Complex division, real part

Average Error: 26.8 → 26.7

Time: 14.6s

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 r6307441 = a;
        double r6307442 = c;
        double r6307443 = r6307441 * r6307442;
        double r6307444 = b;
        double r6307445 = d;
        double r6307446 = r6307444 * r6307445;
        double r6307447 = r6307443 + r6307446;
        double r6307448 = r6307442 * r6307442;
        double r6307449 = r6307445 * r6307445;
        double r6307450 = r6307448 + r6307449;
        double r6307451 = r6307447 / r6307450;
        return r6307451;
}

↓

double f(double a, double b, double c, double d) {
        double r6307452 = b;
        double r6307453 = d;
        double r6307454 = r6307452 * r6307453;
        double r6307455 = a;
        double r6307456 = c;
        double r6307457 = r6307455 * r6307456;
        double r6307458 = r6307454 + r6307457;
        double r6307459 = r6307456 * r6307456;
        double r6307460 = r6307453 * r6307453;
        double r6307461 = r6307459 + r6307460;
        double r6307462 = sqrt(r6307461);
        double r6307463 = r6307458 / r6307462;
        double r6307464 = r6307463 / r6307462;
        return r6307464;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original	26.8
Target	0.4
Herbie	26.7

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

Derivation

1. Initial program 26.8

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

3. Applied add-sqr-sqrt 26.8

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

4. Applied associate-/r* 26.7

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

5. Final simplification 26.7

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

Reproduce

herbie shell --seed 2019174 
(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))))