Complex division, real part

Average Error: 25.6 → 25.6

Time: 10.2s

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 r2477288 = a;
        double r2477289 = c;
        double r2477290 = r2477288 * r2477289;
        double r2477291 = b;
        double r2477292 = d;
        double r2477293 = r2477291 * r2477292;
        double r2477294 = r2477290 + r2477293;
        double r2477295 = r2477289 * r2477289;
        double r2477296 = r2477292 * r2477292;
        double r2477297 = r2477295 + r2477296;
        double r2477298 = r2477294 / r2477297;
        return r2477298;
}

↓

double f(double a, double b, double c, double d) {
        double r2477299 = b;
        double r2477300 = d;
        double r2477301 = r2477299 * r2477300;
        double r2477302 = a;
        double r2477303 = c;
        double r2477304 = r2477302 * r2477303;
        double r2477305 = r2477301 + r2477304;
        double r2477306 = r2477303 * r2477303;
        double r2477307 = r2477300 * r2477300;
        double r2477308 = r2477306 + r2477307;
        double r2477309 = sqrt(r2477308);
        double r2477310 = r2477305 / r2477309;
        double r2477311 = r2477310 / r2477309;
        return r2477311;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original	25.6
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 25.6

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

3. Applied add-sqr-sqrt 25.6

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

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 2019156 
(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))))