Complex division, imag part

Average Error: 25.3 → 25.3

Time: 20.0s

Precision: 64

Math

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

↓

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

C

double f(double a, double b, double c, double d) {
        double r5429293 = b;
        double r5429294 = c;
        double r5429295 = r5429293 * r5429294;
        double r5429296 = a;
        double r5429297 = d;
        double r5429298 = r5429296 * r5429297;
        double r5429299 = r5429295 - r5429298;
        double r5429300 = r5429294 * r5429294;
        double r5429301 = r5429297 * r5429297;
        double r5429302 = r5429300 + r5429301;
        double r5429303 = r5429299 / r5429302;
        return r5429303;
}

↓

double f(double a, double b, double c, double d) {
        double r5429304 = b;
        double r5429305 = c;
        double r5429306 = r5429304 * r5429305;
        double r5429307 = a;
        double r5429308 = d;
        double r5429309 = r5429307 * r5429308;
        double r5429310 = r5429306 - r5429309;
        double r5429311 = r5429305 * r5429305;
        double r5429312 = r5429308 * r5429308;
        double r5429313 = r5429311 + r5429312;
        double r5429314 = sqrt(r5429313);
        double r5429315 = r5429310 / r5429314;
        double r5429316 = r5429315 / r5429314;
        return r5429316;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original	25.3
Target	0.5
Herbie	25.3

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

Derivation

1. Initial program 25.3

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

3. Applied add-sqr-sqrt 25.3

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

4. Applied associate-/r* 25.3

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

5. Final simplification 25.3

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

Reproduce

herbie shell --seed 2019143 
(FPCore (a b c d)
  :name "Complex division, imag part"

  :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))))