Average Error: 0.0 → 0.0
Time: 2.1s
Precision: binary64
\[[x, y] = \mathsf{sort}([x, y]) \\]
\[\left(x + y\right) \cdot \left(z + 1\right) \]
↓
\[\mathsf{fma}\left(z, y, y\right) + \left(x + z \cdot x\right) \]
\left(x + y\right) \cdot \left(z + 1\right)
↓
\mathsf{fma}\left(z, y, y\right) + \left(x + z \cdot x\right)

(FPCore (x y z) :precision binary64 (* (+ x y) (+ z 1.0)))
↓
(FPCore (x y z) :precision binary64 (+ (fma z y y) (+ x (* z x))))
double code(double x, double y, double z) {
	return (x + y) * (z + 1.0);
}
↓
double code(double x, double y, double z) {
	return fma(z, y, y) + (x + (z * x));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 0.0

    \[\left(x + y\right) \cdot \left(z + 1\right) \]

  2. Taylor expanded in x around 0 0.0

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

  3. Applied associate-+r+_binary640.0

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

  4. Simplified0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2022076 
(FPCore (x y z)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, G"
  :precision binary64
  (* (+ x y) (+ z 1.0)))