - a \cdot d}{\mathsf{hypot}\left(c, d\right)}\right)}^{1}}\]

   10. Applied pow1 14.9

       \[\leadsto \color{blue}{{\left(\frac{1}{\mathsf{hypot}\left(c, d\right) \cdot 1}\right)}^{1}} \cdot {\left(\frac{b \cdot c - a \cdot d}{\mathsf{hypot}\left(c, d\right)}\right)}^{1}\]

   11. Applied pow-prod-down 14.9

       \[\leadsto \color{blue}{{\left(\frac{1}{\mathsf{hypot}\left(c, d\right) \cdot 1} \cdot \frac{b \cdot c - a \cdot d}{\mathsf{hypot}\left(c, d\right)}\right)}^{1}}\]

   12. Simplified 14.8

       \[\leadsto {\color{blue}{\left(\frac{\frac{b \cdot c - a \cdot d}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)}\right)}}^{1}\]

   if -5.030681393622023e+154 < d < -7.280462391920087e+83

   1. Initial program 23.7

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
   2. Using strategy rm

   3. Applied div-sub 23.7

      \[\leadsto \color{blue}{\frac{b \cdot c}{c \cdot c + d \cdot d} - \frac{a \cdot d}{c \cdot c + d \cdot d}}\]

   4. Simplified 23.7

      \[\leadsto \color{blue}{\frac{b}{\frac{\mathsf{fma}\left(c, c, d \cdot d\right)}{c}}} - \frac{a \cdot d}{c \cdot c + d \cdot d}\]

   5. Simplified 15.8

      \[\leadsto \frac{b}{\frac{\mathsf{fma}\left(c, c, d \cdot d\right)}{c}} - \color{blue}{\frac{a}{\frac{\mathsf{fma}\left(c, c, d \cdot d\right)}{d}}}\]

   if 3.8050518390474684e-78 < d < 5.2737837352095884e-64

   1. Initial program 18.6

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt 18.6

      \[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]

   4. Applied *-un-lft-identity 18.6

      \[\leadsto \frac{\color{blue}{1 \cdot \left(b \cdot c - a \cdot d\right)}}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}\]

   5. Applied times-frac 18.6

      \[\leadsto \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}}\]

   6. Simplified 18.6

      \[\leadsto \color{blue}{\frac{1}{\mathsf{hypot}\left(c, d\right) \cdot 1}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}\]

   7. Simplified 14.2

      \[\leadsto \frac{1}{\mathsf{hypot}\left(c, d\right) \cdot 1} \cdot \color{blue}{\frac{b \cdot c - a \cdot d}{\mathsf{hypot}\left(c, d\right)}}\]
   8. Using strategy rm

   9. Applied pow1 14.2

      \[\leadsto \frac{1}{\mathsf{hypot}\left(c, d\right) \cdot 1} \cdot \color{blue}{{\left(\frac{b \cdot c - a \cdot d}{\mathsf{hypot}\left(c, d\right)}\right)}^{1}}\]

   10. Applied pow1 14.2

       \[\leadsto \color{blue}{{\left(\frac{1}{\mathsf{hypot}\left(c, d\right) \cdot 1}\right)}^{1}} \cdot {\left(\frac{b \cdot c - a \cdot d}{\mathsf{hypot}\left(c, d\right)}\right)}^{1}\]

   11. Applied pow-prod-down 14.2

       \[\leadsto \color{blue}{{\left(\frac{1}{\mathsf{hypot}\left(c, d\right) \cdot 1} \cdot \frac{b \cdot c - a \cdot d}{\mathsf{hypot}\left(c, d\right)}\right)}^{1}}\]

   12. Simplified 14.1

       \[\leadsto {\color{blue}{\left(\frac{\frac{b \cdot c - a \cdot d}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)}\right)}}^{1}\]

   13. Taylor expanded around inf 40.5

       \[\leadsto {\left(\frac{\color{blue}{b}}{\mathsf{hypot}\left(c, d\right)}\right)}^{1}\]

   if 1.9537314790331482e+152 < d

   1. Initial program 43.9

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt 43.9

      \[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]

