Complex division, imag part

Average Error: 26.6 → 16.6

Time: 3.2s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;d \le -8.29857753110238368 \cdot 10^{124} \lor \neg \left(d \le 7.1886537093075794 \cdot 10^{50}\right):\\ \;\;\;\;\frac{b \cdot c}{c \cdot c + d \cdot d} - \frac{a}{d + \frac{{c}^{2}}{d}}\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{c}{c \cdot c + d \cdot d} - \frac{a \cdot d}{c \cdot c + d \cdot d}\\ \end{array}\]

C

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

↓

double code(double a, double b, double c, double d) {
	double VAR;
	if (((d <= -8.298577531102384e+124) || !(d <= 7.1886537093075794e+50))) {
		VAR = ((double) (((double) (((double) (b * c)) / ((double) (((double) (c * c)) + ((double) (d * d)))))) - ((double) (a / ((double) (d + ((double) (((double) pow(c, 2.0)) / d))))))));
	} else {
		VAR = ((double) (((double) (b * ((double) (c / ((double) (((double) (c * c)) + ((double) (d * d)))))))) - ((double) (((double) (a * d)) / ((double) (((double) (c * c)) + ((double) (d * d))))))));
	}
	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	16.6

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

Derivation

1. Split input into 2 regimes

2. if d < -8.29857753110238368e124 or 7.1886537093075794e50 < d

   1. Initial program 38.1

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

   3. Applied div-sub 38.1

      \[\leadsto \color{blue}{\frac{b \cdot c}{c \cdot c + d \cdot d} - \frac{a \cdot d}{c \cdot c + d \cdot d}}\]
   4. Using strategy rm

   5. Applied associate-/l* 34.8

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

   6. Taylor expanded around 0 16.5

      \[\leadsto \frac{b \cdot c}{c \cdot c + d \cdot d} - \frac{a}{\color{blue}{d + \frac{{c}^{2}}{d}}}\]

   if -8.29857753110238368e124 < d < 7.1886537093075794e50

   1. Initial program 19.1

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

   3. Applied div-sub 19.1

      \[\leadsto \color{blue}{\frac{b \cdot c}{c \cdot c + d \cdot d} - \frac{a \cdot d}{c \cdot c + d \cdot d}}\]
   4. Using strategy rm

   5. Applied *-un-lft-identity 19.1

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

   6. Applied times-frac 16.7

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

   7. Simplified 16.7

      \[\leadsto \color{blue}{b} \cdot \frac{c}{c \cdot c + d \cdot d} - \frac{a \cdot d}{c \cdot c + d \cdot d}\]
3. Recombined 2 regimes into one program.

4. Final simplification 16.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \le -8.29857753110238368 \cdot 10^{124} \lor \neg \left(d \le 7.1886537093075794 \cdot 10^{50}\right):\\ \;\;\;\;\frac{b \cdot c}{c \cdot c + d \cdot d} - \frac{a}{d + \frac{{c}^{2}}{d}}\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{c}{c \cdot c + d \cdot d} - \frac{a \cdot d}{c \cdot c + d \cdot d}\\ \end{array}\]

Reproduce

herbie shell --seed 2020149 
(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)))) (/ (+ (neg a) (* b (/ c d))) (+ d (* c (/ c d)))))

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