Complex division, real part

Average Error: 25.5 → 25.5

Time: 17.0s

Precision: 64

Math

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

↓

\[\frac{(a \cdot c + \left(b \cdot d\right))_*}{(d \cdot d + \left(c \cdot c\right))_*}\]

C

double f(double a, double b, double c, double d) {
        double r26561774 = a;
        double r26561775 = c;
        double r26561776 = r26561774 * r26561775;
        double r26561777 = b;
        double r26561778 = d;
        double r26561779 = r26561777 * r26561778;
        double r26561780 = r26561776 + r26561779;
        double r26561781 = r26561775 * r26561775;
        double r26561782 = r26561778 * r26561778;
        double r26561783 = r26561781 + r26561782;
        double r26561784 = r26561780 / r26561783;
        return r26561784;
}

↓

double f(double a, double b, double c, double d) {
        double r26561785 = a;
        double r26561786 = c;
        double r26561787 = b;
        double r26561788 = d;
        double r26561789 = r26561787 * r26561788;
        double r26561790 = fma(r26561785, r26561786, r26561789);
        double r26561791 = r26561786 * r26561786;
        double r26561792 = fma(r26561788, r26561788, r26561791);
        double r26561793 = r26561790 / r26561792;
        return r26561793;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original	25.5
Target	0.5
Herbie	25.5

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

Derivation

1. Initial program 25.5

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

2. Simplified 25.5

   \[\leadsto \color{blue}{\frac{(a \cdot c + \left(b \cdot d\right))_*}{(d \cdot d + \left(c \cdot c\right))_*}}\]

3. Final simplification 25.5

   \[\leadsto \frac{(a \cdot c + \left(b \cdot d\right))_*}{(d \cdot d + \left(c \cdot c\right))_*}\]

Reproduce

herbie shell --seed 2019112 +o rules:numerics
(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))))