   4. Applied *-un-lft-identity 43.9

      \[\leadsto \frac{\color{blue}{1 \cdot \left(b \cdot c - a \cdot d\right)}}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}\]

   5. Applied times-frac 43.9

      \[\leadsto \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}}\]

   6. Simplified 43.9

      \[\leadsto \color{blue}{\frac{1}{\mathsf{hypot}\left(c, d\right) \cdot 1}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}\]

   7. Simplified 28.2

      \[\leadsto \frac{1}{\mathsf{hypot}\left(c, d\right) \cdot 1} \cdot \color{blue}{\frac{b \cdot c - a \cdot d}{\mathsf{hypot}\left(c, d\right)}}\]
   8. Using strategy rm

   9. Applied pow1 28.2

      \[\leadsto \frac{1}{\mathsf{hypot}\left(c, d\right) \cdot 1} \cdot \color{blue}{{\left(\frac{b \cdot c - a \cdot d}{\mathsf{hypot}\left(c, d\right)}\right)}^{1}}\]

   10. Applied pow1 28.2

       \[\leadsto \color{blue}{{\left(\frac{1}{\mathsf{hypot}\left(c, d\right) \cdot 1}\right)}^{1}} \cdot {\left(\frac{b \cdot c - a \cdot d}{\mathsf{hypot}\left(c, d\right)}\right)}^{1}\]

   11. Applied pow-prod-down 28.2

       \[\leadsto \color{blue}{{\left(\frac{1}{\mathsf{hypot}\left(c, d\right) \cdot 1} \cdot \frac{b \cdot c - a \cdot d}{\mathsf{hypot}\left(c, d\right)}\right)}^{1}}\]

   12. Simplified 28.2

       \[\leadsto {\color{blue}{\left(\frac{\frac{b \cdot c - a \cdot d}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)}\right)}}^{1}\]

   13. Taylor expanded around 0 13.2

       \[\leadsto {\left(\frac{\color{blue}{-1 \cdot a}}{\mathsf{hypot}\left(c, d\right)}\right)}^{1}\]
3. Recombined 4 regimes into one program.

4. Final simplification 14.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \le -5.0306813936220228 \cdot 10^{154}:\\ \;\;\;\;{\left(\frac{\frac{b \cdot c - a \cdot d}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)}\right)}^{1}\\ \mathbf{elif}\;d \le -7.28046239192008739 \cdot 10^{83}:\\ \;\;\;\;\frac{b}{\frac{\mathsf{fma}\left(c, c, d \cdot d\right)}{c}} - \frac{a}{\frac{\mathsf{fma}\left(c, c, d \cdot d\right)}{d}}\\ \mathbf{elif}\;d \le 3.8050518390474684 \cdot 10^{-78}:\\ \;\;\;\;{\left(\frac{\frac{b \cdot c - a \cdot d}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)}\right)}^{1}\\ \mathbf{elif}\;d \le 5.27378373520958845 \cdot 10^{-64}:\\ \;\;\;\;{\left(\frac{b}{\mathsf{hypot}\left(c, d\right)}\right)}^{1}\\ \mathbf{elif}\;d \le 1.9537314790331482 \cdot 10^{152}:\\ \;\;\;\;{\left(\frac{\frac{b \cdot c - a \cdot d}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)}\right)}^{1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{-1 \cdot a}{\mathsf{hypot}\left(c, d\right)}\right)}^{1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020064 +o rules:numerics
(FPCore (a b c d)
  :name "Complex division, imag part"
  :precision binary64

  :herbie-target
  (if (< (fabs d) (fabs c)) (/ (- b (* a (/ d c))) (+ c (* d (/ d c)))) (/ (+ (- a) (* b (/ c d))) (+ d (* c (/ c d)))))

  (/ (- (* b c) (* a d)) (+ (* c c) (* d d))))