Complex division, real part

Average Error: 25.8 → 12.0

Time: 13.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;c \le -2.328453055309952838396386350346556939505 \cdot 10^{184}:\\ \;\;\;\;\frac{-a}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;c \le 1.721692170326450703244997905752579178238 \cdot 10^{180}:\\ \;\;\;\;\frac{\frac{1}{\frac{\mathsf{hypot}\left(c, d\right)}{\mathsf{fma}\left(d, b, a \cdot c\right)}}}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\mathsf{hypot}\left(c, d\right)}\\ \end{array}\]

C

double f(double a, double b, double c, double d) {
        double r137831 = a;
        double r137832 = c;
        double r137833 = r137831 * r137832;
        double r137834 = b;
        double r137835 = d;
        double r137836 = r137834 * r137835;
        double r137837 = r137833 + r137836;
        double r137838 = r137832 * r137832;
        double r137839 = r137835 * r137835;
        double r137840 = r137838 + r137839;
        double r137841 = r137837 / r137840;
        return r137841;
}

↓

double f(double a, double b, double c, double d) {
        double r137842 = c;
        double r137843 = -2.328453055309953e+184;
        bool r137844 = r137842 <= r137843;
        double r137845 = a;
        double r137846 = -r137845;
        double r137847 = d;
        double r137848 = hypot(r137842, r137847);
        double r137849 = r137846 / r137848;
        double r137850 = 1.7216921703264507e+180;
        bool r137851 = r137842 <= r137850;
        double r137852 = 1.0;
        double r137853 = b;
        double r137854 = r137845 * r137842;
        double r137855 = fma(r137847, r137853, r137854);
        double r137856 = r137848 / r137855;
        double r137857 = r137852 / r137856;
        double r137858 = r137857 / r137848;
        double r137859 = r137845 / r137848;
        double r137860 = r137851 ? r137858 : r137859;
        double r137861 = r137844 ? r137849 : r137860;
        return r137861;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original	25.8
Target	0.4
Herbie	12.0

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

Derivation

1. Split input into 3 regimes

2. if c < -2.328453055309953e+184

   1. Initial program 45.4

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

   3. Applied add-sqr-sqrt 45.4

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

   4. Applied *-un-lft-identity 45.4

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

   5. Applied times-frac 45.4

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

   6. Simplified 45.4

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

   7. Simplified 32.6

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

   9. Applied associate-*r/ 32.6

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

   10. Simplified 32.6

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

   11. Taylor expanded around -inf 10.9

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

   12. Simplified 10.9

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

   if -2.328453055309953e+184 < c < 1.7216921703264507e+180

   1. Initial program 20.8

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

   3. Applied add-sqr-sqrt 20.8

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

   4. Applied *-un-lft-identity 20.8

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

   5. Applied times-frac 20.8

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

   6. Simplified 20.8

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

   7. Simplified 12.3

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

   9. Applied associate-*r/ 12.2

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

   10. Simplified 12.1

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

   12. Applied clear-num 12.2

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

   13. Simplified 12.2

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

   if 1.7216921703264507e+180 < c

   1. Initial program 42.8

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

   3. Applied add-sqr-sqrt 42.8

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

   4. Applied *-un-lft-identity 42.8

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

   5. Applied times-frac 42.8

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

   6. Simplified 42.8

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

   7. Simplified 29.2

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

   9. Applied associate-*r/ 29.1

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

   10. Simplified 29.1

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

   11. Taylor expanded around inf 11.6

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

4. Final simplification 12.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \le -2.328453055309952838396386350346556939505 \cdot 10^{184}:\\ \;\;\;\;\frac{-a}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;c \le 1.721692170326450703244997905752579178238 \cdot 10^{180}:\\ \;\;\;\;\frac{\frac{1}{\frac{\mathsf{hypot}\left(c, d\right)}{\mathsf{fma}\left(d, b, a \cdot c\right)}}}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\mathsf{hypot}\left(c, d\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 +o rules:numerics
(FPCore (a b c d)
  :name "Complex division, real part"

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

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