bug366 (missed optimization)

Average Accuracy: 41.7% → 100.0%

Time: 3.8s

Precision: binary64

Cost: 12992

Math

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

↓

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

FPCore

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

↓

(FPCore (x y z) :precision binary64 (hypot (hypot z y) 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(hypot(z, y), 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(Math.hypot(z, y), x);
}

Python

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

↓

def code(x, y, z):
	return math.hypot(math.hypot(z, y), 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(hypot(z, y), 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(hypot(z, y), 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[N[Sqrt[z ^ 2 + y ^ 2], $MachinePrecision] ^ 2 + x ^ 2], $MachinePrecision]

Target

Original	41.7%
Target	100.0%
Herbie	100.0%

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

Derivation

1. Initial program 41.7%

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

2. Applied egg-rr 100.0%

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

   [Start] 41.7	\[ \sqrt{x \cdot x + \left(y \cdot y + z \cdot z\right)} \]
   +-commutative [=>] 41.7	\[ \sqrt{\color{blue}{\left(y \cdot y + z \cdot z\right) + x \cdot x}} \]
   add-sqr-sqrt [=>] 41.7	\[ \sqrt{\color{blue}{\sqrt{y \cdot y + z \cdot z} \cdot \sqrt{y \cdot y + z \cdot z}} + x \cdot x} \]
   hypot-def [=>] 56.1	\[ \color{blue}{\mathsf{hypot}\left(\sqrt{y \cdot y + z \cdot z}, x\right)} \]
   hypot-def [=>] 100.0	\[ \mathsf{hypot}\left(\color{blue}{\mathsf{hypot}\left(y, z\right)}, x\right) \]

3. Simplified 100.0%

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

   [Start] 100.0	\[ \mathsf{hypot}\left(\mathsf{hypot}\left(y, z\right), x\right) \]
   hypot-def [<=] 56.1	\[ \mathsf{hypot}\left(\color{blue}{\sqrt{y \cdot y + z \cdot z}}, x\right) \]
   +-commutative [<=] 56.1	\[ \mathsf{hypot}\left(\sqrt{\color{blue}{z \cdot z + y \cdot y}}, x\right) \]
   hypot-def [=>] 100.0	\[ \mathsf{hypot}\left(\color{blue}{\mathsf{hypot}\left(z, y\right)}, x\right) \]

4. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	74.5%
Cost	6660

\[\begin{array}{l} \mathbf{if}\;z \leq 1.02 \cdot 10^{+70}:\\ \;\;\;\;\mathsf{hypot}\left(y, x\right)\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]

Alternative 2
Accuracy	68.6%
Cost	6528

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

Alternative 3
Accuracy	30.4%
Cost	260

\[\begin{array}{l} \mathbf{if}\;z \leq 3.2 \cdot 10^{+51}:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]

Alternative 4
Accuracy	19.1%
Cost	64

\[z \]

Reproduce

herbie shell --seed 2023151 
(FPCore (x y z)
  :name "bug366 (missed optimization)"
  :precision binary64

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

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