Complex division, real part

Average Error: 26.5 → 26.3

Time: 3.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \le 5.1588593744494812 \cdot 10^{245}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \left(-1 \cdot a\right)\\ \end{array}\]

C

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

↓

double code(double a, double b, double c, double d) {
	double temp;
	if (((((a * c) + (b * d)) / ((c * c) + (d * d))) <= 5.158859374449481e+245)) {
		temp = (((a * c) + (b * d)) / ((c * c) + (d * d)));
	} else {
		temp = ((1.0 / sqrt(((c * c) + (d * d)))) * (-1.0 * a));
	}
	return temp;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original	26.5
Target	0.5
Herbie	26.3

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

Derivation

1. Split input into 2 regimes

2. if (/ (+ (* a c) (* b d)) (+ (* c c) (* d d))) < 5.158859374449481e+245

   1. Initial program 14.6

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

   if 5.158859374449481e+245 < (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))

   1. Initial program 60.6

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

   3. Applied add-sqr-sqrt 60.6

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

   4. Applied *-un-lft-identity 60.6

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

   5. Applied times-frac 60.6

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

   6. Taylor expanded around -inf 60.1

      \[\leadsto \frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \color{blue}{\left(-1 \cdot a\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 26.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \le 5.1588593744494812 \cdot 10^{245}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \left(-1 \cdot a\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020057 
(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))))