Complex division, real part

Average Error: 25.6 → 25.5

Time: 15.2s

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 r5969900 = a;
        double r5969901 = c;
        double r5969902 = r5969900 * r5969901;
        double r5969903 = b;
        double r5969904 = d;
        double r5969905 = r5969903 * r5969904;
        double r5969906 = r5969902 + r5969905;
        double r5969907 = r5969901 * r5969901;
        double r5969908 = r5969904 * r5969904;
        double r5969909 = r5969907 + r5969908;
        double r5969910 = r5969906 / r5969909;
        return r5969910;
}

↓

double f(double a, double b, double c, double d) {
        double r5969911 = b;
        double r5969912 = d;
        double r5969913 = r5969911 * r5969912;
        double r5969914 = a;
        double r5969915 = c;
        double r5969916 = r5969914 * r5969915;
        double r5969917 = r5969913 + r5969916;
        double r5969918 = r5969915 * r5969915;
        double r5969919 = r5969912 * r5969912;
        double r5969920 = r5969918 + r5969919;
        double r5969921 = sqrt(r5969920);
        double r5969922 = r5969917 / r5969921;
        double r5969923 = r5969922 / r5969921;
        return r5969923;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original	25.6
Target	0.4
Herbie	25.5

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

Derivation

1. Initial program 25.6

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

3. Applied add-sqr-sqrt 25.6

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

4. Applied associate-/r* 25.5

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

5. Final simplification 25.5

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

Reproduce

herbie shell --seed 2019164 
(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))))