Average Error: 26.1 → 26.1
Time: 15.8s
Precision: 64
\[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\frac{(a \cdot c + \left(b \cdot d\right))_*}{(d \cdot d + \left(c \cdot c\right))_*}\]
double f(double a, double b, double c, double d) {
        double r12366432 = a;
        double r12366433 = c;
        double r12366434 = r12366432 * r12366433;
        double r12366435 = b;
        double r12366436 = d;
        double r12366437 = r12366435 * r12366436;
        double r12366438 = r12366434 + r12366437;
        double r12366439 = r12366433 * r12366433;
        double r12366440 = r12366436 * r12366436;
        double r12366441 = r12366439 + r12366440;
        double r12366442 = r12366438 / r12366441;
        return r12366442;
}
↓
double f(double a, double b, double c, double d) {
        double r12366443 = a;
        double r12366444 = c;
        double r12366445 = b;
        double r12366446 = d;
        double r12366447 = r12366445 * r12366446;
        double r12366448 = fma(r12366443, r12366444, r12366447);
        double r12366449 = r12366444 * r12366444;
        double r12366450 = fma(r12366446, r12366446, r12366449);
        double r12366451 = r12366448 / r12366450;
        return r12366451;
}


\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))_*}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original26.1
Target0.4
Herbie26.1
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Initial program 26.1

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

  2. Simplified26.1

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

  3. Final simplification26.1

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

Reproduce

herbie shell --seed 2019102 +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))))