Complex division, real part

Average Error: 25.9 → 25.8

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 r3930896 = a;
        double r3930897 = c;
        double r3930898 = r3930896 * r3930897;
        double r3930899 = b;
        double r3930900 = d;
        double r3930901 = r3930899 * r3930900;
        double r3930902 = r3930898 + r3930901;
        double r3930903 = r3930897 * r3930897;
        double r3930904 = r3930900 * r3930900;
        double r3930905 = r3930903 + r3930904;
        double r3930906 = r3930902 / r3930905;
        return r3930906;
}

↓

double f(double a, double b, double c, double d) {
        double r3930907 = b;
        double r3930908 = d;
        double r3930909 = r3930907 * r3930908;
        double r3930910 = a;
        double r3930911 = c;
        double r3930912 = r3930910 * r3930911;
        double r3930913 = r3930909 + r3930912;
        double r3930914 = r3930911 * r3930911;
        double r3930915 = r3930908 * r3930908;
        double r3930916 = r3930914 + r3930915;
        double r3930917 = sqrt(r3930916);
        double r3930918 = r3930913 / r3930917;
        double r3930919 = r3930918 / r3930917;
        return r3930919;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original	25.9
Target	0.5
Herbie	25.8

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

Derivation

1. Initial program 25.9

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

3. Applied add-sqr-sqrt 25.9

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

4. Applied associate-/r* 25.8

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

5. Final simplification 25.8

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

Reproduce

herbie shell --seed 2019151 
(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))))