Complex division, real part

Average Error: 25.7 → 25.6

Time: 12.4s

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 r3861745 = a;
        double r3861746 = c;
        double r3861747 = r3861745 * r3861746;
        double r3861748 = b;
        double r3861749 = d;
        double r3861750 = r3861748 * r3861749;
        double r3861751 = r3861747 + r3861750;
        double r3861752 = r3861746 * r3861746;
        double r3861753 = r3861749 * r3861749;
        double r3861754 = r3861752 + r3861753;
        double r3861755 = r3861751 / r3861754;
        return r3861755;
}

↓

double f(double a, double b, double c, double d) {
        double r3861756 = b;
        double r3861757 = d;
        double r3861758 = r3861756 * r3861757;
        double r3861759 = a;
        double r3861760 = c;
        double r3861761 = r3861759 * r3861760;
        double r3861762 = r3861758 + r3861761;
        double r3861763 = r3861760 * r3861760;
        double r3861764 = r3861757 * r3861757;
        double r3861765 = r3861763 + r3861764;
        double r3861766 = sqrt(r3861765);
        double r3861767 = r3861762 / r3861766;
        double r3861768 = r3861767 / r3861766;
        return r3861768;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original	25.7
Target	0.4
Herbie	25.6

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

Derivation

1. Initial program 25.7

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

3. Applied add-sqr-sqrt 25.7

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

4. Applied associate-/r* 25.6

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

5. Final simplification 25.6

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

Reproduce

herbie shell --seed 2019132 
(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))))