Complex division, real part

Average Error: 26.6 → 14.2

Time: 10.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;c \le -6.67809890291080594 \cdot 10^{97}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{\mathsf{hypot}\left(c, d\right)}}}{\frac{\mathsf{hypot}\left(c, d\right)}{\mathsf{fma}\left(a, \sqrt{\frac{-1}{c}} \cdot c, d \cdot \left(\sqrt{\frac{-1}{c}} \cdot b\right)\right)}}\\ \mathbf{elif}\;c \le 3.37035099458032735 \cdot 10^{73}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right) \cdot 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\mathsf{hypot}\left(c, d\right) \cdot 1}\\ \end{array}\]

C

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

↓

double code(double a, double b, double c, double d) {
	double VAR;
	if ((c <= -6.678098902910806e+97)) {
		VAR = ((double) (((double) (1.0 / ((double) sqrt(((double) hypot(c, d)))))) / ((double) (((double) hypot(c, d)) / ((double) fma(a, ((double) (((double) sqrt(((double) (-1.0 / c)))) * c)), ((double) (d * ((double) (((double) sqrt(((double) (-1.0 / c)))) * b))))))))));
	} else {
		double VAR_1;
		if ((c <= 3.3703509945803273e+73)) {
			VAR_1 = ((double) (((double) (((double) fma(a, c, ((double) (b * d)))) / ((double) hypot(c, d)))) / ((double) (((double) hypot(c, d)) * 1.0))));
		} else {
			VAR_1 = ((double) (a / ((double) (((double) hypot(c, d)) * 1.0))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original	26.6
Target	0.4
Herbie	14.2

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

Derivation

1. Split input into 3 regimes

2. if c < -6.678098902910806e+97

   1. Initial program 40.4

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

   3. Applied add-sqr-sqrt 40.4

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

   4. Applied *-un-lft-identity 40.4

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

   5. Applied times-frac 40.4

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

   6. Simplified 40.4

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

   7. Simplified 26.9

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

   9. Applied associate-*r/ 26.8

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

   10. Simplified 26.8

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

   12. Applied add-sqr-sqrt 26.9

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

   13. Applied *-un-lft-identity 26.9

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

   14. Applied times-frac 26.9

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

   15. Applied associate-/l* 26.9

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

   16. Simplified 26.9

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

   17. Taylor expanded around -inf 19.3

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

   18. Simplified 19.3

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

   if -6.678098902910806e+97 < c < 3.3703509945803273e+73

   1. Initial program 18.8

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

   3. Applied add-sqr-sqrt 18.8

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

   4. Applied *-un-lft-identity 18.8

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

   5. Applied times-frac 18.8

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

   6. Simplified 18.8

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

   7. Simplified 11.6

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

   9. Applied associate-*r/ 11.6

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

   10. Simplified 11.5

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

   if 3.3703509945803273e+73 < c

   1. Initial program 37.6

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

   3. Applied add-sqr-sqrt 37.6

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

   4. Applied *-un-lft-identity 37.6

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

   5. Applied times-frac 37.6

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

   6. Simplified 37.6

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

   7. Simplified 26.1

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

   9. Applied associate-*r/ 26.1

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

   10. Simplified 26.0

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

   11. Taylor expanded around inf 17.7

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

4. Final simplification 14.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \le -6.67809890291080594 \cdot 10^{97}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{\mathsf{hypot}\left(c, d\right)}}}{\frac{\mathsf{hypot}\left(c, d\right)}{\mathsf{fma}\left(a, \sqrt{\frac{-1}{c}} \cdot c, d \cdot \left(\sqrt{\frac{-1}{c}} \cdot b\right)\right)}}\\ \mathbf{elif}\;c \le 3.37035099458032735 \cdot 10^{73}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right) \cdot 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\mathsf{hypot}\left(c, d\right) \cdot 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020120 +o rules:numerics
(FPCore (a b c d)
  :name "Complex division, real part"
  :precision binary64

  :herbie-target
  (if (< (fabs d) (fabs c)) (/ (+ a (* b (/ d c))) (+ c (* d (/ d c)))) (/ (+ b (* a (/ c d))) (+ d (* c (/ c d)))))

  (/ (+ (* a c) (* b d)) (+ (* c c) (* d d))))