Data.Array.Repa.Algorithms.Pixel:doubleRmsOfRGB8 from repa-algorithms-3.4.0.1

Percentage Accurate: 44.0% → 99.4%

Time: 5.9s

Alternatives: 2

Speedup: 1.1×

Specification

Math

\[\begin{array}{l} \\ \sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (sqrt (/ (+ (+ (* x x) (* y y)) (* z z)) 3.0)))

C

double code(double x, double y, double z) {
	return sqrt(((((x * x) + (y * y)) + (z * z)) / 3.0));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = sqrt(((((x * x) + (y * y)) + (z * z)) / 3.0d0))
end function

Java

public static double code(double x, double y, double z) {
	return Math.sqrt(((((x * x) + (y * y)) + (z * z)) / 3.0));
}

Python

def code(x, y, z):
	return math.sqrt(((((x * x) + (y * y)) + (z * z)) / 3.0))

Julia

function code(x, y, z)
	return sqrt(Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) + Float64(z * z)) / 3.0))
end

MATLAB

function tmp = code(x, y, z)
	tmp = sqrt(((((x * x) + (y * y)) + (z * z)) / 3.0));
end

Wolfram

code[x_, y_, z_] := N[Sqrt[N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] + N[(z * z), $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision]], $MachinePrecision]

Initial Program: 44.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (sqrt (/ (+ (+ (* x x) (* y y)) (* z z)) 3.0)))

C

double code(double x, double y, double z) {
	return sqrt(((((x * x) + (y * y)) + (z * z)) / 3.0));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = sqrt(((((x * x) + (y * y)) + (z * z)) / 3.0d0))
end function

Java

public static double code(double x, double y, double z) {
	return Math.sqrt(((((x * x) + (y * y)) + (z * z)) / 3.0));
}

Python

def code(x, y, z):
	return math.sqrt(((((x * x) + (y * y)) + (z * z)) / 3.0))

Julia

function code(x, y, z)
	return sqrt(Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) + Float64(z * z)) / 3.0))
end

MATLAB

function tmp = code(x, y, z)
	tmp = sqrt(((((x * x) + (y * y)) + (z * z)) / 3.0));
end

Wolfram

code[x_, y_, z_] := N[Sqrt[N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] + N[(z * z), $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision]], $MachinePrecision]

Alternative 1: 99.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} x = |x|\\ y = |y|\\ z = |z|\\ [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \sqrt{0.3333333333333333} \cdot \mathsf{hypot}\left(y, z\right) \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
NOTE: z should be positive before calling this function
NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z) :precision binary64 (* (sqrt 0.3333333333333333) (hypot y z)))

C

x = abs(x);
y = abs(y);
z = abs(z);
assert(x < y && y < z);
double code(double x, double y, double z) {
	return sqrt(0.3333333333333333) * hypot(y, z);
}

Java

x = Math.abs(x);
y = Math.abs(y);
z = Math.abs(z);
assert x < y && y < z;
public static double code(double x, double y, double z) {
	return Math.sqrt(0.3333333333333333) * Math.hypot(y, z);
}

Python

x = abs(x)
y = abs(y)
z = abs(z)
[x, y, z] = sort([x, y, z])
def code(x, y, z):
	return math.sqrt(0.3333333333333333) * math.hypot(y, z)

Julia

x = abs(x)
y = abs(y)
z = abs(z)
x, y, z = sort([x, y, z])
function code(x, y, z)
	return Float64(sqrt(0.3333333333333333) * hypot(y, z))
end

MATLAB

x = abs(x)
y = abs(y)
z = abs(z)
x, y, z = num2cell(sort([x, y, z])){:}
function tmp = code(x, y, z)
	tmp = sqrt(0.3333333333333333) * hypot(y, z);
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
NOTE: z should be positive before calling this function
NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := N[(N[Sqrt[0.3333333333333333], $MachinePrecision] * N[Sqrt[y ^ 2 + z ^ 2], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 37.1%

   \[\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}} \]

3. Simplified 37.1%

   \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(x, x, \mathsf{fma}\left(y, y, z \cdot z\right)\right) \cdot 0.3333333333333333}} \]

4. Taylor expanded in x around 0 24.9%

   \[\leadsto \color{blue}{\sqrt{0.3333333333333333} \cdot \sqrt{{y}^{2} + {z}^{2}}} \]
5. Step-by-step derivation

   1. unpow2 24.9%

      \[\leadsto \sqrt{0.3333333333333333} \cdot \sqrt{\color{blue}{y \cdot y} + {z}^{2}} \]

   2. unpow2 24.9%

      \[\leadsto \sqrt{0.3333333333333333} \cdot \sqrt{y \cdot y + \color{blue}{z \cdot z}} \]

   3. hypot-def 66.6%

      \[\leadsto \sqrt{0.3333333333333333} \cdot \color{blue}{\mathsf{hypot}\left(y, z\right)} \]

6. Simplified 66.6%

   \[\leadsto \color{blue}{\sqrt{0.3333333333333333} \cdot \mathsf{hypot}\left(y, z\right)} \]

7. Final simplification 66.6%

   \[\leadsto \sqrt{0.3333333333333333} \cdot \mathsf{hypot}\left(y, z\right) \]

Alternative 2: 98.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} x = |x|\\ y = |y|\\ z = |z|\\ [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \sqrt{0.3333333333333333} \cdot z \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
NOTE: z should be positive before calling this function
NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z) :precision binary64 (* (sqrt 0.3333333333333333) z))

C

x = abs(x);
y = abs(y);
z = abs(z);
assert(x < y && y < z);
double code(double x, double y, double z) {
	return sqrt(0.3333333333333333) * z;
}

