Average Error: 0.0 → 0.0
Time: 19.5s
Precision: 64
\[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i\]
↓
\[\mathsf{fma}\left(c, i, \mathsf{fma}\left(t, z, \mathsf{fma}\left(a, b, x \cdot y\right)\right)\right)\]
\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i
↓
\mathsf{fma}\left(c, i, \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 r54194 = x;
        double r54195 = y;
        double r54196 = r54194 * r54195;
        double r54197 = z;
        double r54198 = t;
        double r54199 = r54197 * r54198;
        double r54200 = r54196 + r54199;
        double r54201 = a;
        double r54202 = b;
        double r54203 = r54201 * r54202;
        double r54204 = r54200 + r54203;
        double r54205 = c;
        double r54206 = i;
        double r54207 = r54205 * r54206;
        double r54208 = r54204 + r54207;
        return r54208;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r54209 = c;
        double r54210 = i;
        double r54211 = t;
        double r54212 = z;
        double r54213 = a;
        double r54214 = b;
        double r54215 = x;
        double r54216 = y;
        double r54217 = r54215 * r54216;
        double r54218 = fma(r54213, r54214, r54217);
        double r54219 = fma(r54211, r54212, r54218);
        double r54220 = fma(r54209, r54210, r54219);
        return r54220;
}

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

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

  3. Taylor expanded around inf 0.0

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

  4. Simplified0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2019322 +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)))