Complex division, real part

Average Error: 26.7 → 26.6

Time: 27.5s

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 r6327249 = a;
        double r6327250 = c;
        double r6327251 = r6327249 * r6327250;
        double r6327252 = b;
        double r6327253 = d;
        double r6327254 = r6327252 * r6327253;
        double r6327255 = r6327251 + r6327254;
        double r6327256 = r6327250 * r6327250;
        double r6327257 = r6327253 * r6327253;
        double r6327258 = r6327256 + r6327257;
        double r6327259 = r6327255 / r6327258;
        return r6327259;
}

↓

double f(double a, double b, double c, double d) {
        double r6327260 = b;
        double r6327261 = d;
        double r6327262 = r6327260 * r6327261;
        double r6327263 = a;
        double r6327264 = c;
        double r6327265 = r6327263 * r6327264;
        double r6327266 = r6327262 + r6327265;
        double r6327267 = r6327264 * r6327264;
        double r6327268 = r6327261 * r6327261;
        double r6327269 = r6327267 + r6327268;
        double r6327270 = sqrt(r6327269);
        double r6327271 = r6327266 / r6327270;
        double r6327272 = r6327271 / r6327270;
        return r6327272;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original	26.7
Target	0.5
Herbie	26.6

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

Derivation

1. Initial program 26.7

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

3. Applied add-sqr-sqrt 26.7

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

4. Applied associate-/r* 26.6

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

5. Final simplification 26.6

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

Reproduce

herbie shell --seed 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))))