Average Error: 26.0 → 25.9
Time: 15.2s
Precision: 64
\[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\frac{\frac{\mathsf{fma}\left(a, c, \left(b \cdot d\right)\right)}{\sqrt{\mathsf{fma}\left(d, d, \left(c \cdot c\right)\right)}}}{\sqrt{\mathsf{fma}\left(d, d, \left(c \cdot c\right)\right)}}\]
\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}
↓
\frac{\frac{\mathsf{fma}\left(a, c, \left(b \cdot d\right)\right)}{\sqrt{\mathsf{fma}\left(d, d, \left(c \cdot c\right)\right)}}}{\sqrt{\mathsf{fma}\left(d, d, \left(c \cdot c\right)\right)}}
double f(double a, double b, double c, double d) {
        double r21523975 = a;
        double r21523976 = c;
        double r21523977 = r21523975 * r21523976;
        double r21523978 = b;
        double r21523979 = d;
        double r21523980 = r21523978 * r21523979;
        double r21523981 = r21523977 + r21523980;
        double r21523982 = r21523976 * r21523976;
        double r21523983 = r21523979 * r21523979;
        double r21523984 = r21523982 + r21523983;
        double r21523985 = r21523981 / r21523984;
        return r21523985;
}
↓
double f(double a, double b, double c, double d) {
        double r21523986 = a;
        double r21523987 = c;
        double r21523988 = b;
        double r21523989 = d;
        double r21523990 = r21523988 * r21523989;
        double r21523991 = fma(r21523986, r21523987, r21523990);
        double r21523992 = r21523987 * r21523987;
        double r21523993 = fma(r21523989, r21523989, r21523992);
        double r21523994 = sqrt(r21523993);
        double r21523995 = r21523991 / r21523994;
        double r21523996 = r21523995 / r21523994;
        return r21523996;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original26.0
Target0.4
Herbie25.9
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Initial program 26.0

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

  2. Simplified26.0

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

  4. Applied add-sqr-sqrt26.0

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

  5. Applied associate-/r*25.9

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

  6. Taylor expanded around -inf 25.9

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

  7. Simplified25.9

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

  8. Final simplification25.9

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

Reproduce

herbie shell --seed 2019120 +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))))