Complex division, imag part

Average Error: 26.1 → 12.4

Time: 5.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;c \le -5.24214216730030625 \cdot 10^{138}:\\ \;\;\;\;\frac{-1 \cdot b}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;c \le 3.2259502147858646 \cdot 10^{139}:\\ \;\;\;\;\frac{\frac{b \cdot c - a \cdot d}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;c \le 1.9590575859470631 \cdot 10^{185}:\\ \;\;\;\;\frac{\frac{c}{\frac{{\left(\sqrt{\mathsf{hypot}\left(c, d\right)}\right)}^{3}}{b}} - \frac{d}{\frac{{\left(\sqrt{\mathsf{hypot}\left(c, d\right)}\right)}^{3}}{a}}}{\sqrt{\mathsf{hypot}\left(c, d\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{\mathsf{hypot}\left(c, d\right)}\\ \end{array}\]

C

double code(double a, double b, double c, double d) {
	return (((b * c) - (a * d)) / ((c * c) + (d * d)));
}

↓

double code(double a, double b, double c, double d) {
	double VAR;
	if ((c <= -5.242142167300306e+138)) {
		VAR = ((-1.0 * b) / hypot(c, d));
	} else {
		double VAR_1;
		if ((c <= 3.2259502147858646e+139)) {
			VAR_1 = ((((b * c) - (a * d)) / hypot(c, d)) / hypot(c, d));
		} else {
			double VAR_2;
			if ((c <= 1.959057585947063e+185)) {
				VAR_2 = (((c / (pow(sqrt(hypot(c, d)), 3.0) / b)) - (d / (pow(sqrt(hypot(c, d)), 3.0) / a))) / sqrt(hypot(c, d)));
			} else {
				VAR_2 = (b / hypot(c, d));
			}
			VAR_1 = VAR_2;
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original	26.1
Target	0.4
Herbie	12.4

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

Derivation

1. Split input into 4 regimes

2. if c < -5.242142167300306e+138

   1. Initial program 43.7

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

   3. Applied add-sqr-sqrt 43.7

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

   4. Applied *-un-lft-identity 43.7

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

   5. Applied times-frac 43.7

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

   6. Simplified 43.7

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

   7. Simplified 28.3

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

   9. Applied associate-*r/ 28.3

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

   10. Simplified 28.3

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

   11. Taylor expanded around -inf 13.9

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

   if -5.242142167300306e+138 < c < 3.2259502147858646e+139

   1. Initial program 19.1

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

   3. Applied add-sqr-sqrt 19.1

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

   4. Applied *-un-lft-identity 19.1

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

   5. Applied times-frac 19.1

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

   6. Simplified 19.1

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

   7. Simplified 12.2

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

   9. Applied associate-*r/ 12.2

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

   10. Simplified 12.1

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

   if 3.2259502147858646e+139 < c < 1.959057585947063e+185

   1. Initial program 41.8

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

   3. Applied add-sqr-sqrt 41.8

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

   4. Applied *-un-lft-identity 41.8

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

   5. Applied times-frac 41.9

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

   6. Simplified 41.9

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

   7. Simplified 21.7

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

   9. Applied associate-*r/ 21.7

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

   10. Simplified 21.6

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

   12. Applied add-sqr-sqrt 21.8

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

   13. Applied associate-/r* 21.8

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

   15. Applied div-sub 21.8

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

   16. Applied div-sub 21.8

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

   17. Simplified 15.6

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

   18. Simplified 13.0

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

   if 1.959057585947063e+185 < c

   1. Initial program 43.0

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

   3. Applied add-sqr-sqrt 43.0

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

   4. Applied *-un-lft-identity 43.0

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

   5. Applied times-frac 43.0

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

   6. Simplified 43.0

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

   7. Simplified 31.0

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

   9. Applied associate-*r/ 31.0

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

   10. Simplified 31.0

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

   11. Taylor expanded around inf 12.2

       \[\leadsto \frac{\color{blue}{b}}{\mathsf{hypot}\left(c, d\right)}\]
3. Recombined 4 regimes into one program.

4. Final simplification 12.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \le -5.24214216730030625 \cdot 10^{138}:\\ \;\;\;\;\frac{-1 \cdot b}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;c \le 3.2259502147858646 \cdot 10^{139}:\\ \;\;\;\;\frac{\frac{b \cdot c - a \cdot d}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;c \le 1.9590575859470631 \cdot 10^{185}:\\ \;\;\;\;\frac{\frac{c}{\frac{{\left(\sqrt{\mathsf{hypot}\left(c, d\right)}\right)}^{3}}{b}} - \frac{d}{\frac{{\left(\sqrt{\mathsf{hypot}\left(c, d\right)}\right)}^{3}}{a}}}{\sqrt{\mathsf{hypot}\left(c, d\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{\mathsf{hypot}\left(c, d\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020100 +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))))