Complex division, real part

Average Error: 25.5 → 24.2

Time: 3.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} = -\infty \lor \neg \left(\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \le 1.483779865357748 \cdot 10^{302}\right):\\ \;\;\;\;\frac{b}{\sqrt{c \cdot c + d \cdot d}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\\ \end{array}\]

C

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

↓

double code(double a, double b, double c, double d) {
	double VAR;
	if ((((((a * c) + (b * d)) / ((c * c) + (d * d))) <= -inf.0) || !((((a * c) + (b * d)) / ((c * c) + (d * d))) <= 1.4837798653577477e+302))) {
		VAR = (b / sqrt(((c * c) + (d * d))));
	} else {
		VAR = ((((a * c) + (b * d)) / sqrt(((c * c) + (d * d)))) / sqrt(((c * c) + (d * d))));
	}
	return VAR;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original	25.5
Target	0.6
Herbie	24.2

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

Derivation

1. Split input into 2 regimes

2. if (/ (+ (* a c) (* b d)) (+ (* c c) (* d d))) < -inf.0 or 1.4837798653577477e+302 < (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))

   1. Initial program 63.7

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

   3. Applied add-sqr-sqrt 63.7

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

   4. Applied associate-/r* 63.7

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

   5. Taylor expanded around 0 59.0

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

   if -inf.0 < (/ (+ (* a c) (* b d)) (+ (* c c) (* d d))) < 1.4837798653577477e+302

   1. Initial program 11.1

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

   3. Applied add-sqr-sqrt 11.1

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

   4. Applied associate-/r* 11.0

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

4. Final simplification 24.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} = -\infty \lor \neg \left(\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \le 1.483779865357748 \cdot 10^{302}\right):\\ \;\;\;\;\frac{b}{\sqrt{c \cdot c + d \cdot d}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020075 
(FPCore (a b c d)
  :name "Complex division, real part"
  :precision binary64

  :herbie-target
  (if (< (fabs d) (fabs c)) (/ (+ a (* b (/ d c))) (+ c (* d (/ d c)))) (/ (+ b (* a (/ c d))) (+ d (* c (/ c d)))))

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