Complex division, imag part

Average Error: 25.7 → 12.5

Time: 11.3s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;c \le -5.9876509466672635 \cdot 10^{162}:\\ \;\;\;\;\frac{-b}{\mathsf{hypot}\left(d, c\right)} \cdot \left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right)\\ \mathbf{elif}\;c \le 2.5019882417216274 \cdot 10^{136}:\\ \;\;\;\;\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \frac{\frac{1}{\frac{\mathsf{hypot}\left(d, c\right)}{\mathsf{fma}\left(b, c, -d \cdot a\right)}}}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \frac{b}{\mathsf{hypot}\left(d, c\right)}\\ \end{array}\]

C

double f(double a, double b, double c, double d) {
        double r124979 = b;
        double r124980 = c;
        double r124981 = r124979 * r124980;
        double r124982 = a;
        double r124983 = d;
        double r124984 = r124982 * r124983;
        double r124985 = r124981 - r124984;
        double r124986 = r124980 * r124980;
        double r124987 = r124983 * r124983;
        double r124988 = r124986 + r124987;
        double r124989 = r124985 / r124988;
        return r124989;
}

↓

double f(double a, double b, double c, double d) {
        double r124990 = c;
        double r124991 = -5.9876509466672635e+162;
        bool r124992 = r124990 <= r124991;
        double r124993 = b;
        double r124994 = -r124993;
        double r124995 = d;
        double r124996 = hypot(r124995, r124990);
        double r124997 = r124994 / r124996;
        double r124998 = 1.0;
        double r124999 = cbrt(r124998);
        double r125000 = r124999 * r124999;
        double r125001 = r124997 * r125000;
        double r125002 = 2.5019882417216274e+136;
        bool r125003 = r124990 <= r125002;
        double r125004 = a;
        double r125005 = r124995 * r125004;
        double r125006 = -r125005;
        double r125007 = fma(r124993, r124990, r125006);
        double r125008 = r124996 / r125007;
        double r125009 = r124998 / r125008;
        double r125010 = r125009 / r124996;
        double r125011 = r125000 * r125010;
        double r125012 = r124993 / r124996;
        double r125013 = r125000 * r125012;
        double r125014 = r125003 ? r125011 : r125013;
        double r125015 = r124992 ? r125001 : r125014;
        return r125015;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original	25.7
Target	0.5
Herbie	12.5

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

Derivation

1. Split input into 3 regimes

2. if c < -5.9876509466672635e+162

   1. Initial program 43.8

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

   2. Simplified 43.8

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

   4. Applied add-sqr-sqrt 43.8

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

   5. Applied *-un-lft-identity 43.8

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

   6. Applied times-frac 43.8

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

   7. Simplified 43.8

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

   8. Simplified 29.5

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

   10. Applied *-un-lft-identity 29.5

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

   11. Applied add-cube-cbrt 29.5

       \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{1 \cdot \mathsf{hypot}\left(d, c\right)} \cdot \frac{\mathsf{fma}\left(b, c, -d \cdot a\right)}{\mathsf{hypot}\left(d, c\right)}\]

   12. Applied times-frac 29.5

       \[\leadsto \color{blue}{\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1} \cdot \frac{\sqrt[3]{1}}{\mathsf{hypot}\left(d, c\right)}\right)} \cdot \frac{\mathsf{fma}\left(b, c, -d \cdot a\right)}{\mathsf{hypot}\left(d, c\right)}\]

   13. Applied associate-*l* 29.5

       \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1} \cdot \left(\frac{\sqrt[3]{1}}{\mathsf{hypot}\left(d, c\right)} \cdot \frac{\mathsf{fma}\left(b, c, -d \cdot a\right)}{\mathsf{hypot}\left(d, c\right)}\right)}\]

   14. Simplified 29.5

       \[\leadsto \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1} \cdot \color{blue}{\frac{\frac{\mathsf{fma}\left(b, c, -d \cdot a\right)}{\mathsf{hypot}\left(d, c\right)}}{\mathsf{hypot}\left(d, c\right)}}\]

   15. Taylor expanded around -inf 12.6

       \[\leadsto \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1} \cdot \frac{\color{blue}{-1 \cdot b}}{\mathsf{hypot}\left(d, c\right)}\]

