Complex division, imag part

Average Error: 26.0 → 24.5

Time: 10.3s

Precision: 64

Math

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

↓

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

C

double f(double a, double b, double c, double d) {
        double r1992261 = b;
        double r1992262 = c;
        double r1992263 = r1992261 * r1992262;
        double r1992264 = a;
        double r1992265 = d;
        double r1992266 = r1992264 * r1992265;
        double r1992267 = r1992263 - r1992266;
        double r1992268 = r1992262 * r1992262;
        double r1992269 = r1992265 * r1992265;
        double r1992270 = r1992268 + r1992269;
        double r1992271 = r1992267 / r1992270;
        return r1992271;
}

↓

double f(double a, double b, double c, double d) {
        double r1992272 = b;
        double r1992273 = c;
        double r1992274 = r1992272 * r1992273;
        double r1992275 = r1992273 * r1992273;
        double r1992276 = d;
        double r1992277 = r1992276 * r1992276;
        double r1992278 = r1992275 + r1992277;
        double r1992279 = r1992274 / r1992278;
        double r1992280 = a;
        double r1992281 = r1992278 / r1992276;
        double r1992282 = r1992280 / r1992281;
        double r1992283 = r1992279 - r1992282;
        return r1992283;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original	26.0
Target	0.4
Herbie	24.5

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

Derivation

1. Initial program 26.0

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

3. Applied div-sub 26.0

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

5. Applied associate-/l* 24.5

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

6. Final simplification 24.5

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

Reproduce

herbie shell --seed 2019128 
(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))))