Complex division, real part

Average Error: 25.5 → 25.5

Time: 3.2s

Precision: 64

Math

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

↓

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

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

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original	25.5
Target	0.5
Herbie	25.5

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

Derivation

1. Initial program 25.5

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

3. Applied div-inv 25.6

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

5. Applied *-un-lft-identity 25.6

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

6. Applied associate-*l* 25.6

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

7. Simplified 25.5

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

8. Final simplification 25.5

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

Reproduce

herbie shell --seed 2020049 
(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))))