FRP.Yampa.Vector3:vector3Rho from Yampa-0.10.2

Percentage Accurate: 44.6% → 99.1%

Time: 2.6s

Precision: binary64

Cost: 6528

• Unsound rule application detected in e-graph. Results may not be sound.

Math

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

↓

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \mathsf{hypot}\left(z, x\right) \end{array} \]

FPCore

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

↓

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(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)));
}

↓

assert(x < y && y < 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)));
}

↓

assert x < y && y < 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)))

↓

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	return math.hypot(z, x)

Julia

function code(x, y, z)
	return sqrt(Float64(Float64(Float64(x * x) + Float64(y * y)) + Float64(z * z)))
end

↓

x, y, z = sort([x, y, z])
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

↓

x, y, z = num2cell(sort([x, y, z])){:}
function tmp = code(x, y, z)
	tmp = hypot(z, x);
end

Wolfram

code[x_, y_, z_] := N[Sqrt[N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] + N[(z * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

↓

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := N[Sqrt[z ^ 2 + x ^ 2], $MachinePrecision]

Target

Original	44.6%
Target	71.2%
Herbie	99.1%

\[\begin{array}{l} \mathbf{if}\;z < -6.396479394109776 \cdot 10^{+136}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z < 7.320293694404182 \cdot 10^{+117}:\\ \;\;\;\;\sqrt{\left(z \cdot z + x \cdot x\right) + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]

Derivation

1. Initial program 48.0%

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

2. Taylor expanded in y around 0 32.0%

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

3. Simplified 60.4%

   \[\leadsto \color{blue}{\mathsf{hypot}\left(z, x\right)} \]
   Step-by-step derivation

   [Start] 32.0%	\[ \sqrt{{z}^{2} + {x}^{2}} \]
   unpow2 [=>] 32.0%	\[ \sqrt{\color{blue}{z \cdot z} + {x}^{2}} \]
   unpow2 [=>] 32.0%	\[ \sqrt{z \cdot z + \color{blue}{x \cdot x}} \]
   hypot-def [=>] 60.4%	\[ \color{blue}{\mathsf{hypot}\left(z, x\right)} \]

4. Final simplification 60.4%

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

Alternatives

Alternative 1
Accuracy	99.1%
Cost	6528

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

Alternative 2
Accuracy	80.0%
Cost	6924

\[\begin{array}{l} \mathbf{if}\;z \leq 350:\\ \;\;\;\;\mathsf{hypot}\left(y, x\right)\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{+56}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 8.8 \cdot 10^{+80}:\\ \;\;\;\;\mathsf{hypot}\left(y, x\right)\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]

Alternative 3
Accuracy	79.4%
Cost	524

\[\begin{array}{l} \mathbf{if}\;z \leq 122:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq 1.5 \cdot 10^{+62}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 4 \cdot 10^{+80}:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]

Alternative 4
Accuracy	51.6%
Cost	64

\[z \]

Reproduce

herbie shell --seed 2023167 
(FPCore (x y z)
  :name "FRP.Yampa.Vector3:vector3Rho from Yampa-0.10.2"
  :precision binary64

  :herbie-target
  (if (< z -6.396479394109776e+136) (- z) (if (< z 7.320293694404182e+117) (sqrt (+ (+ (* z z) (* x x)) (* y y))) z))

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