Complex division, imag part

Average Error: 25.9 → 24.1

Time: 3.8s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;c \leq -2.870423058279664 \cdot 10^{-77} \lor \neg \left(c \leq 1.9014227410894348 \cdot 10^{-237}\right):\\ \;\;\;\;\frac{c}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{b}{\sqrt{c \cdot c + d \cdot d}} - d \cdot \frac{a}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c \cdot b - d \cdot a}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\\ \end{array}\]

C

double code(double a, double b, double c, double d) {
	return (((double) (((double) (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 (((c <= -2.870423058279664e-77) || !(c <= 1.9014227410894348e-237))) {
		VAR = ((double) (((double) ((c / ((double) sqrt(((double) (((double) (c * c)) + ((double) (d * d))))))) * (b / ((double) sqrt(((double) (((double) (c * c)) + ((double) (d * d))))))))) - ((double) (d * (a / ((double) (((double) (c * c)) + ((double) (d * d)))))))));
	} else {
		VAR = ((((double) (((double) (c * b)) - ((double) (d * a)))) / ((double) sqrt(((double) (((double) (c * c)) + ((double) (d * d))))))) / ((double) sqrt(((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	25.9
Target	0.4
Herbie	24.1

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

Derivation

1. Split input into 2 regimes

2. if c < -2.87042305827966395e-77 or 1.90142274108943484e-237 < c

   1. Initial program 27.4

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

   3. Applied div-sub 27.4

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

   4. Simplified 27.0

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

   5. Simplified 26.7

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

   7. Applied add-sqr-sqrt 26.7

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

   8. Applied *-un-lft-identity 26.7

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

   9. Applied times-frac 26.7

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

   10. Applied associate-*r* 25.2

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

   11. Simplified 25.2

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

   if -2.87042305827966395e-77 < c < 1.90142274108943484e-237

   1. Initial program 20.2

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

   3. Applied add-sqr-sqrt 20.2

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

   4. Applied associate-/r* 20.1

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

4. Final simplification 24.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.870423058279664 \cdot 10^{-77} \lor \neg \left(c \leq 1.9014227410894348 \cdot 10^{-237}\right):\\ \;\;\;\;\frac{c}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{b}{\sqrt{c \cdot c + d \cdot d}} - d \cdot \frac{a}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c \cdot b - d \cdot a}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020196 
(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))))