bug366 (missed optimization)

Average Error: 38.8 → 10.9

Time: 3.1s

Precision: binary64

Cost: 7368

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \leq 6.5 \cdot 10^{-42}:\\ \;\;\;\;\mathsf{hypot}\left(y, x\right)\\ \mathbf{elif}\;z \leq 1.56 \cdot 10^{+97}:\\ \;\;\;\;\sqrt{x \cdot x + \left(y \cdot y + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(z, y\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= z 6.5e-42)
   (hypot y x)
   (if (<= z 1.56e+97) (sqrt (+ (* x x) (+ (* y y) (* z z)))) (hypot z y))))

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) {
	double tmp;
	if (z <= 6.5e-42) {
		tmp = hypot(y, x);
	} else if (z <= 1.56e+97) {
		tmp = sqrt(((x * x) + ((y * y) + (z * z))));
	} else {
		tmp = hypot(z, y);
	}
	return tmp;
}

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) {
	double tmp;
	if (z <= 6.5e-42) {
		tmp = Math.hypot(y, x);
	} else if (z <= 1.56e+97) {
		tmp = Math.sqrt(((x * x) + ((y * y) + (z * z))));
	} else {
		tmp = Math.hypot(z, y);
	}
	return tmp;
}

Python

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

↓

def code(x, y, z):
	tmp = 0
	if z <= 6.5e-42:
		tmp = math.hypot(y, x)
	elif z <= 1.56e+97:
		tmp = math.sqrt(((x * x) + ((y * y) + (z * z))))
	else:
		tmp = math.hypot(z, y)
	return tmp

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)
	tmp = 0.0
	if (z <= 6.5e-42)
		tmp = hypot(y, x);
	elseif (z <= 1.56e+97)
		tmp = sqrt(Float64(Float64(x * x) + Float64(Float64(y * y) + Float64(z * z))));
	else
		tmp = hypot(z, y);
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 6.5e-42)
		tmp = hypot(y, x);
	elseif (z <= 1.56e+97)
		tmp = sqrt(((x * x) + ((y * y) + (z * z))));
	else
		tmp = hypot(z, y);
	end
	tmp_2 = tmp;
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_] := If[LessEqual[z, 6.5e-42], N[Sqrt[y ^ 2 + x ^ 2], $MachinePrecision], If[LessEqual[z, 1.56e+97], N[Sqrt[N[(N[(x * x), $MachinePrecision] + N[(N[(y * y), $MachinePrecision] + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[z ^ 2 + y ^ 2], $MachinePrecision]]]

Target

Original	38.8
Target	0.0
Herbie	10.9

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

Derivation

1. Split input into 3 regimes

2. if z < 6.4999999999999998e-42

   1. Initial program 31.6

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

   2. Taylor expanded in z around 0 37.1

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

   3. Simplified 6.7

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

      [Start] 37.1	\[ \sqrt{{y}^{2} + {x}^{2}} \]
      unpow2 [=>] 37.1	\[ \sqrt{\color{blue}{y \cdot y} + {x}^{2}} \]
      unpow2 [=>] 37.1	\[ \sqrt{y \cdot y + \color{blue}{x \cdot x}} \]
      hypot-def [=>] 6.7	\[ \color{blue}{\mathsf{hypot}\left(y, x\right)} \]

   if 6.4999999999999998e-42 < z < 1.56e97

   1. Initial program 19.7

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

   if 1.56e97 < z

   1. Initial program 53.0

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

   2. Taylor expanded in x around 0 53.4

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

   3. Simplified 10.6

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

      [Start] 53.4	\[ \sqrt{{z}^{2} + {y}^{2}} \]
      unpow2 [=>] 53.4	\[ \sqrt{\color{blue}{z \cdot z} + {y}^{2}} \]
      unpow2 [=>] 53.4	\[ \sqrt{z \cdot z + \color{blue}{y \cdot y}} \]
      hypot-def [=>] 10.6	\[ \color{blue}{\mathsf{hypot}\left(z, y\right)} \]

3. Recombined 3 regimes into one program.

4. Final simplification 10.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 6.5 \cdot 10^{-42}:\\ \;\;\;\;\mathsf{hypot}\left(y, x\right)\\ \mathbf{elif}\;z \leq 1.56 \cdot 10^{+97}:\\ \;\;\;\;\sqrt{x \cdot x + \left(y \cdot y + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(z, y\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	12.8
Cost	6660

\[\begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{hypot}\left(y, x\right)\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]

Alternative 2
Error	12.5
Cost	6660

\[\begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{hypot}\left(y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(z, y\right)\\ \end{array} \]

Alternative 3
Error	13.0
Cost	260

\[\begin{array}{l} \mathbf{if}\;x \leq -7.6 \cdot 10^{-9}:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]

Alternative 4
Error	31.2
Cost	64

\[z \]

Reproduce

herbie shell --seed 2023002 
(FPCore (x y z)
  :name "bug366 (missed optimization)"
  :precision binary64

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

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