Complex division, imag part

Average Error: 25.6 → 25.6

Time: 17.7s

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 r2369078 = b;
        double r2369079 = c;
        double r2369080 = r2369078 * r2369079;
        double r2369081 = a;
        double r2369082 = d;
        double r2369083 = r2369081 * r2369082;
        double r2369084 = r2369080 - r2369083;
        double r2369085 = r2369079 * r2369079;
        double r2369086 = r2369082 * r2369082;
        double r2369087 = r2369085 + r2369086;
        double r2369088 = r2369084 / r2369087;
        return r2369088;
}

↓

double f(double a, double b, double c, double d) {
        double r2369089 = b;
        double r2369090 = c;
        double r2369091 = r2369089 * r2369090;
        double r2369092 = a;
        double r2369093 = d;
        double r2369094 = r2369092 * r2369093;
        double r2369095 = r2369091 - r2369094;
        double r2369096 = r2369090 * r2369090;
        double r2369097 = r2369093 * r2369093;
        double r2369098 = r2369096 + r2369097;
        double r2369099 = sqrt(r2369098);
        double r2369100 = r2369095 / r2369099;
        double r2369101 = r2369100 / r2369099;
        return r2369101;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original	25.6
Target	0.4
Herbie	25.6

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

Derivation

1. Initial program 25.6

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

3. Applied add-sqr-sqrt 25.6

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

4. Applied associate-/r* 25.6

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

5. Final simplification 25.6

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

Reproduce

herbie shell --seed 2019153 
(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))))