Average Error: 25.8 → 25.8
Time: 16.6s
Precision: 64
\[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{fma}\left(d, d, c \cdot c\right)}\]
\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}
↓
\frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{fma}\left(d, d, c \cdot c\right)}
double f(double a, double b, double c, double d) {
        double r4133589 = a;
        double r4133590 = c;
        double r4133591 = r4133589 * r4133590;
        double r4133592 = b;
        double r4133593 = d;
        double r4133594 = r4133592 * r4133593;
        double r4133595 = r4133591 + r4133594;
        double r4133596 = r4133590 * r4133590;
        double r4133597 = r4133593 * r4133593;
        double r4133598 = r4133596 + r4133597;
        double r4133599 = r4133595 / r4133598;
        return r4133599;
}
↓
double f(double a, double b, double c, double d) {
        double r4133600 = a;
        double r4133601 = c;
        double r4133602 = b;
        double r4133603 = d;
        double r4133604 = r4133602 * r4133603;
        double r4133605 = fma(r4133600, r4133601, r4133604);
        double r4133606 = r4133601 * r4133601;
        double r4133607 = fma(r4133603, r4133603, r4133606);
        double r4133608 = r4133605 / r4133607;
        return r4133608;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original25.8
Target0.4
Herbie25.8
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Initial program 25.8

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

  2. Simplified25.8

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{fma}\left(d, d, c \cdot c\right)}}\]

  3. Final simplification25.8

    \[\leadsto \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{fma}\left(d, d, c \cdot c\right)}\]

Reproduce

herbie shell --seed 2019163 +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))))