Complex division, imag part

Average Error: 25.4 → 25.3

Time: 12.6s

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 r4258295 = b;
        double r4258296 = c;
        double r4258297 = r4258295 * r4258296;
        double r4258298 = a;
        double r4258299 = d;
        double r4258300 = r4258298 * r4258299;
        double r4258301 = r4258297 - r4258300;
        double r4258302 = r4258296 * r4258296;
        double r4258303 = r4258299 * r4258299;
        double r4258304 = r4258302 + r4258303;
        double r4258305 = r4258301 / r4258304;
        return r4258305;
}

↓

double f(double a, double b, double c, double d) {
        double r4258306 = b;
        double r4258307 = c;
        double r4258308 = r4258306 * r4258307;
        double r4258309 = a;
        double r4258310 = d;
        double r4258311 = r4258309 * r4258310;
        double r4258312 = r4258308 - r4258311;
        double r4258313 = r4258307 * r4258307;
        double r4258314 = r4258310 * r4258310;
        double r4258315 = r4258313 + r4258314;
        double r4258316 = sqrt(r4258315);
        double r4258317 = r4258312 / r4258316;
        double r4258318 = r4258317 / r4258316;
        return r4258318;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original	25.4
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 25.4

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

3. Applied add-sqr-sqrt 25.4

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

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 2019134 
(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))))