Complex division, imag part

Average Error: 25.4 → 25.3

Time: 10.9s

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 r2125538 = b;
        double r2125539 = c;
        double r2125540 = r2125538 * r2125539;
        double r2125541 = a;
        double r2125542 = d;
        double r2125543 = r2125541 * r2125542;
        double r2125544 = r2125540 - r2125543;
        double r2125545 = r2125539 * r2125539;
        double r2125546 = r2125542 * r2125542;
        double r2125547 = r2125545 + r2125546;
        double r2125548 = r2125544 / r2125547;
        return r2125548;
}

↓

double f(double a, double b, double c, double d) {
        double r2125549 = b;
        double r2125550 = c;
        double r2125551 = r2125549 * r2125550;
        double r2125552 = a;
        double r2125553 = d;
        double r2125554 = r2125552 * r2125553;
        double r2125555 = r2125551 - r2125554;
        double r2125556 = r2125550 * r2125550;
        double r2125557 = r2125553 * r2125553;
        double r2125558 = r2125556 + r2125557;
        double r2125559 = sqrt(r2125558);
        double r2125560 = r2125555 / r2125559;
        double r2125561 = r2125560 / r2125559;
        return r2125561;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original	25.4
Target	0.4
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 2019154 
(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))))