   16. Simplified 12.6

       \[\leadsto \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1} \cdot \frac{\color{blue}{-b}}{\mathsf{hypot}\left(d, c\right)}\]

   if -5.9876509466672635e+162 < c < 2.5019882417216274e+136

   1. Initial program 19.2

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

   2. Simplified 19.2

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

   4. Applied add-sqr-sqrt 19.2

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

   5. Applied *-un-lft-identity 19.2

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

   6. Applied times-frac 19.2

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

   7. Simplified 19.2

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

   8. Simplified 12.1

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

   10. Applied *-un-lft-identity 12.1

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

   11. Applied add-cube-cbrt 12.1

       \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{1 \cdot \mathsf{hypot}\left(d, c\right)} \cdot \frac{\mathsf{fma}\left(b, c, -d \cdot a\right)}{\mathsf{hypot}\left(d, c\right)}\]

   12. Applied times-frac 12.1

       \[\leadsto \color{blue}{\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1} \cdot \frac{\sqrt[3]{1}}{\mathsf{hypot}\left(d, c\right)}\right)} \cdot \frac{\mathsf{fma}\left(b, c, -d \cdot a\right)}{\mathsf{hypot}\left(d, c\right)}\]

   13. Applied associate-*l* 12.1

       \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1} \cdot \left(\frac{\sqrt[3]{1}}{\mathsf{hypot}\left(d, c\right)} \cdot \frac{\mathsf{fma}\left(b, c, -d \cdot a\right)}{\mathsf{hypot}\left(d, c\right)}\right)}\]

   14. Simplified 12.0

       \[\leadsto \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1} \cdot \color{blue}{\frac{\frac{\mathsf{fma}\left(b, c, -d \cdot a\right)}{\mathsf{hypot}\left(d, c\right)}}{\mathsf{hypot}\left(d, c\right)}}\]
   15. Using strategy rm

   16. Applied clear-num 12.1

       \[\leadsto \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1} \cdot \frac{\color{blue}{\frac{1}{\frac{\mathsf{hypot}\left(d, c\right)}{\mathsf{fma}\left(b, c, -d \cdot a\right)}}}}{\mathsf{hypot}\left(d, c\right)}\]

   if 2.5019882417216274e+136 < c

   1. Initial program 42.2

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

   2. Simplified 42.2

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

   4. Applied add-sqr-sqrt 42.2

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

   5. Applied *-un-lft-identity 42.2

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

   6. Applied times-frac 42.2

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

   7. Simplified 42.2

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

   8. Simplified 27.3

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

   10. Applied *-un-lft-identity 27.3

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

   11. Applied add-cube-cbrt 27.3

       \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{1 \cdot \mathsf{hypot}\left(d, c\right)} \cdot \frac{\mathsf{fma}\left(b, c, -d \cdot a\right)}{\mathsf{hypot}\left(d, c\right)}\]

   12. Applied times-frac 27.3

       \[\leadsto \color{blue}{\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1} \cdot \frac{\sqrt[3]{1}}{\mathsf{hypot}\left(d, c\right)}\right)} \cdot \frac{\mathsf{fma}\left(b, c, -d \cdot a\right)}{\mathsf{hypot}\left(d, c\right)}\]

   13. Applied associate-*l* 27.3

       \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1} \cdot \left(\frac{\sqrt[3]{1}}{\mathsf{hypot}\left(d, c\right)} \cdot \frac{\mathsf{fma}\left(b, c, -d \cdot a\right)}{\mathsf{hypot}\left(d, c\right)}\right)}\]

   14. Simplified 27.3

       \[\leadsto \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1} \cdot \color{blue}{\frac{\frac{\mathsf{fma}\left(b, c, -d \cdot a\right)}{\mathsf{hypot}\left(d, c\right)}}{\mathsf{hypot}\left(d, c\right)}}\]

   15. Taylor expanded around inf 14.6

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

4. Final simplification 12.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \le -5.9876509466672635 \cdot 10^{162}:\\ \;\;\;\;\frac{-b}{\mathsf{hypot}\left(d, c\right)} \cdot \left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right)\\ \mathbf{elif}\;c \le 2.5019882417216274 \cdot 10^{136}:\\ \;\;\;\;\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \frac{\frac{1}{\frac{\mathsf{hypot}\left(d, c\right)}{\mathsf{fma}\left(b, c, -d \cdot a\right)}}}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \frac{b}{\mathsf{hypot}\left(d, c\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +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))))