Complex division, real part

Average Error: 25.6 → 25.5

Time: 10.9s

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 r2396074 = a;
        double r2396075 = c;
        double r2396076 = r2396074 * r2396075;
        double r2396077 = b;
        double r2396078 = d;
        double r2396079 = r2396077 * r2396078;
        double r2396080 = r2396076 + r2396079;
        double r2396081 = r2396075 * r2396075;
        double r2396082 = r2396078 * r2396078;
        double r2396083 = r2396081 + r2396082;
        double r2396084 = r2396080 / r2396083;
        return r2396084;
}

↓

double f(double a, double b, double c, double d) {
        double r2396085 = b;
        double r2396086 = d;
        double r2396087 = r2396085 * r2396086;
        double r2396088 = a;
        double r2396089 = c;
        double r2396090 = r2396088 * r2396089;
        double r2396091 = r2396087 + r2396090;
        double r2396092 = r2396089 * r2396089;
        double r2396093 = r2396086 * r2396086;
        double r2396094 = r2396092 + r2396093;
        double r2396095 = sqrt(r2396094);
        double r2396096 = r2396091 / r2396095;
        double r2396097 = r2396096 / r2396095;
        return r2396097;
}

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 2019154 
(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))))