Average Error: 0.0 → 0.0
Time: 5.2s
Precision: 64
\[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i\]
↓
\[\mathsf{fma}\left(i, c, \mathsf{fma}\left(t, z, \mathsf{fma}\left(a, b, x \cdot y\right)\right)\right)\]
\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i
↓
\mathsf{fma}\left(i, c, \mathsf{fma}\left(t, z, \mathsf{fma}\left(a, b, 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 r186024 = x;
        double r186025 = y;
        double r186026 = r186024 * r186025;
        double r186027 = z;
        double r186028 = t;
        double r186029 = r186027 * r186028;
        double r186030 = r186026 + r186029;
        double r186031 = a;
        double r186032 = b;
        double r186033 = r186031 * r186032;
        double r186034 = r186030 + r186033;
        double r186035 = c;
        double r186036 = i;
        double r186037 = r186035 * r186036;
        double r186038 = r186034 + r186037;
        return r186038;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r186039 = i;
        double r186040 = c;
        double r186041 = t;
        double r186042 = z;
        double r186043 = a;
        double r186044 = b;
        double r186045 = x;
        double r186046 = y;
        double r186047 = r186045 * r186046;
        double r186048 = fma(r186043, r186044, r186047);
        double r186049 = fma(r186041, r186042, r186048);
        double r186050 = fma(r186039, r186040, r186049);
        return r186050;
}

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

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

  4. Simplified0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2020034 +o rules:numerics
(FPCore (x y z t a b c i)
  :name "Linear.V4:$cdot from linear-1.19.1.3, C"
  :precision binary64
  (+ (+ (+ (* x y) (* z t)) (* a b)) (* c i)))