Complex division, imag part

Average Error: 25.9 → 25.9

Time: 13.7s

Precision: 64

Math

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

↓

\[\frac{\frac{1}{\frac{\sqrt{c \cdot c + d \cdot d}}{b \cdot c - a \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\]

C

double f(double a, double b, double c, double d) {
        double r87287 = b;
        double r87288 = c;
        double r87289 = r87287 * r87288;
        double r87290 = a;
        double r87291 = d;
        double r87292 = r87290 * r87291;
        double r87293 = r87289 - r87292;
        double r87294 = r87288 * r87288;
        double r87295 = r87291 * r87291;
        double r87296 = r87294 + r87295;
        double r87297 = r87293 / r87296;
        return r87297;
}

↓

double f(double a, double b, double c, double d) {
        double r87298 = 1.0;
        double r87299 = c;
        double r87300 = r87299 * r87299;
        double r87301 = d;
        double r87302 = r87301 * r87301;
        double r87303 = r87300 + r87302;
        double r87304 = sqrt(r87303);
        double r87305 = b;
        double r87306 = r87305 * r87299;
        double r87307 = a;
        double r87308 = r87307 * r87301;
        double r87309 = r87306 - r87308;
        double r87310 = r87304 / r87309;
        double r87311 = r87298 / r87310;
        double r87312 = r87311 / r87304;
        return r87312;
}

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	25.9

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

Derivation

1. Initial program 25.9

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

3. Applied add-sqr-sqrt 25.9

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

4. Applied associate-/r* 25.8

   \[\leadsto \color{blue}{\frac{\frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}}\]
5. Using strategy rm

6. Applied clear-num 25.9

   \[\leadsto \frac{\color{blue}{\frac{1}{\frac{\sqrt{c \cdot c + d \cdot d}}{b \cdot c - a \cdot d}}}}{\sqrt{c \cdot c + d \cdot d}}\]

7. Final simplification 25.9

   \[\leadsto \frac{\frac{1}{\frac{\sqrt{c \cdot c + d \cdot d}}{b \cdot c - a \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\]

Reproduce

herbie shell --seed 2019326 
(FPCore (a b c d)
  :name "Complex division, imag part"
  :precision binary64

  :herbie-target
  (if (< (fabs d) (fabs c)) (/ (- b (* a (/ d c))) (+ c (* d (/ d c)))) (/ (+ (- a) (* b (/ c d))) (+ d (* c (/ c d)))))

  (/ (- (* b c) (* a d)) (+ (* c c) (* d d))))