Average Error: 26.4 → 26.4
Time: 7.6s
Precision: 64
\[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}\]
\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}
↓
\frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}
double f(double a, double b, double c, double d) {
        double r99276 = a;
        double r99277 = c;
        double r99278 = r99276 * r99277;
        double r99279 = b;
        double r99280 = d;
        double r99281 = r99279 * r99280;
        double r99282 = r99278 + r99281;
        double r99283 = r99277 * r99277;
        double r99284 = r99280 * r99280;
        double r99285 = r99283 + r99284;
        double r99286 = r99282 / r99285;
        return r99286;
}
↓
double f(double a, double b, double c, double d) {
        double r99287 = a;
        double r99288 = c;
        double r99289 = r99287 * r99288;
        double r99290 = b;
        double r99291 = d;
        double r99292 = r99290 * r99291;
        double r99293 = r99289 + r99292;
        double r99294 = r99288 * r99288;
        double r99295 = r99291 * r99291;
        double r99296 = r99294 + r99295;
        double r99297 = sqrt(r99296);
        double r99298 = r99297 * r99297;
        double r99299 = r99293 / r99298;
        return r99299;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original26.4
Target0.5
Herbie26.4
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Initial program 26.4

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

  3. Applied add-sqr-sqrt26.4

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

  4. Final simplification26.4

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

Reproduce

herbie shell --seed 2020046 
(FPCore (a b c d)
  :name "Complex division, real part"
  :precision binary64

  :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))))