Complex division, real part

Average Error: 26.1 → 26.0

Time: 14.3s

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 r5848232 = a;
        double r5848233 = c;
        double r5848234 = r5848232 * r5848233;
        double r5848235 = b;
        double r5848236 = d;
        double r5848237 = r5848235 * r5848236;
        double r5848238 = r5848234 + r5848237;
        double r5848239 = r5848233 * r5848233;
        double r5848240 = r5848236 * r5848236;
        double r5848241 = r5848239 + r5848240;
        double r5848242 = r5848238 / r5848241;
        return r5848242;
}

↓

double f(double a, double b, double c, double d) {
        double r5848243 = b;
        double r5848244 = d;
        double r5848245 = r5848243 * r5848244;
        double r5848246 = a;
        double r5848247 = c;
        double r5848248 = r5848246 * r5848247;
        double r5848249 = r5848245 + r5848248;
        double r5848250 = r5848247 * r5848247;
        double r5848251 = r5848244 * r5848244;
        double r5848252 = r5848250 + r5848251;
        double r5848253 = sqrt(r5848252);
        double r5848254 = r5848249 / r5848253;
        double r5848255 = r5848254 / r5848253;
        return r5848255;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original	26.1
Target	0.4
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 26.1

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

3. Applied add-sqr-sqrt 26.1

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

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 2019179 
(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))))