Average Error: 0.0 → 0.0
Time: 2.2s
Precision: binary64
\[[x, y] = \mathsf{sort}([x, y]) \\]
\[\left(x + y\right) \cdot \left(z + 1\right) \]
↓
\[y + \mathsf{fma}\left(y, z, x + z \cdot x\right) \]
\left(x + y\right) \cdot \left(z + 1\right)
↓
y + \mathsf{fma}\left(y, z, x + z \cdot x\right)

(FPCore (x y z) :precision binary64 (* (+ x y) (+ z 1.0)))
↓
(FPCore (x y z) :precision binary64 (+ y (fma y z (+ 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 y + fma(y, z, (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 fma-def_binary640.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2022067 
(FPCore (x y z)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, G"
  :precision binary64
  (* (+ x y) (+ z 1.0)))