Data.Octree.Internal:octantDistance from Octree-0.5.4.2

Average Error: 32.1 → 7.8

Time: 2.2s

Precision: binary64

Cost: 7112

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

Math

\[\sqrt{x \cdot x + y \cdot y} \]

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -2.25 \cdot 10^{+110}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq -1.5 \cdot 10^{-146}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (if (<= x -2.25e+110)
   (- x)
   (if (<= x -1.5e-146) (sqrt (+ (* x x) (* y y))) y)))

C

double code(double x, double y) {
	return sqrt(((x * x) + (y * y)));
}

↓

double code(double x, double y) {
	double tmp;
	if (x <= -2.25e+110) {
		tmp = -x;
	} else if (x <= -1.5e-146) {
		tmp = sqrt(((x * x) + (y * y)));
	} else {
		tmp = y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = sqrt(((x * x) + (y * y)))
end function

↓

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-2.25d+110)) then
        tmp = -x
    else if (x <= (-1.5d-146)) then
        tmp = sqrt(((x * x) + (y * y)))
    else
        tmp = y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	return Math.sqrt(((x * x) + (y * y)));
}

↓

public static double code(double x, double y) {
	double tmp;
	if (x <= -2.25e+110) {
		tmp = -x;
	} else if (x <= -1.5e-146) {
		tmp = Math.sqrt(((x * x) + (y * y)));
	} else {
		tmp = y;
	}
	return tmp;
}

Python

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

↓

def code(x, y):
	tmp = 0
	if x <= -2.25e+110:
		tmp = -x
	elif x <= -1.5e-146:
		tmp = math.sqrt(((x * x) + (y * y)))
	else:
		tmp = y
	return tmp

Julia

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

↓

function code(x, y)
	tmp = 0.0
	if (x <= -2.25e+110)
		tmp = Float64(-x);
	elseif (x <= -1.5e-146)
		tmp = sqrt(Float64(Float64(x * x) + Float64(y * y)));
	else
		tmp = y;
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.25e+110)
		tmp = -x;
	elseif (x <= -1.5e-146)
		tmp = sqrt(((x * x) + (y * y)));
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[x_, y_] := If[LessEqual[x, -2.25e+110], (-x), If[LessEqual[x, -1.5e-146], N[Sqrt[N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], y]]

Target

Original	32.1
Target	18.2
Herbie	7.8

\[\begin{array}{l} \mathbf{if}\;x < -1.1236950826599826 \cdot 10^{+145}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x < 1.116557621183362 \cdot 10^{+93}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Derivation

1. Split input into 3 regimes

2. if x < -2.2500000000000001e110

   1. Initial program 53.5

      \[\sqrt{x \cdot x + y \cdot y} \]

   2. Taylor expanded in x around -inf 6.4

      \[\leadsto \color{blue}{-1 \cdot x} \]

   3. Simplified 6.4

      \[\leadsto \color{blue}{-x} \]
      Proof

      [Start] 6.4	\[ -1 \cdot x \]
      rational.json-simplify-2 [=>] 6.4	\[ \color{blue}{x \cdot -1} \]
      rational.json-simplify-9 [=>] 6.4	\[ \color{blue}{-x} \]

   if -2.2500000000000001e110 < x < -1.50000000000000009e-146

   1. Initial program 11.3

      \[\sqrt{x \cdot x + y \cdot y} \]

   if -1.50000000000000009e-146 < x

   1. Initial program 32.7

      \[\sqrt{x \cdot x + y \cdot y} \]

   2. Taylor expanded in x around 0 6.1

      \[\leadsto \color{blue}{y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 7.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.25 \cdot 10^{+110}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq -1.5 \cdot 10^{-146}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternatives

Alternative 1
Error	11.4
Cost	524

\[\begin{array}{l} \mathbf{if}\;y \leq 6.5 \cdot 10^{-97}:\\ \;\;\;\;-x\\ \mathbf{elif}\;y \leq 5.6 \cdot 10^{-16}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{+19}:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 2
Error	30.8
Cost	64

\[y \]

Reproduce

herbie shell --seed 2023063 
(FPCore (x y)
  :name "Data.Octree.Internal:octantDistance  from Octree-0.5.4.2"
  :precision binary64

  :herbie-target
  (if (< x -1.1236950826599826e+145) (- x) (if (< x 1.116557621183362e+93) (sqrt (+ (* x x) (* y y))) x))

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