Complex division, real part

Average Error: 25.9 → 25.8

Time: 11.9s

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 r2192222 = a;
        double r2192223 = c;
        double r2192224 = r2192222 * r2192223;
        double r2192225 = b;
        double r2192226 = d;
        double r2192227 = r2192225 * r2192226;
        double r2192228 = r2192224 + r2192227;
        double r2192229 = r2192223 * r2192223;
        double r2192230 = r2192226 * r2192226;
        double r2192231 = r2192229 + r2192230;
        double r2192232 = r2192228 / r2192231;
        return r2192232;
}

↓

double f(double a, double b, double c, double d) {
        double r2192233 = b;
        double r2192234 = d;
        double r2192235 = r2192233 * r2192234;
        double r2192236 = a;
        double r2192237 = c;
        double r2192238 = r2192236 * r2192237;
        double r2192239 = r2192235 + r2192238;
        double r2192240 = r2192237 * r2192237;
        double r2192241 = r2192234 * r2192234;
        double r2192242 = r2192240 + r2192241;
        double r2192243 = sqrt(r2192242);
        double r2192244 = r2192239 / r2192243;
        double r2192245 = r2192244 / r2192243;
        return r2192245;
}

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))))