Complex division, imag part

Average Error: 25.4 → 25.3

Time: 13.6s

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 r2294840 = b;
        double r2294841 = c;
        double r2294842 = r2294840 * r2294841;
        double r2294843 = a;
        double r2294844 = d;
        double r2294845 = r2294843 * r2294844;
        double r2294846 = r2294842 - r2294845;
        double r2294847 = r2294841 * r2294841;
        double r2294848 = r2294844 * r2294844;
        double r2294849 = r2294847 + r2294848;
        double r2294850 = r2294846 / r2294849;
        return r2294850;
}

↓

double f(double a, double b, double c, double d) {
        double r2294851 = b;
        double r2294852 = c;
        double r2294853 = r2294851 * r2294852;
        double r2294854 = a;
        double r2294855 = d;
        double r2294856 = r2294854 * r2294855;
        double r2294857 = r2294853 - r2294856;
        double r2294858 = r2294852 * r2294852;
        double r2294859 = r2294855 * r2294855;
        double r2294860 = r2294858 + r2294859;
        double r2294861 = sqrt(r2294860);
        double r2294862 = r2294857 / r2294861;
        double r2294863 = r2294862 / r2294861;
        return r2294863;
}

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 2019152 
(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))))