Complex division, imag part

Average Error: 25.8 → 12.8

Time: 15.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;c \le -1.4037038847053747 \cdot 10^{+154}:\\ \;\;\;\;\frac{-b}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{elif}\;c \le 1.8584161242035747 \cdot 10^{+93}:\\ \;\;\;\;\frac{\frac{1}{\frac{\mathsf{hypot}\left(d, c\right)}{b \cdot c - a \cdot d}}}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{\mathsf{hypot}\left(d, c\right)}\\ \end{array}\]

C

double f(double a, double b, double c, double d) {
        double r2964086 = b;
        double r2964087 = c;
        double r2964088 = r2964086 * r2964087;
        double r2964089 = a;
        double r2964090 = d;
        double r2964091 = r2964089 * r2964090;
        double r2964092 = r2964088 - r2964091;
        double r2964093 = r2964087 * r2964087;
        double r2964094 = r2964090 * r2964090;
        double r2964095 = r2964093 + r2964094;
        double r2964096 = r2964092 / r2964095;
        return r2964096;
}

↓

double f(double a, double b, double c, double d) {
        double r2964097 = c;
        double r2964098 = -1.4037038847053747e+154;
        bool r2964099 = r2964097 <= r2964098;
        double r2964100 = b;
        double r2964101 = -r2964100;
        double r2964102 = d;
        double r2964103 = hypot(r2964102, r2964097);
        double r2964104 = r2964101 / r2964103;
        double r2964105 = 1.8584161242035747e+93;
        bool r2964106 = r2964097 <= r2964105;
        double r2964107 = 1.0;
        double r2964108 = r2964100 * r2964097;
        double r2964109 = a;
        double r2964110 = r2964109 * r2964102;
        double r2964111 = r2964108 - r2964110;
        double r2964112 = r2964103 / r2964111;
        double r2964113 = r2964107 / r2964112;
        double r2964114 = r2964113 / r2964103;
        double r2964115 = r2964100 / r2964103;
        double r2964116 = r2964106 ? r2964114 : r2964115;
        double r2964117 = r2964099 ? r2964104 : r2964116;
        return r2964117;
}

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	12.8

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

Derivation

1. Split input into 3 regimes

2. if c < -1.4037038847053747e+154

   1. Initial program 43.9

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

   2. Simplified 43.9

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

   4. Applied add-sqr-sqrt 43.9

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

   5. Applied associate-/r* 43.9

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

   7. Applied fma-udef 43.9

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

   8. Applied hypot-def 43.9

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

   10. Applied fma-udef 43.9

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

   11. Applied hypot-def 27.1

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

   12. Taylor expanded around -inf 13.2

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

   13. Simplified 13.2

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

   if -1.4037038847053747e+154 < c < 1.8584161242035747e+93

   1. Initial program 18.9

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

   2. Simplified 18.9

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

   4. Applied add-sqr-sqrt 18.9

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

   5. Applied associate-/r* 18.8

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

   7. Applied fma-udef 18.8

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

   8. Applied hypot-def 18.8

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

   10. Applied fma-udef 18.8

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

   11. Applied hypot-def 11.7

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

   13. Applied *-un-lft-identity 11.7

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

   14. Applied associate-/l* 11.8

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

   if 1.8584161242035747e+93 < c

   1. Initial program 37.5

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

   2. Simplified 37.5

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

   4. Applied add-sqr-sqrt 37.5

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

   5. Applied associate-/r* 37.5

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

   7. Applied fma-udef 37.5

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

   8. Applied hypot-def 37.5

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

   10. Applied fma-udef 37.5

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

   11. Applied hypot-def 24.3

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

   12. Taylor expanded around inf 15.9

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

4. Final simplification 12.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \le -1.4037038847053747 \cdot 10^{+154}:\\ \;\;\;\;\frac{-b}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{elif}\;c \le 1.8584161242035747 \cdot 10^{+93}:\\ \;\;\;\;\frac{\frac{1}{\frac{\mathsf{hypot}\left(d, c\right)}{b \cdot c - a \cdot d}}}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{\mathsf{hypot}\left(d, c\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019151 +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))))