bug366 (missed optimization)

Average Accuracy: 40.8% → 99.3%

Time: 3.4s

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	40.8%
Target	100.0%
Herbie	99.3%

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

Derivation

1. Initial program 40.8%

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

2. Taylor expanded in y around 0 40.4%

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

3. Simplified 99.3%

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

   [Start] 40.4	\[ \sqrt{{z}^{2} + {x}^{2}} \]
   unpow2 [=>] 40.4	\[ \sqrt{\color{blue}{z \cdot z} + {x}^{2}} \]
   unpow2 [=>] 40.4	\[ \sqrt{z \cdot z + \color{blue}{x \cdot x}} \]
   hypot-def [=>] 99.3	\[ \color{blue}{\mathsf{hypot}\left(z, x\right)} \]

4. Final simplification 99.3%

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

Alternatives

Alternative 1
Accuracy	79.8%
Cost	6924

\[\begin{array}{l} \mathbf{if}\;z \leq 1.7 \cdot 10^{-21}:\\ \;\;\;\;\mathsf{hypot}\left(y, x\right)\\ \mathbf{elif}\;z \leq 10000000:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 8.2 \cdot 10^{+34}:\\ \;\;\;\;\mathsf{hypot}\left(y, x\right)\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]

Alternative 2
Accuracy	79.4%
Cost	524

\[\begin{array}{l} \mathbf{if}\;z \leq 1.7 \cdot 10^{-21}:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq 11000000:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{+34}:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]

Alternative 3
Accuracy	52.4%
Cost	64

\[z \]

Reproduce

herbie shell --seed 2023137 
(FPCore (x y z)
  :name "bug366 (missed optimization)"
  :precision binary64

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

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