Complex division, real part

Average Error: 25.6 → 12.7

Time: 5.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;c \le -1.6740965059493151 \cdot 10^{204}:\\ \;\;\;\;{\left(\frac{-1 \cdot a}{\mathsf{hypot}\left(c, d\right)}\right)}^{1}\\ \mathbf{elif}\;c \le 3.3012917942497675 \cdot 10^{146}:\\ \;\;\;\;{\left(\frac{\frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)}\right)}^{1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{a}{\mathsf{hypot}\left(c, d\right)}\right)}^{1}\\ \end{array}\]

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 <= -1.674096505949315e+204)) {
		VAR = ((double) pow(((double) (((double) (-1.0 * a)) / ((double) hypot(c, d)))), 1.0));
	} else {
		double VAR_1;
		if ((c <= 3.3012917942497675e+146)) {
			VAR_1 = ((double) pow(((double) (((double) (((double) fma(a, c, ((double) (b * d)))) / ((double) hypot(c, d)))) / ((double) hypot(c, d)))), 1.0));
		} else {
			VAR_1 = ((double) pow(((double) (a / ((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	25.6
Target	0.4
Herbie	12.7

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

Derivation

1. Split input into 3 regimes

2. if c < -1.674096505949315e+204

   1. Initial program 41.9

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

   3. Applied add-sqr-sqrt 41.9

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

   4. Applied *-un-lft-identity 41.9

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

   5. Applied times-frac 41.9

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

   6. Simplified 41.9

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

   7. Simplified 30.0

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

   9. Applied pow1 30.0

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

   10. Applied pow1 30.0

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

   11. Applied pow-prod-down 30.0

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

   12. Simplified 30.0

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

   13. Taylor expanded around -inf 11.4

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

   if -1.674096505949315e+204 < c < 3.3012917942497675e+146

   1. Initial program 20.6

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

   3. Applied add-sqr-sqrt 20.6

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

   4. Applied *-un-lft-identity 20.6

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

   5. Applied times-frac 20.6

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

   6. Simplified 20.6

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

   7. Simplified 12.9

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

   9. Applied pow1 12.9

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

   10. Applied pow1 12.9

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

   11. Applied pow-prod-down 12.9

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

   12. Simplified 12.8

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

   if 3.3012917942497675e+146 < c

   1. Initial program 43.4

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

   3. Applied add-sqr-sqrt 43.4

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

   4. Applied *-un-lft-identity 43.4

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

   5. Applied times-frac 43.4

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

   6. Simplified 43.4

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

   7. Simplified 28.5

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

   9. Applied pow1 28.5

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

   10. Applied pow1 28.5

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

   11. Applied pow-prod-down 28.5

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

   12. Simplified 28.4

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

   13. Taylor expanded around inf 13.4

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

4. Final simplification 12.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \le -1.6740965059493151 \cdot 10^{204}:\\ \;\;\;\;{\left(\frac{-1 \cdot a}{\mathsf{hypot}\left(c, d\right)}\right)}^{1}\\ \mathbf{elif}\;c \le 3.3012917942497675 \cdot 10^{146}:\\ \;\;\;\;{\left(\frac{\frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)}\right)}^{1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{a}{\mathsf{hypot}\left(c, d\right)}\right)}^{1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020113 +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))))