Complex division, real part

Average Error: 26.7 → 26.7

Time: 14.5s

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 r6622053 = a;
        double r6622054 = c;
        double r6622055 = r6622053 * r6622054;
        double r6622056 = b;
        double r6622057 = d;
        double r6622058 = r6622056 * r6622057;
        double r6622059 = r6622055 + r6622058;
        double r6622060 = r6622054 * r6622054;
        double r6622061 = r6622057 * r6622057;
        double r6622062 = r6622060 + r6622061;
        double r6622063 = r6622059 / r6622062;
        return r6622063;
}

↓

double f(double a, double b, double c, double d) {
        double r6622064 = b;
        double r6622065 = d;
        double r6622066 = r6622064 * r6622065;
        double r6622067 = a;
        double r6622068 = c;
        double r6622069 = r6622067 * r6622068;
        double r6622070 = r6622066 + r6622069;
        double r6622071 = r6622068 * r6622068;
        double r6622072 = r6622065 * r6622065;
        double r6622073 = r6622071 + r6622072;
        double r6622074 = sqrt(r6622073);
        double r6622075 = r6622070 / r6622074;
        double r6622076 = r6622075 / r6622074;
        return r6622076;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original	26.7
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 26.7

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

3. Applied add-sqr-sqrt 26.7

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

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 2019169 
(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))))