Complex division, real part

Average Error: 25.3 → 25.2

Time: 14.8s

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 r5581338 = a;
        double r5581339 = c;
        double r5581340 = r5581338 * r5581339;
        double r5581341 = b;
        double r5581342 = d;
        double r5581343 = r5581341 * r5581342;
        double r5581344 = r5581340 + r5581343;
        double r5581345 = r5581339 * r5581339;
        double r5581346 = r5581342 * r5581342;
        double r5581347 = r5581345 + r5581346;
        double r5581348 = r5581344 / r5581347;
        return r5581348;
}

↓

double f(double a, double b, double c, double d) {
        double r5581349 = b;
        double r5581350 = d;
        double r5581351 = r5581349 * r5581350;
        double r5581352 = a;
        double r5581353 = c;
        double r5581354 = r5581352 * r5581353;
        double r5581355 = r5581351 + r5581354;
        double r5581356 = r5581353 * r5581353;
        double r5581357 = r5581350 * r5581350;
        double r5581358 = r5581356 + r5581357;
        double r5581359 = sqrt(r5581358);
        double r5581360 = r5581355 / r5581359;
        double r5581361 = r5581360 / r5581359;
        return r5581361;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original	25.3
Target	0.4
Herbie	25.2

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

Derivation

1. Initial program 25.3

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

3. Applied add-sqr-sqrt 25.3

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

4. Applied associate-/r* 25.2

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

5. Final simplification 25.2

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

Reproduce

herbie shell --seed 2019158 
(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))))