# ?

Average Error: 7.3 → 1.0
Time: 7.9s
Precision: binary64
Cost: 6528

# ?

$\left(\left(-10 \leq x \land x \leq 10\right) \land \left(-10 \leq y \land y \leq 10\right)\right) \land \left(-10 \leq z \land z \leq 10\right)$
$\begin{array}{c}[x, y, z] = \mathsf{sort}([x, y, z])\\ \end{array}$
$\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}$
$\mathsf{hypot}\left(z, x\right)$
(FPCore (x y z) :precision binary64 (sqrt (+ (+ (* x x) (* y y)) (* z z))))
(FPCore (x y z) :precision binary64 (hypot z x))
double code(double x, double y, double z) {
return sqrt((((x * x) + (y * y)) + (z * z)));
}

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

public static double code(double x, double y, double z) {
return Math.sqrt((((x * x) + (y * y)) + (z * z)));
}

public static double code(double x, double y, double z) {
return Math.hypot(z, x);
}

def code(x, y, z):
return math.sqrt((((x * x) + (y * y)) + (z * z)))

def code(x, y, z):
return math.hypot(z, x)

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

function code(x, y, z)
return hypot(z, x)
end

function tmp = code(x, y, z)
tmp = sqrt((((x * x) + (y * y)) + (z * z)));
end

function tmp = code(x, y, z)
tmp = hypot(z, x);
end

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

code[x_, y_, z_] := N[Sqrt[z ^ 2 + x ^ 2], \$MachinePrecision]

\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}

\mathsf{hypot}\left(z, x\right)


# Try it out?

Results

 In Out
Enter valid numbers for all inputs

# Derivation?

1. Initial program 7.3

$\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}$
2. Simplified7.3

$\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(x, x, \mathsf{fma}\left(y, y, z \cdot z\right)\right)}}$
Proof
[Start]7.3 $\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}$ $\sqrt{\color{blue}{x \cdot x + \left(y \cdot y + z \cdot z\right)}}$ $\sqrt{\color{blue}{\mathsf{fma}\left(x, x, y \cdot y + z \cdot z\right)}}$ $\sqrt{\mathsf{fma}\left(x, x, \color{blue}{\mathsf{fma}\left(y, y, z \cdot z\right)}\right)}$
3. Taylor expanded in y around 0 8.1

$\leadsto \color{blue}{\sqrt{{z}^{2} + {x}^{2}}}$
4. Simplified1.0

$\leadsto \color{blue}{\mathsf{hypot}\left(z, x\right)}$
Proof
[Start]8.1 $\sqrt{{z}^{2} + {x}^{2}}$ $\sqrt{\color{blue}{z \cdot z} + {x}^{2}}$ $\sqrt{z \cdot z + \color{blue}{x \cdot x}}$ $\color{blue}{\mathsf{hypot}\left(z, x\right)}$
5. Final simplification1.0

$\leadsto \mathsf{hypot}\left(z, x\right)$

# Alternatives

Alternative 1
Error12.3
Cost6660
$\begin{array}{l} \mathbf{if}\;z \leq 6.2 \cdot 10^{-141}:\\ \;\;\;\;\mathsf{hypot}\left(y, x\right)\\ \mathbf{else}:\\ \;\;\;\;z + 0.5 \cdot \frac{y \cdot y}{z}\\ \end{array}$
Alternative 2
Error12.9
Cost708
$\begin{array}{l} \mathbf{if}\;z \leq 7.8 \cdot 10^{-141}:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;z + 0.5 \cdot \frac{y \cdot y}{z}\\ \end{array}$
Alternative 3
Error12.7
Cost708
$\begin{array}{l} \mathbf{if}\;z \leq 2 \cdot 10^{-140}:\\ \;\;\;\;\frac{y \cdot y}{x} \cdot -0.5 - x\\ \mathbf{else}:\\ \;\;\;\;z + 0.5 \cdot \frac{y \cdot y}{z}\\ \end{array}$
Alternative 4
Error13.1
Cost260
$\begin{array}{l} \mathbf{if}\;z \leq 1.2 \cdot 10^{-141}:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array}$
Alternative 5
Error30.9
Cost64
$z$

# Reproduce?

herbie shell --seed 1
(FPCore (x y z)
:name "sqrt(x*x + y*y + z*z)"
:precision binary64
:pre (and (and (and (<= -10.0 x) (<= x 10.0)) (and (<= -10.0 y) (<= y 10.0))) (and (<= -10.0 z) (<= z 10.0)))
(sqrt (+ (+ (* x x) (* y y)) (* z z))))