Complex division, real part

Average Error: 26.2 → 26.1

Time: 16.4s

Precision: 64

Math

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

↓

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

C

double f(double a, double b, double c, double d) {
        double r6359674 = a;
        double r6359675 = c;
        double r6359676 = r6359674 * r6359675;
        double r6359677 = b;
        double r6359678 = d;
        double r6359679 = r6359677 * r6359678;
        double r6359680 = r6359676 + r6359679;
        double r6359681 = r6359675 * r6359675;
        double r6359682 = r6359678 * r6359678;
        double r6359683 = r6359681 + r6359682;
        double r6359684 = r6359680 / r6359683;
        return r6359684;
}

↓

double f(double a, double b, double c, double d) {
        double r6359685 = b;
        double r6359686 = d;
        double r6359687 = r6359685 * r6359686;
        double r6359688 = a;
        double r6359689 = c;
        double r6359690 = r6359688 * r6359689;
        double r6359691 = r6359687 + r6359690;
        double r6359692 = r6359689 * r6359689;
        double r6359693 = r6359686 * r6359686;
        double r6359694 = r6359692 + r6359693;
        double r6359695 = sqrt(r6359694);
        double r6359696 = r6359691 / r6359695;
        double r6359697 = r6359696 / r6359695;
        return r6359697;
}

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	26.1

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

Derivation

1. Initial program 26.2

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

3. Applied add-sqr-sqrt 26.2

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

4. Applied associate-/r* 26.1

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

5. Final simplification 26.1

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

Reproduce

herbie shell --seed 2019168 
(FPCore (a b c d)
  :name "Complex division, real part"

  :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))))