Fortran

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
NOTE: z should be positive before calling this function
NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = sqrt(0.3333333333333333d0) * z
end function

Java

x = Math.abs(x);
y = Math.abs(y);
z = Math.abs(z);
assert x < y && y < z;
public static double code(double x, double y, double z) {
	return Math.sqrt(0.3333333333333333) * z;
}

Python

x = abs(x)
y = abs(y)
z = abs(z)
[x, y, z] = sort([x, y, z])
def code(x, y, z):
	return math.sqrt(0.3333333333333333) * z

Julia

x = abs(x)
y = abs(y)
z = abs(z)
x, y, z = sort([x, y, z])
function code(x, y, z)
	return Float64(sqrt(0.3333333333333333) * z)
end

MATLAB

x = abs(x)
y = abs(y)
z = abs(z)
x, y, z = num2cell(sort([x, y, z])){:}
function tmp = code(x, y, z)
	tmp = sqrt(0.3333333333333333) * z;
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
NOTE: z should be positive before calling this function
NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := N[(N[Sqrt[0.3333333333333333], $MachinePrecision] * z), $MachinePrecision]

Derivation

1. Initial program 37.1%

   \[\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}} \]

3. Simplified 37.1%

   \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(x, x, \mathsf{fma}\left(y, y, z \cdot z\right)\right) \cdot 0.3333333333333333}} \]

4. Taylor expanded in z around inf 19.9%

   \[\leadsto \color{blue}{z \cdot \sqrt{0.3333333333333333}} \]
5. Step-by-step derivation

   1. *-commutative 19.9%

      \[\leadsto \color{blue}{\sqrt{0.3333333333333333} \cdot z} \]

6. Simplified 19.9%

   \[\leadsto \color{blue}{\sqrt{0.3333333333333333} \cdot z} \]

7. Final simplification 19.9%

   \[\leadsto \sqrt{0.3333333333333333} \cdot z \]

Developer target: 97.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z < -6.396479394109776 \cdot 10^{+136}:\\ \;\;\;\;\frac{-z}{\sqrt{3}}\\ \mathbf{elif}\;z < 7.320293694404182 \cdot 10^{+117}:\\ \;\;\;\;\frac{\sqrt{\left(z \cdot z + x \cdot x\right) + y \cdot y}}{\sqrt{3}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.3333333333333333} \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (< z -6.396479394109776e+136)
   (/ (- z) (sqrt 3.0))
   (if (< z 7.320293694404182e+117)
     (/ (sqrt (+ (+ (* z z) (* x x)) (* y y))) (sqrt 3.0))
     (* (sqrt 0.3333333333333333) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z < -6.396479394109776e+136) {
		tmp = -z / sqrt(3.0);
	} else if (z < 7.320293694404182e+117) {
		tmp = sqrt((((z * z) + (x * x)) + (y * y))) / sqrt(3.0);
	} else {
		tmp = sqrt(0.3333333333333333) * z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z < (-6.396479394109776d+136)) then
        tmp = -z / sqrt(3.0d0)
    else if (z < 7.320293694404182d+117) then
        tmp = sqrt((((z * z) + (x * x)) + (y * y))) / sqrt(3.0d0)
    else
        tmp = sqrt(0.3333333333333333d0) * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z < -6.396479394109776e+136) {
		tmp = -z / Math.sqrt(3.0);
	} else if (z < 7.320293694404182e+117) {
		tmp = Math.sqrt((((z * z) + (x * x)) + (y * y))) / Math.sqrt(3.0);
	} else {
		tmp = Math.sqrt(0.3333333333333333) * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z < -6.396479394109776e+136:
		tmp = -z / math.sqrt(3.0)
	elif z < 7.320293694404182e+117:
		tmp = math.sqrt((((z * z) + (x * x)) + (y * y))) / math.sqrt(3.0)
	else:
		tmp = math.sqrt(0.3333333333333333) * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z < -6.396479394109776e+136)
		tmp = Float64(Float64(-z) / sqrt(3.0));
	elseif (z < 7.320293694404182e+117)
		tmp = Float64(sqrt(Float64(Float64(Float64(z * z) + Float64(x * x)) + Float64(y * y))) / sqrt(3.0));
	else
		tmp = Float64(sqrt(0.3333333333333333) * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z < -6.396479394109776e+136)
		tmp = -z / sqrt(3.0);
	elseif (z < 7.320293694404182e+117)
		tmp = sqrt((((z * z) + (x * x)) + (y * y))) / sqrt(3.0);
	else
		tmp = sqrt(0.3333333333333333) * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Less[z, -6.396479394109776e+136], N[((-z) / N[Sqrt[3.0], $MachinePrecision]), $MachinePrecision], If[Less[z, 7.320293694404182e+117], N[(N[Sqrt[N[(N[(N[(z * z), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[3.0], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[0.3333333333333333], $MachinePrecision] * z), $MachinePrecision]]]

Reproduce

herbie shell --seed 2023291 
(FPCore (x y z)
  :name "Data.Array.Repa.Algorithms.Pixel:doubleRmsOfRGB8 from repa-algorithms-3.4.0.1"
  :precision binary64

  :herbie-target
  (if (< z -6.396479394109776e+136) (/ (- z) (sqrt 3.0)) (if (< z 7.320293694404182e+117) (/ (sqrt (+ (+ (* z z) (* x x)) (* y y))) (sqrt 3.0)) (* (sqrt 0.3333333333333333) z)))

  (sqrt (/ (+ (+ (* x x) (* y y)) (* z z)) 3.0)))