Complex division, real part

Average Error: 26.8 → 26.7

Time: 17.1s

Precision: 64

Math

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

↓

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

C

double f(double a, double b, double c, double d) {
        double r9684588 = a;
        double r9684589 = c;
        double r9684590 = r9684588 * r9684589;
        double r9684591 = b;
        double r9684592 = d;
        double r9684593 = r9684591 * r9684592;
        double r9684594 = r9684590 + r9684593;
        double r9684595 = r9684589 * r9684589;
        double r9684596 = r9684592 * r9684592;
        double r9684597 = r9684595 + r9684596;
        double r9684598 = r9684594 / r9684597;
        return r9684598;
}

↓

double f(double a, double b, double c, double d) {
        double r9684599 = a;
        double r9684600 = c;
        double r9684601 = r9684599 * r9684600;
        double r9684602 = b;
        double r9684603 = d;
        double r9684604 = r9684602 * r9684603;
        double r9684605 = r9684601 + r9684604;
        double r9684606 = r9684600 * r9684600;
        double r9684607 = r9684603 * r9684603;
        double r9684608 = r9684606 + r9684607;
        double r9684609 = sqrt(r9684608);
        double r9684610 = r9684605 / r9684609;
        double r9684611 = r9684610 / r9684609;
        return r9684611;
}

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

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

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))))