Complex division, real part

Average Error: 26.0 → 25.9

Time: 19.9s

Precision: 64

Math

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

↓

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

C

double f(double a, double b, double c, double d) {
        double r31827733 = a;
        double r31827734 = c;
        double r31827735 = r31827733 * r31827734;
        double r31827736 = b;
        double r31827737 = d;
        double r31827738 = r31827736 * r31827737;
        double r31827739 = r31827735 + r31827738;
        double r31827740 = r31827734 * r31827734;
        double r31827741 = r31827737 * r31827737;
        double r31827742 = r31827740 + r31827741;
        double r31827743 = r31827739 / r31827742;
        return r31827743;
}

↓

double f(double a, double b, double c, double d) {
        double r31827744 = b;
        double r31827745 = d;
        double r31827746 = r31827744 * r31827745;
        double r31827747 = a;
        double r31827748 = c;
        double r31827749 = r31827747 * r31827748;
        double r31827750 = r31827746 + r31827749;
        double r31827751 = r31827748 * r31827748;
        double r31827752 = r31827745 * r31827745;
        double r31827753 = r31827751 + r31827752;
        double r31827754 = sqrt(r31827753);
        double r31827755 = r31827750 / r31827754;
        double r31827756 = r31827755 / r31827754;
        return r31827756;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original	26.0
Target	0.4
Herbie	25.9

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

Derivation

1. Initial program 26.0

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

3. Applied add-sqr-sqrt 26.0

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

4. Applied associate-/r* 25.9

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

5. Final simplification 25.9

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

Reproduce

herbie shell --seed 2019120 
(FPCore (a b c d)
  :name "Complex division, real part"

  :herbie-target
  (if (< (fabs d) (fabs c)) (/ (+ a (* b (/ d c))) (+ c (* d (/ d c)))) (/ (+ b (* a (/ c d))) (+ d (* c (/ c d)))))

  (/ (+ (* a c) (* b d)) (+ (* c c) (* d d))))