Average Error: 0.0 → 0.0
Time: 1.8s
Precision: 64
\[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i\]
↓
\[\mathsf{fma}\left(i, c, \mathsf{fma}\left(b, a, \mathsf{fma}\left(t, z, x \cdot y\right)\right)\right)\]
\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i
↓
\mathsf{fma}\left(i, c, \mathsf{fma}\left(b, a, \mathsf{fma}\left(t, z, x \cdot y\right)\right)\right)
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r61884 = x;
        double r61885 = y;
        double r61886 = r61884 * r61885;
        double r61887 = z;
        double r61888 = t;
        double r61889 = r61887 * r61888;
        double r61890 = r61886 + r61889;
        double r61891 = a;
        double r61892 = b;
        double r61893 = r61891 * r61892;
        double r61894 = r61890 + r61893;
        double r61895 = c;
        double r61896 = i;
        double r61897 = r61895 * r61896;
        double r61898 = r61894 + r61897;
        return r61898;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r61899 = i;
        double r61900 = c;
        double r61901 = b;
        double r61902 = a;
        double r61903 = t;
        double r61904 = z;
        double r61905 = x;
        double r61906 = y;
        double r61907 = r61905 * r61906;
        double r61908 = fma(r61903, r61904, r61907);
        double r61909 = fma(r61901, r61902, r61908);
        double r61910 = fma(r61899, r61900, r61909);
        return r61910;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus i

Derivation

  1. Initial program 0.0

    \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i\]

  2. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(i, c, \mathsf{fma}\left(b, a, \mathsf{fma}\left(x, y, z \cdot t\right)\right)\right)}\]

  3. Taylor expanded around inf 0.0

    \[\leadsto \mathsf{fma}\left(i, c, \mathsf{fma}\left(b, a, \color{blue}{t \cdot z + x \cdot y}\right)\right)\]

  4. Simplified0.0

    \[\leadsto \mathsf{fma}\left(i, c, \mathsf{fma}\left(b, a, \color{blue}{\mathsf{fma}\left(t, z, x \cdot y\right)}\right)\right)\]

  5. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(i, c, \mathsf{fma}\left(b, a, \mathsf{fma}\left(t, z, x \cdot y\right)\right)\right)\]

Reproduce

herbie shell --seed 2019356 +o rules:numerics
(FPCore (x y z t a b c i)
  :name "Linear.V4:$cdot from linear-1.19.1.3"
  :precision binary64
  (+ (+ (+ (* x y) (* z t)) (* a b)) (* c i)))