Complex division, imag part

Average Error: 26.2 → 23.9

Time: 3.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;d \le 1.39961108870036364 \cdot 10^{39}:\\ \;\;\;\;\frac{b}{\frac{{c}^{2} + {d}^{2}}{c}} - \frac{a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot c}{c \cdot c + d \cdot d} - \frac{a}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{d}{\sqrt{c \cdot c + d \cdot d}}\\ \end{array}\]

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 ((d <= 1.3996110887003636e+39)) {
		VAR = ((double) (((double) (b / ((double) (((double) (((double) pow(c, 2.0)) + ((double) pow(d, 2.0)))) / c)))) - ((double) (((double) (a * d)) / ((double) (((double) (c * c)) + ((double) (d * d))))))));
	} else {
		VAR = ((double) (((double) (((double) (b * c)) / ((double) (((double) (c * c)) + ((double) (d * d)))))) - ((double) (((double) (a / ((double) sqrt(((double) (((double) (c * c)) + ((double) (d * d)))))))) * ((double) (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	26.2
Target	0.5
Herbie	23.9

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

Derivation

1. Split input into 2 regimes

2. if d < 1.3996110887003636e+39

   1. Initial program 23.7

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

   3. Applied div-sub 23.7

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

   5. Applied associate-/l* 21.7

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

   6. Simplified 21.7

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

   if 1.3996110887003636e+39 < d

   1. Initial program 34.8

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

   3. Applied div-sub 34.8

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

   5. Applied add-sqr-sqrt 34.8

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

   6. Applied times-frac 31.6

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

4. Final simplification 23.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \le 1.39961108870036364 \cdot 10^{39}:\\ \;\;\;\;\frac{b}{\frac{{c}^{2} + {d}^{2}}{c}} - \frac{a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot c}{c \cdot c + d \cdot d} - \frac{a}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{d}{\sqrt{c \cdot c + d \cdot d}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020129 
(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))))