Complex division, imag part

Average Error: 26.0 → 13.2

Time: 12.3s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;c \le -8.156571467291585296848265053911461752012 \cdot 10^{143}:\\ \;\;\;\;\frac{-b}{\mathsf{hypot}\left(c, d\right)} \cdot \sqrt{1}\\ \mathbf{elif}\;c \le 1.514468218294920189983723932644127325328 \cdot 10^{69}:\\ \;\;\;\;\sqrt{1} \cdot \frac{\frac{\mathsf{fma}\left(b, c, -d \cdot a\right)}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1} \cdot \frac{b}{\mathsf{hypot}\left(c, d\right)}\\ \end{array}\]

C

double f(double a, double b, double c, double d) {
        double r97761 = b;
        double r97762 = c;
        double r97763 = r97761 * r97762;
        double r97764 = a;
        double r97765 = d;
        double r97766 = r97764 * r97765;
        double r97767 = r97763 - r97766;
        double r97768 = r97762 * r97762;
        double r97769 = r97765 * r97765;
        double r97770 = r97768 + r97769;
        double r97771 = r97767 / r97770;
        return r97771;
}

↓

double f(double a, double b, double c, double d) {
        double r97772 = c;
        double r97773 = -8.156571467291585e+143;
        bool r97774 = r97772 <= r97773;
        double r97775 = b;
        double r97776 = -r97775;
        double r97777 = d;
        double r97778 = hypot(r97772, r97777);
        double r97779 = r97776 / r97778;
        double r97780 = 1.0;
        double r97781 = sqrt(r97780);
        double r97782 = r97779 * r97781;
        double r97783 = 1.5144682182949202e+69;
        bool r97784 = r97772 <= r97783;
        double r97785 = a;
        double r97786 = r97777 * r97785;
        double r97787 = -r97786;
        double r97788 = fma(r97775, r97772, r97787);
        double r97789 = r97788 / r97778;
        double r97790 = r97789 / r97778;
        double r97791 = r97781 * r97790;
        double r97792 = r97775 / r97778;
        double r97793 = r97781 * r97792;
        double r97794 = r97784 ? r97791 : r97793;
        double r97795 = r97774 ? r97782 : r97794;
        return r97795;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original	26.0
Target	0.4
Herbie	13.2

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

Derivation

1. Split input into 3 regimes

2. if c < -8.156571467291585e+143

   1. Initial program 44.8

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

   3. Applied add-sqr-sqrt 44.8

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

   4. Applied *-un-lft-identity 44.8

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

   5. Applied times-frac 44.8

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

   6. Simplified 44.8

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

   7. Simplified 29.5

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

   9. Applied *-un-lft-identity 29.5

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

   10. Applied add-sqr-sqrt 29.5

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

   11. Applied times-frac 29.5

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

   12. Applied associate-*l* 29.5

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

   13. Simplified 29.4

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

   14. Taylor expanded around -inf 14.9

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

   15. Simplified 14.9

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

   if -8.156571467291585e+143 < c < 1.5144682182949202e+69

   1. Initial program 18.6

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

   3. Applied add-sqr-sqrt 18.6

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

   4. Applied *-un-lft-identity 18.6

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

   5. Applied times-frac 18.6

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

   6. Simplified 18.6

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

   7. Simplified 11.9

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

   9. Applied *-un-lft-identity 11.9

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

   10. Applied add-sqr-sqrt 11.9

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

   11. Applied times-frac 11.9

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

   12. Applied associate-*l* 11.9

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

   13. Simplified 11.8

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

   if 1.5144682182949202e+69 < c

   1. Initial program 36.9

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

   3. Applied add-sqr-sqrt 36.9

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

   4. Applied *-un-lft-identity 36.9

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

   5. Applied times-frac 36.9

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

   6. Simplified 36.9

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

   7. Simplified 24.8

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

   9. Applied *-un-lft-identity 24.8

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

   10. Applied add-sqr-sqrt 24.8

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

   11. Applied times-frac 24.8

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

   12. Applied associate-*l* 24.8

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

   13. Simplified 24.7

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

   14. Taylor expanded around inf 16.6

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

4. Final simplification 13.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \le -8.156571467291585296848265053911461752012 \cdot 10^{143}:\\ \;\;\;\;\frac{-b}{\mathsf{hypot}\left(c, d\right)} \cdot \sqrt{1}\\ \mathbf{elif}\;c \le 1.514468218294920189983723932644127325328 \cdot 10^{69}:\\ \;\;\;\;\sqrt{1} \cdot \frac{\frac{\mathsf{fma}\left(b, c, -d \cdot a\right)}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1} \cdot \frac{b}{\mathsf{hypot}\left(c, d\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019350 +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))))