Complex division, imag part

Average Error: 26.2 → 26.5

Time: 3.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \le -2.89329598299196714 \cdot 10^{246}:\\ \;\;\;\;\frac{1}{\frac{c \cdot c + d \cdot d}{b \cdot c - a \cdot d}}\\ \mathbf{elif}\;a \le -1.5666623877864712 \cdot 10^{220}:\\ \;\;\;\;\frac{-1 \cdot a}{\sqrt{c \cdot c + d \cdot d}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\\ \end{array}\]

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 ((a <= -2.893295982991967e+246)) {
		VAR = ((double) (1.0 / ((double) (((double) (((double) (c * c)) + ((double) (d * d)))) / ((double) (((double) (b * c)) - ((double) (a * d))))))));
	} else {
		double VAR_1;
		if ((a <= -1.5666623877864712e+220)) {
			VAR_1 = ((double) (((double) (-1.0 * a)) / ((double) sqrt(((double) (((double) (c * c)) + ((double) (d * d))))))));
		} else {
			VAR_1 = ((double) (((double) (((double) (((double) (b * c)) - ((double) (a * d)))) / ((double) sqrt(((double) (((double) (c * c)) + ((double) (d * d)))))))) / ((double) sqrt(((double) (((double) (c * c)) + ((double) (d * d))))))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original	26.2
Target	0.4
Herbie	26.5

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

Derivation

1. Split input into 3 regimes

2. if a < -2.893295982991967e+246

   1. Initial program 45.3

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

   3. Applied clear-num 45.4

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

   if -2.893295982991967e+246 < a < -1.5666623877864712e+220

   1. Initial program 34.7

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

   3. Applied add-sqr-sqrt 34.7

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

   4. Applied associate-/r* 34.7

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

   5. Taylor expanded around 0 51.6

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

   if -1.5666623877864712e+220 < a

   1. Initial program 25.1

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

   3. Applied add-sqr-sqrt 25.1

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

   4. Applied associate-/r* 25.0

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

4. Final simplification 26.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -2.89329598299196714 \cdot 10^{246}:\\ \;\;\;\;\frac{1}{\frac{c \cdot c + d \cdot d}{b \cdot c - a \cdot d}}\\ \mathbf{elif}\;a \le -1.5666623877864712 \cdot 10^{220}:\\ \;\;\;\;\frac{-1 \cdot a}{\sqrt{c \cdot c + d \cdot d}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020122 
(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))))