Average Error: 38.0 → 0.5

Time: 2.2s

Precision: binary64

Cost: 6528

\[ \begin{array}{c}[x, y, z] = \mathsf{sort}([x, y, z])\\ \end{array} \]

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

↓

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

(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(x * x) + Float64(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[(x * x), $MachinePrecision] + N[(N[(y * y), $MachinePrecision] + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

↓

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

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

↓

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

Try it out

In
Out

Enter valid numbers for all inputs

Target

Original	38.0
Target	0.0
Herbie	0.5

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

Derivation

Initial program 38.0
\[\sqrt{x \cdot x + \left(y \cdot y + z \cdot z\right)} \]
Taylor expanded in y around 0 38.2
\[\leadsto \color{blue}{\sqrt{{z}^{2} + {x}^{2}}} \]
Simplified0.5
\[\leadsto \color{blue}{\mathsf{hypot}\left(z, x\right)} \]
Proof
(hypot.f64 z x): 0 points increase in error, 0 points decrease in error
(Rewrite<= hypot-def_binary64 (sqrt.f64 (+.f64 (*.f64 z z) (*.f64 x x)))): 148 points increase in error, 0 points decrease in error
(sqrt.f64 (+.f64 (Rewrite<= unpow2_binary64 (pow.f64 z 2)) (*.f64 x x))): 0 points increase in error, 0 points decrease in error
(sqrt.f64 (+.f64 (pow.f64 z 2) (Rewrite<= unpow2_binary64 (pow.f64 x 2)))): 0 points increase in error, 0 points decrease in error
Final simplification0.5
\[\leadsto \mathsf{hypot}\left(z, x\right) \]

Alternatives

Alternative 1
Error	12.6
Cost	6660

\[\begin{array}{l} \mathbf{if}\;z \leq 3600:\\ \;\;\;\;\mathsf{hypot}\left(y, x\right)\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]

Alternative 2
Error	12.9
Cost	260

\[\begin{array}{l} \mathbf{if}\;z \leq 1850:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]

Alternative 3
Error	30.7
Cost	64

\[z \]

Reproduce

herbie shell --seed 2022337 
(FPCore (x y z)
  :name "bug366 (missed optimization)"
  :precision binary64

  :herbie-target
  (hypot x (hypot y z))

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