Average Error: 37.2 → 0.5
Time: 2.4s
Precision: binary64
\[ \begin{array}{c}[x, y, z] = \mathsf{sort}([x, y, z])\\ \end{array} \]
\[\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z} \]
↓
\[\mathsf{hypot}\left(z, x\right) \]
(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)

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.2
Target19.1
Herbie0.5
\[\begin{array}{l} \mathbf{if}\;z < -6.396479394109776 \cdot 10^{+136}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z < 7.320293694404182 \cdot 10^{+117}:\\ \;\;\;\;\sqrt{\left(z \cdot z + x \cdot x\right) + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]

Derivation

  1. Initial program 37.2

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

  2. Simplified37.2

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

  3. Taylor expanded in y around 0 37.4

    \[\leadsto \color{blue}{\sqrt{{z}^{2} + {x}^{2}}} \]

  4. Simplified0.5

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

  5. Final simplification0.5

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

Reproduce

herbie shell --seed 2022156 
(FPCore (x y z)
  :name "FRP.Yampa.Vector3:vector3Rho from Yampa-0.10.2"
  :precision binary64

  :herbie-target
  (if (< z -6.396479394109776e+136) (- z) (if (< z 7.320293694404182e+117) (sqrt (+ (+ (* z z) (* x x)) (* y y))) z))

  (sqrt (+ (+ (* x x) (* y y)) (* z z))))