Complex division, real part

Average Error: 25.8 → 25.7

Time: 13.6s

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 r4987385 = a;
        double r4987386 = c;
        double r4987387 = r4987385 * r4987386;
        double r4987388 = b;
        double r4987389 = d;
        double r4987390 = r4987388 * r4987389;
        double r4987391 = r4987387 + r4987390;
        double r4987392 = r4987386 * r4987386;
        double r4987393 = r4987389 * r4987389;
        double r4987394 = r4987392 + r4987393;
        double r4987395 = r4987391 / r4987394;
        return r4987395;
}

↓

double f(double a, double b, double c, double d) {
        double r4987396 = b;
        double r4987397 = d;
        double r4987398 = r4987396 * r4987397;
        double r4987399 = a;
        double r4987400 = c;
        double r4987401 = r4987399 * r4987400;
        double r4987402 = r4987398 + r4987401;
        double r4987403 = r4987400 * r4987400;
        double r4987404 = r4987397 * r4987397;
        double r4987405 = r4987403 + r4987404;
        double r4987406 = sqrt(r4987405);
        double r4987407 = r4987402 / r4987406;
        double r4987408 = r4987407 / r4987406;
        return r4987408;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original	25.8
Target	0.4
Herbie	25.7

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

Derivation

1. Initial program 25.8

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

3. Applied add-sqr-sqrt 25.8

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

4. Applied associate-/r* 25.7

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

5. Final simplification 25.7

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

Reproduce

herbie shell --seed 2019163 
(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))))