Complex division, imag part

Average Error: 25.9 → 25.9

Time: 14.7s

Precision: 64

Math

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

↓

\[\frac{\left(b \cdot c - a \cdot d\right) \cdot \frac{1}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\]

C

double f(double a, double b, double c, double d) {
        double r88758 = b;
        double r88759 = c;
        double r88760 = r88758 * r88759;
        double r88761 = a;
        double r88762 = d;
        double r88763 = r88761 * r88762;
        double r88764 = r88760 - r88763;
        double r88765 = r88759 * r88759;
        double r88766 = r88762 * r88762;
        double r88767 = r88765 + r88766;
        double r88768 = r88764 / r88767;
        return r88768;
}

↓

double f(double a, double b, double c, double d) {
        double r88769 = b;
        double r88770 = c;
        double r88771 = r88769 * r88770;
        double r88772 = a;
        double r88773 = d;
        double r88774 = r88772 * r88773;
        double r88775 = r88771 - r88774;
        double r88776 = 1.0;
        double r88777 = r88770 * r88770;
        double r88778 = r88773 * r88773;
        double r88779 = r88777 + r88778;
        double r88780 = sqrt(r88779);
        double r88781 = r88776 / r88780;
        double r88782 = r88775 * r88781;
        double r88783 = r88782 / r88780;
        return r88783;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original	25.9
Target	0.4
Herbie	25.9

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

Derivation

1. Initial program 25.9

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

3. Applied add-sqr-sqrt 25.9

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

4. Applied associate-/r* 25.8

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

6. Applied div-inv 25.9

   \[\leadsto \frac{\color{blue}{\left(b \cdot c - a \cdot d\right) \cdot \frac{1}{\sqrt{c \cdot c + d \cdot d}}}}{\sqrt{c \cdot c + d \cdot d}}\]

7. Final simplification 25.9

   \[\leadsto \frac{\left(b \cdot c - a \cdot d\right) \cdot \frac{1}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\]

Reproduce

herbie shell --seed 2019323 
(FPCore (a b c d)
  :name "Complex division, imag part"
  :precision binary64

  :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))))