Average Error: 0.0 → 0.0
Time: 3.6s
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 r149502 = x;
        double r149503 = y;
        double r149504 = r149502 * r149503;
        double r149505 = z;
        double r149506 = t;
        double r149507 = r149505 * r149506;
        double r149508 = r149504 + r149507;
        double r149509 = a;
        double r149510 = b;
        double r149511 = r149509 * r149510;
        double r149512 = r149508 + r149511;
        double r149513 = c;
        double r149514 = i;
        double r149515 = r149513 * r149514;
        double r149516 = r149512 + r149515;
        return r149516;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r149517 = i;
        double r149518 = c;
        double r149519 = t;
        double r149520 = z;
        double r149521 = a;
        double r149522 = b;
        double r149523 = x;
        double r149524 = y;
        double r149525 = r149523 * r149524;
        double r149526 = fma(r149521, r149522, r149525);
        double r149527 = fma(r149519, r149520, r149526);
        double r149528 = fma(r149517, r149518, r149527);
        return r149528;
}

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 2019354 +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)))