Average Error: 25.8 → 25.8
Time: 16.5s
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 r4001586 = a;
        double r4001587 = c;
        double r4001588 = r4001586 * r4001587;
        double r4001589 = b;
        double r4001590 = d;
        double r4001591 = r4001589 * r4001590;
        double r4001592 = r4001588 + r4001591;
        double r4001593 = r4001587 * r4001587;
        double r4001594 = r4001590 * r4001590;
        double r4001595 = r4001593 + r4001594;
        double r4001596 = r4001592 / r4001595;
        return r4001596;
}
↓
double f(double a, double b, double c, double d) {
        double r4001597 = a;
        double r4001598 = c;
        double r4001599 = b;
        double r4001600 = d;
        double r4001601 = r4001599 * r4001600;
        double r4001602 = fma(r4001597, r4001598, r4001601);
        double r4001603 = r4001598 * r4001598;
        double r4001604 = fma(r4001600, r4001600, r4001603);
        double r4001605 = r4001602 / r4001604;
        return r4001605;
}

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))))