# ?

Average Error: 0.0 → 0.0
Time: 6.1s
Precision: binary64
Cost: 13248

# ?

$\left(\left(-1000 \leq x \land x \leq 1000\right) \land \left(-1000 \leq y \land y \leq 1000\right)\right) \land \left(-1000 \leq z \land z \leq 1000\right)$
$\begin{array}{c}[x, y, z] = \mathsf{sort}([x, y, z])\\ \end{array}$
$\left(x \cdot x + y \cdot y\right) + z \cdot z$
$\mathsf{fma}\left(x, x, \mathsf{fma}\left(y, y, z \cdot z\right)\right)$
(FPCore (x y z) :precision binary64 (+ (+ (* x x) (* y y)) (* z z)))
(FPCore (x y z) :precision binary64 (fma x x (fma y y (* z z))))
double code(double x, double y, double z) {
return ((x * x) + (y * y)) + (z * z);
}

double code(double x, double y, double z) {
return fma(x, x, fma(y, y, (z * z)));
}

function code(x, y, z)
return Float64(Float64(Float64(x * x) + Float64(y * y)) + Float64(z * z))
end

function code(x, y, z)
return fma(x, x, fma(y, y, Float64(z * z)))
end

code[x_, y_, z_] := N[(N[(N[(x * x), $MachinePrecision] + N[(y * y),$MachinePrecision]), $MachinePrecision] + N[(z * z),$MachinePrecision]), $MachinePrecision]  code[x_, y_, z_] := N[(x * x + N[(y * y + N[(z * z),$MachinePrecision]), $MachinePrecision]),$MachinePrecision]

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

\mathsf{fma}\left(x, x, \mathsf{fma}\left(y, y, z \cdot z\right)\right)


# Derivation?

1. Initial program 0.0

$\left(x \cdot x + y \cdot y\right) + z \cdot z$
2. Simplified0.0

$\leadsto \color{blue}{\mathsf{fma}\left(x, x, \mathsf{fma}\left(y, y, z \cdot z\right)\right)}$
Proof
[Start]0.0 $\left(x \cdot x + y \cdot y\right) + z \cdot z$ $\color{blue}{x \cdot x + \left(y \cdot y + z \cdot z\right)}$ $\color{blue}{\mathsf{fma}\left(x, x, y \cdot y + z \cdot z\right)}$ $\mathsf{fma}\left(x, x, \color{blue}{\mathsf{fma}\left(y, y, z \cdot z\right)}\right)$
3. Final simplification0.0

$\leadsto \mathsf{fma}\left(x, x, \mathsf{fma}\left(y, y, z \cdot z\right)\right)$

# Alternatives

Alternative 1
Error0.0
Cost6976
$z \cdot z + \mathsf{fma}\left(x, x, y \cdot y\right)$
Alternative 2
Error9.2
Cost845
$\begin{array}{l} \mathbf{if}\;z \leq 3.7 \cdot 10^{-152} \lor \neg \left(z \leq 8.4 \cdot 10^{-141}\right) \land z \leq 1.8 \cdot 10^{-109}:\\ \;\;\;\;y \cdot y + x \cdot x\\ \mathbf{else}:\\ \;\;\;\;z \cdot z + y \cdot y\\ \end{array}$
Alternative 3
Error0.0
Cost704
$z \cdot z + \left(y \cdot y + x \cdot x\right)$
Alternative 4
Error9.2
Cost580
$\begin{array}{l} \mathbf{if}\;x \leq -2.65 \cdot 10^{-129}:\\ \;\;\;\;y \cdot y + x \cdot x\\ \mathbf{else}:\\ \;\;\;\;z \cdot z\\ \end{array}$
Alternative 5
Error9.6
Cost324
$\begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-129}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;z \cdot z\\ \end{array}$
Alternative 6
Error27.7
Cost192
$x \cdot x$

# Reproduce?

herbie shell --seed 1
(FPCore (x y z)
:name "x*x+y*y+z*z"
:precision binary64
:pre (and (and (and (<= -1000.0 x) (<= x 1000.0)) (and (<= -1000.0 y) (<= y 1000.0))) (and (<= -1000.0 z) (<= z 1000.0)))
(+ (+ (* x x) (* y y)) (* z z)))