Complex division, imag part

Average Error: 26.0 → 13.4

Time: 15.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;c \le -9.293826114497670869139658346683561592978 \cdot 10^{210}:\\ \;\;\;\;\frac{-1}{\mathsf{hypot}\left(d, c\right)} \cdot b\\ \mathbf{elif}\;c \le 1.15056779942975516580562280806905910555 \cdot 10^{182}:\\ \;\;\;\;\frac{\frac{b \cdot c - d \cdot a}{\mathsf{hypot}\left(d, c\right)}}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{1}{\mathsf{hypot}\left(d, c\right)}\\ \end{array}\]

C

double f(double a, double b, double c, double d) {
        double r5727973 = b;
        double r5727974 = c;
        double r5727975 = r5727973 * r5727974;
        double r5727976 = a;
        double r5727977 = d;
        double r5727978 = r5727976 * r5727977;
        double r5727979 = r5727975 - r5727978;
        double r5727980 = r5727974 * r5727974;
        double r5727981 = r5727977 * r5727977;
        double r5727982 = r5727980 + r5727981;
        double r5727983 = r5727979 / r5727982;
        return r5727983;
}

↓

double f(double a, double b, double c, double d) {
        double r5727984 = c;
        double r5727985 = -9.293826114497671e+210;
        bool r5727986 = r5727984 <= r5727985;
        double r5727987 = -1.0;
        double r5727988 = d;
        double r5727989 = hypot(r5727988, r5727984);
        double r5727990 = r5727987 / r5727989;
        double r5727991 = b;
        double r5727992 = r5727990 * r5727991;
        double r5727993 = 1.1505677994297552e+182;
        bool r5727994 = r5727984 <= r5727993;
        double r5727995 = r5727991 * r5727984;
        double r5727996 = a;
        double r5727997 = r5727988 * r5727996;
        double r5727998 = r5727995 - r5727997;
        double r5727999 = r5727998 / r5727989;
        double r5728000 = r5727999 / r5727989;
        double r5728001 = 1.0;
        double r5728002 = r5728001 / r5727989;
        double r5728003 = r5727991 * r5728002;
        double r5728004 = r5727994 ? r5728000 : r5728003;
        double r5728005 = r5727986 ? r5727992 : r5728004;
        return r5728005;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original	26.0
Target	0.5
Herbie	13.4

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

Derivation

1. Split input into 3 regimes

2. if c < -9.293826114497671e+210

   1. Initial program 41.6

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

   2. Simplified 41.6

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

   4. Applied clear-num 41.6

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

   6. Applied *-un-lft-identity 41.6

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

   7. Applied add-sqr-sqrt 41.6

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

   8. Applied times-frac 41.6

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

   9. Applied add-cube-cbrt 41.6

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

   10. Applied times-frac 41.6

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

   11. Simplified 41.6

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

   12. Simplified 29.9

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

   13. Taylor expanded around -inf 10.2

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

   14. Simplified 10.2

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

   if -9.293826114497671e+210 < c < 1.1505677994297552e+182

   1. Initial program 22.3

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

   2. Simplified 22.3

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

   4. Applied clear-num 22.4

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

   6. Applied *-un-lft-identity 22.4

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

   7. Applied add-sqr-sqrt 22.4

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

   8. Applied times-frac 22.4

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

   9. Applied add-cube-cbrt 22.4

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

   10. Applied times-frac 22.3

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

   11. Simplified 22.3

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

   12. Simplified 14.0

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

   14. Applied associate-*r/ 14.0

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

   15. Simplified 13.9

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

   if 1.1505677994297552e+182 < c

   1. Initial program 42.1

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

   2. Simplified 42.1

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

   4. Applied clear-num 42.1

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

   6. Applied *-un-lft-identity 42.1

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

   7. Applied add-sqr-sqrt 42.1

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

   8. Applied times-frac 42.1

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

   9. Applied add-cube-cbrt 42.1

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

   10. Applied times-frac 42.1

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

   11. Simplified 42.1

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

   12. Simplified 29.5

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

   13. Taylor expanded around inf 12.1

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

4. Final simplification 13.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \le -9.293826114497670869139658346683561592978 \cdot 10^{210}:\\ \;\;\;\;\frac{-1}{\mathsf{hypot}\left(d, c\right)} \cdot b\\ \mathbf{elif}\;c \le 1.15056779942975516580562280806905910555 \cdot 10^{182}:\\ \;\;\;\;\frac{\frac{b \cdot c - d \cdot a}{\mathsf{hypot}\left(d, c\right)}}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{1}{\mathsf{hypot}\left(d, c\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019192 +o rules:numerics
(FPCore (a b c d)
  :name "Complex division, imag part"

  :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))))