Complex division, real part

Average Error: 26.1 → 26.0

Time: 12.6s

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 r4334144 = a;
        double r4334145 = c;
        double r4334146 = r4334144 * r4334145;
        double r4334147 = b;
        double r4334148 = d;
        double r4334149 = r4334147 * r4334148;
        double r4334150 = r4334146 + r4334149;
        double r4334151 = r4334145 * r4334145;
        double r4334152 = r4334148 * r4334148;
        double r4334153 = r4334151 + r4334152;
        double r4334154 = r4334150 / r4334153;
        return r4334154;
}

↓

double f(double a, double b, double c, double d) {
        double r4334155 = b;
        double r4334156 = d;
        double r4334157 = r4334155 * r4334156;
        double r4334158 = a;
        double r4334159 = c;
        double r4334160 = r4334158 * r4334159;
        double r4334161 = r4334157 + r4334160;
        double r4334162 = r4334159 * r4334159;
        double r4334163 = r4334156 * r4334156;
        double r4334164 = r4334162 + r4334163;
        double r4334165 = sqrt(r4334164);
        double r4334166 = r4334161 / r4334165;
        double r4334167 = r4334166 / r4334165;
        return r4334167;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original	26.1
Target	0.4
Herbie	26.0

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

Derivation

1. Initial program 26.1

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

3. Applied add-sqr-sqrt 26.1

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

4. Applied associate-/r* 26.0

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

5. Final simplification 26.0

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

Reproduce

herbie shell --seed 2019133 
(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))))