Average Error: 0.0 → 0.0
Time: 17.0s
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 r87821 = x;
        double r87822 = y;
        double r87823 = r87821 * r87822;
        double r87824 = z;
        double r87825 = t;
        double r87826 = r87824 * r87825;
        double r87827 = r87823 + r87826;
        double r87828 = a;
        double r87829 = b;
        double r87830 = r87828 * r87829;
        double r87831 = r87827 + r87830;
        double r87832 = c;
        double r87833 = i;
        double r87834 = r87832 * r87833;
        double r87835 = r87831 + r87834;
        return r87835;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r87836 = c;
        double r87837 = i;
        double r87838 = t;
        double r87839 = z;
        double r87840 = a;
        double r87841 = b;
        double r87842 = x;
        double r87843 = y;
        double r87844 = r87842 * r87843;
        double r87845 = fma(r87840, r87841, r87844);
        double r87846 = fma(r87838, r87839, r87845);
        double r87847 = fma(r87836, r87837, r87846);
        return r87847;
}

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, C"
  :precision binary64
  (+ (+ (+ (* x y) (* z t)) (* a b)) (* c i)))