Average Error: 0.0 → 0.0
Time: 810.0ms
Precision: 64
\[x + y \cdot \left(z + x\right)\]
↓
\[\mathsf{fma}\left(y, z, \mathsf{fma}\left(y, x, x\right)\right)\]
x + y \cdot \left(z + x\right)
↓
\mathsf{fma}\left(y, z, \mathsf{fma}\left(y, x, x\right)\right)
double f(double x, double y, double z) {
        double r92205 = x;
        double r92206 = y;
        double r92207 = z;
        double r92208 = r92207 + r92205;
        double r92209 = r92206 * r92208;
        double r92210 = r92205 + r92209;
        return r92210;
}
↓
double f(double x, double y, double z) {
        double r92211 = y;
        double r92212 = z;
        double r92213 = x;
        double r92214 = fma(r92211, r92213, r92213);
        double r92215 = fma(r92211, r92212, r92214);
        return r92215;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 0.0

    \[x + y \cdot \left(z + x\right)\]

  2. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(y, z + x, x\right)}\]

  3. Taylor expanded around 0 0.0

    \[\leadsto \color{blue}{z \cdot y + \left(x + x \cdot y\right)}\]

  4. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(y, z, \mathsf{fma}\left(y, x, x\right)\right)}\]

  5. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(y, z, \mathsf{fma}\left(y, x, x\right)\right)\]

Reproduce

herbie shell --seed 2020001 +o rules:numerics
(FPCore (x y z)
  :name "Main:bigenough2 from A"
  :precision binary64
  (+ x (* y (+ z x))))