bug366 (missed optimization)

Average Error: 38.6 → 0.4

Time: 3.5s

Precision: binary64

Cost: 6528

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

Math

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

↓

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

FPCore

(FPCore (x y z) :precision binary64 (sqrt (+ (* x x) (+ (* y y) (* z z)))))

↓

(FPCore (x y z) :precision binary64 (hypot z x))

C

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);
}

Java

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);
}

Python

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

↓

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

Julia

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

MATLAB

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

Wolfram

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]

Target

Original	38.6
Target	0
Herbie	0.4

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

Derivation

1. Initial program 38.6

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

2. Taylor expanded in y around 0 38.8

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

3. Simplified 0.4

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

   [Start] 38.8	\[ \sqrt{{z}^{2} + {x}^{2}} \]
   unpow2 [=>] 38.8	\[ \sqrt{\color{blue}{z \cdot z} + {x}^{2}} \]
   unpow2 [=>] 38.8	\[ \sqrt{z \cdot z + \color{blue}{x \cdot x}} \]
   hypot-def [=>] 0.4	\[ \color{blue}{\mathsf{hypot}\left(z, x\right)} \]

4. Final simplification 0.4

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

Alternatives

Alternative 1
Error	12.7
Cost	6660

\[\begin{array}{l} \mathbf{if}\;z \leq 4.1 \cdot 10^{+40}:\\ \;\;\;\;\mathsf{hypot}\left(y, x\right)\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]

Alternative 2
Error	13.0
Cost	260

\[\begin{array}{l} \mathbf{if}\;z \leq 3.6 \cdot 10^{+40}:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]

Alternative 3
Error	30.8
Cost	64

\[z \]

Reproduce

herbie shell --seed 2023054 
(FPCore (x y z)
  :name "bug366 (missed optimization)"
  :precision binary64

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

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