Complex division, imag part

Average Error: 26.1 → 26.0

Time: 13.0s

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 r5290659 = b;
        double r5290660 = c;
        double r5290661 = r5290659 * r5290660;
        double r5290662 = a;
        double r5290663 = d;
        double r5290664 = r5290662 * r5290663;
        double r5290665 = r5290661 - r5290664;
        double r5290666 = r5290660 * r5290660;
        double r5290667 = r5290663 * r5290663;
        double r5290668 = r5290666 + r5290667;
        double r5290669 = r5290665 / r5290668;
        return r5290669;
}

↓

double f(double a, double b, double c, double d) {
        double r5290670 = b;
        double r5290671 = c;
        double r5290672 = r5290670 * r5290671;
        double r5290673 = a;
        double r5290674 = d;
        double r5290675 = r5290673 * r5290674;
        double r5290676 = r5290672 - r5290675;
        double r5290677 = r5290671 * r5290671;
        double r5290678 = r5290674 * r5290674;
        double r5290679 = r5290677 + r5290678;
        double r5290680 = sqrt(r5290679);
        double r5290681 = r5290676 / r5290680;
        double r5290682 = r5290681 / r5290680;
        return r5290682;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original	26.1
Target	0.4
Herbie	26.0

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

Derivation

1. Initial program 26.1

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

3. Applied add-sqr-sqrt 26.1

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

4. Applied associate-/r* 26.0

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

5. Final simplification 26.0

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

Reproduce

herbie shell --seed 2019163 
(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))))