Complex division, imag part

Average Error: 26.0 → 8.8

Time: 3.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;d \le -1.32733267246339254 \cdot 10^{154} \lor \neg \left(d \le 1.33619704567748062 \cdot 10^{154}\right):\\ \;\;\;\;\frac{b}{\frac{c \cdot c + d \cdot d}{c}} - \frac{a}{d + \frac{{c}^{2}}{d}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{\frac{{d}^{2}}{c} + c} - \frac{a}{\frac{c \cdot c + d \cdot d}{d}}\\ \end{array}\]

C

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

↓

double code(double a, double b, double c, double d) {
	double temp;
	if (((d <= -1.3273326724633925e+154) || !(d <= 1.3361970456774806e+154))) {
		temp = ((b / (((c * c) + (d * d)) / c)) - (a / (d + (pow(c, 2.0) / d))));
	} else {
		temp = ((b / ((pow(d, 2.0) / c) + c)) - (a / (((c * c) + (d * d)) / d)));
	}
	return temp;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original	26.0
Target	0.5
Herbie	8.8

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

Derivation

1. Split input into 2 regimes

2. if d < -1.3273326724633925e+154 or 1.3361970456774806e+154 < d

   1. Initial program 45.2

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

   3. Applied *-un-lft-identity 45.2

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

   5. Applied div-sub 45.2

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

   6. Simplified 45.0

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

   7. Simplified 44.1

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

   8. Taylor expanded around 0 17.1

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

   if -1.3273326724633925e+154 < d < 1.3361970456774806e+154

   1. Initial program 19.3

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

   3. Applied *-un-lft-identity 19.3

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

   5. Applied div-sub 19.3

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

   6. Simplified 17.3

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

   7. Simplified 15.5

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

   8. Taylor expanded around 0 5.8

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

4. Final simplification 8.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \le -1.32733267246339254 \cdot 10^{154} \lor \neg \left(d \le 1.33619704567748062 \cdot 10^{154}\right):\\ \;\;\;\;\frac{b}{\frac{c \cdot c + d \cdot d}{c}} - \frac{a}{d + \frac{{c}^{2}}{d}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{\frac{{d}^{2}}{c} + c} - \frac{a}{\frac{c \cdot c + d \cdot d}{d}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020053 
(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))))