Average Error: 0.0 → 0.0
Time: 17.9s
Precision: 64
\[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i\]
↓
\[\mathsf{fma}\left(z, t, \mathsf{fma}\left(x, y, \mathsf{fma}\left(a, b, i \cdot c\right)\right)\right)\]
\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i
↓
\mathsf{fma}\left(z, t, \mathsf{fma}\left(x, y, \mathsf{fma}\left(a, b, i \cdot c\right)\right)\right)
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r2305648 = x;
        double r2305649 = y;
        double r2305650 = r2305648 * r2305649;
        double r2305651 = z;
        double r2305652 = t;
        double r2305653 = r2305651 * r2305652;
        double r2305654 = r2305650 + r2305653;
        double r2305655 = a;
        double r2305656 = b;
        double r2305657 = r2305655 * r2305656;
        double r2305658 = r2305654 + r2305657;
        double r2305659 = c;
        double r2305660 = i;
        double r2305661 = r2305659 * r2305660;
        double r2305662 = r2305658 + r2305661;
        return r2305662;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r2305663 = z;
        double r2305664 = t;
        double r2305665 = x;
        double r2305666 = y;
        double r2305667 = a;
        double r2305668 = b;
        double r2305669 = i;
        double r2305670 = c;
        double r2305671 = r2305669 * r2305670;
        double r2305672 = fma(r2305667, r2305668, r2305671);
        double r2305673 = fma(r2305665, r2305666, r2305672);
        double r2305674 = fma(r2305663, r2305664, r2305673);
        return r2305674;
}

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

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

  4. Applied add-cbrt-cube27.4

    \[\leadsto \mathsf{fma}\left(z, t, \mathsf{fma}\left(x, y, \color{blue}{\sqrt[3]{\left(\mathsf{fma}\left(c, i, a \cdot b\right) \cdot \mathsf{fma}\left(c, i, a \cdot b\right)\right) \cdot \mathsf{fma}\left(c, i, a \cdot b\right)}}\right)\right)\]
  5. Using strategy rm

  6. Applied add-cube-cbrt27.4

    \[\leadsto \mathsf{fma}\left(z, t, \mathsf{fma}\left(x, y, \sqrt[3]{\left(\mathsf{fma}\left(c, i, a \cdot b\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\mathsf{fma}\left(c, i, a \cdot b\right)} \cdot \sqrt[3]{\mathsf{fma}\left(c, i, a \cdot b\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(c, i, a \cdot b\right)}\right)}\right) \cdot \mathsf{fma}\left(c, i, a \cdot b\right)}\right)\right)\]

  7. Applied associate-*r*27.4

    \[\leadsto \mathsf{fma}\left(z, t, \mathsf{fma}\left(x, y, \sqrt[3]{\color{blue}{\left(\left(\mathsf{fma}\left(c, i, a \cdot b\right) \cdot \left(\sqrt[3]{\mathsf{fma}\left(c, i, a \cdot b\right)} \cdot \sqrt[3]{\mathsf{fma}\left(c, i, a \cdot b\right)}\right)\right) \cdot \sqrt[3]{\mathsf{fma}\left(c, i, a \cdot b\right)}\right)} \cdot \mathsf{fma}\left(c, i, a \cdot b\right)}\right)\right)\]

  8. Taylor expanded around inf 0.0

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

  9. Simplified0.0

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

  10. Final simplification0.0

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

Reproduce

herbie shell --seed 2019151 +o rules:numerics
(FPCore (x y z t a b c i)
  :name "Linear.V4:$cdot from linear-1.19.1.3"
  (+ (+ (+ (* x y) (* z t)) (* a b)) (* c i)))