Average Error: 25.3 → 25.5
Time: 27.2s
Precision: 64
\[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\frac{1}{\mathsf{fma}\left(d, d, \left(c \cdot c\right)\right)} \cdot \mathsf{fma}\left(a, c, \left(b \cdot d\right)\right)\]
\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}
↓
\frac{1}{\mathsf{fma}\left(d, d, \left(c \cdot c\right)\right)} \cdot \mathsf{fma}\left(a, c, \left(b \cdot d\right)\right)
double f(double a, double b, double c, double d) {
        double r22830887 = a;
        double r22830888 = c;
        double r22830889 = r22830887 * r22830888;
        double r22830890 = b;
        double r22830891 = d;
        double r22830892 = r22830890 * r22830891;
        double r22830893 = r22830889 + r22830892;
        double r22830894 = r22830888 * r22830888;
        double r22830895 = r22830891 * r22830891;
        double r22830896 = r22830894 + r22830895;
        double r22830897 = r22830893 / r22830896;
        return r22830897;
}
↓
double f(double a, double b, double c, double d) {
        double r22830898 = 1.0;
        double r22830899 = d;
        double r22830900 = c;
        double r22830901 = r22830900 * r22830900;
        double r22830902 = fma(r22830899, r22830899, r22830901);
        double r22830903 = r22830898 / r22830902;
        double r22830904 = a;
        double r22830905 = b;
        double r22830906 = r22830905 * r22830899;
        double r22830907 = fma(r22830904, r22830900, r22830906);
        double r22830908 = r22830903 * r22830907;
        return r22830908;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original25.3
Target0.5
Herbie25.5
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Initial program 25.3

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

  2. Simplified25.3

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

  4. Applied div-inv25.5

    \[\leadsto \color{blue}{\mathsf{fma}\left(a, c, \left(b \cdot d\right)\right) \cdot \frac{1}{\mathsf{fma}\left(d, d, \left(c \cdot c\right)\right)}}\]

  5. Final simplification25.5

    \[\leadsto \frac{1}{\mathsf{fma}\left(d, d, \left(c \cdot c\right)\right)} \cdot \mathsf{fma}\left(a, c, \left(b \cdot d\right)\right)\]

Reproduce

herbie shell --seed 2019125 +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))))