Complex division, real part

Average Error: 26.0 → 26.0

Time: 18.9s

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 r5224065 = a;
        double r5224066 = c;
        double r5224067 = r5224065 * r5224066;
        double r5224068 = b;
        double r5224069 = d;
        double r5224070 = r5224068 * r5224069;
        double r5224071 = r5224067 + r5224070;
        double r5224072 = r5224066 * r5224066;
        double r5224073 = r5224069 * r5224069;
        double r5224074 = r5224072 + r5224073;
        double r5224075 = r5224071 / r5224074;
        return r5224075;
}

↓

double f(double a, double b, double c, double d) {
        double r5224076 = b;
        double r5224077 = d;
        double r5224078 = r5224076 * r5224077;
        double r5224079 = a;
        double r5224080 = c;
        double r5224081 = r5224079 * r5224080;
        double r5224082 = r5224078 + r5224081;
        double r5224083 = r5224080 * r5224080;
        double r5224084 = r5224077 * r5224077;
        double r5224085 = r5224083 + r5224084;
        double r5224086 = sqrt(r5224085);
        double r5224087 = r5224082 / r5224086;
        double r5224088 = r5224087 / r5224086;
        return r5224088;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original	26.0
Target	0.5
Herbie	26.0

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

Derivation

1. Initial program 26.0

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

3. Applied add-sqr-sqrt 26.0

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

4. Applied associate-/r* 26.0

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

5. Final simplification 26.0

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

Reproduce

herbie shell --seed 2019144 
(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))))