Complex division, imag part

Average Error: 25.7 → 12.9

Time: 16.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;c \le -1.4815058757062313 \cdot 10^{+190}:\\ \;\;\;\;-\frac{b}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{elif}\;c \le 4.948399038442073 \cdot 10^{+94}:\\ \;\;\;\;\frac{\frac{b \cdot c - d \cdot a}{\mathsf{hypot}\left(d, c\right)}}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{\mathsf{hypot}\left(d, c\right)}\\ \end{array}\]

C

double f(double a, double b, double c, double d) {
        double r3559903 = b;
        double r3559904 = c;
        double r3559905 = r3559903 * r3559904;
        double r3559906 = a;
        double r3559907 = d;
        double r3559908 = r3559906 * r3559907;
        double r3559909 = r3559905 - r3559908;
        double r3559910 = r3559904 * r3559904;
        double r3559911 = r3559907 * r3559907;
        double r3559912 = r3559910 + r3559911;
        double r3559913 = r3559909 / r3559912;
        return r3559913;
}

↓

double f(double a, double b, double c, double d) {
        double r3559914 = c;
        double r3559915 = -1.4815058757062313e+190;
        bool r3559916 = r3559914 <= r3559915;
        double r3559917 = b;
        double r3559918 = d;
        double r3559919 = hypot(r3559918, r3559914);
        double r3559920 = r3559917 / r3559919;
        double r3559921 = -r3559920;
        double r3559922 = 4.948399038442073e+94;
        bool r3559923 = r3559914 <= r3559922;
        double r3559924 = r3559917 * r3559914;
        double r3559925 = a;
        double r3559926 = r3559918 * r3559925;
        double r3559927 = r3559924 - r3559926;
        double r3559928 = r3559927 / r3559919;
        double r3559929 = r3559928 / r3559919;
        double r3559930 = r3559923 ? r3559929 : r3559920;
        double r3559931 = r3559916 ? r3559921 : r3559930;
        return r3559931;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original	25.7
Target	0.4
Herbie	12.9

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

Derivation

1. Split input into 3 regimes

2. if c < -1.4815058757062313e+190

   1. Initial program 41.5

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

   2. Simplified 41.5

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

   4. Applied add-sqr-sqrt 41.5

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

   5. Applied associate-/r* 41.5

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

   7. Applied fma-udef 41.5

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

   8. Applied hypot-def 41.5

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

   10. Applied *-un-lft-identity 41.5

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

   11. Applied associate-/r* 41.5

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

   12. Simplified 30.3

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

   13. Taylor expanded around -inf 11.0

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

   14. Simplified 11.0

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

   if -1.4815058757062313e+190 < c < 4.948399038442073e+94

   1. Initial program 20.1

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

   2. Simplified 20.1

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

   4. Applied add-sqr-sqrt 20.1

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

   5. Applied associate-/r* 20.0

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

   7. Applied fma-udef 20.0

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

   8. Applied hypot-def 20.0

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

   10. Applied *-un-lft-identity 20.0

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

   11. Applied associate-/r* 20.0

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

   12. Simplified 12.3

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

   if 4.948399038442073e+94 < c

   1. Initial program 38.5

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

   2. Simplified 38.5

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

   4. Applied add-sqr-sqrt 38.5

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

   5. Applied associate-/r* 38.5

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

   7. Applied fma-udef 38.5

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

   8. Applied hypot-def 38.5

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

   9. Taylor expanded around inf 16.6

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

4. Final simplification 12.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \le -1.4815058757062313 \cdot 10^{+190}:\\ \;\;\;\;-\frac{b}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{elif}\;c \le 4.948399038442073 \cdot 10^{+94}:\\ \;\;\;\;\frac{\frac{b \cdot c - d \cdot a}{\mathsf{hypot}\left(d, c\right)}}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{\mathsf{hypot}\left(d, c\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019142 +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))))