Average Error: 0.0 → 0.0
Time: 3.4s
Precision: 64
\[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i\]
↓
\[y \cdot x + \mathsf{fma}\left(t, z, \mathsf{fma}\left(a, b, i \cdot c\right)\right)\]
\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i
↓
y \cdot x + \mathsf{fma}\left(t, z, \mathsf{fma}\left(a, b, i \cdot c\right)\right)
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r69745 = x;
        double r69746 = y;
        double r69747 = r69745 * r69746;
        double r69748 = z;
        double r69749 = t;
        double r69750 = r69748 * r69749;
        double r69751 = r69747 + r69750;
        double r69752 = a;
        double r69753 = b;
        double r69754 = r69752 * r69753;
        double r69755 = r69751 + r69754;
        double r69756 = c;
        double r69757 = i;
        double r69758 = r69756 * r69757;
        double r69759 = r69755 + r69758;
        return r69759;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r69760 = y;
        double r69761 = x;
        double r69762 = r69760 * r69761;
        double r69763 = t;
        double r69764 = z;
        double r69765 = a;
        double r69766 = b;
        double r69767 = i;
        double r69768 = c;
        double r69769 = r69767 * r69768;
        double r69770 = fma(r69765, r69766, r69769);
        double r69771 = fma(r69763, r69764, r69770);
        double r69772 = r69762 + r69771;
        return r69772;
}

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. Taylor expanded around inf 0.0

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

  3. Simplified0.0

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

  5. Applied fma-udef0.0

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

  6. Applied associate-+l+0.0

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

  7. Simplified0.0

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2020001 +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)))