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

Percentage Accurate: 44.6% → 99.4%

Time: 4.7s

Alternatives: 5

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: 44.6% 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

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

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 z y)))

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(z, y);
}

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(z, y);
}

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(z, y)

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(z, y))
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(z, y);
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[z ^ 2 + y ^ 2], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 47.7%

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

2. Taylor expanded in x around 0 31.2%

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

   1. *-commutative 31.2%

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

   2. unpow2 31.2%

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

   3. unpow2 31.2%

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

   4. hypot-def 64.2%

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

4. Simplified 64.2%

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

5. Final simplification 64.2%

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

Alternative 2: 97.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} x = |x|\\ y = |y|\\ z = |z|\\ [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+303}:\\ \;\;\;\;\sqrt{\frac{z \cdot z + \left(x \cdot x + y \cdot y\right)}{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{3}}{z}}\\ \end{array} \end{array} \]

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
 (if (<= (* z z) 2e+303)
   (sqrt (/ (+ (* z z) (+ (* x x) (* y y))) 3.0))
   (/ 1.0 (/ (sqrt 3.0) z))))

C

x = abs(x);
y = abs(y);
z = abs(z);
assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e+303) {
		tmp = sqrt((((z * z) + ((x * x) + (y * y))) / 3.0));
	} else {
		tmp = 1.0 / (sqrt(3.0) / z);
	}
	return tmp;
}

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
    real(8) :: tmp
    if ((z * z) <= 2d+303) then
        tmp = sqrt((((z * z) + ((x * x) + (y * y))) / 3.0d0))
    else
        tmp = 1.0d0 / (sqrt(3.0d0) / z)
    end if
    code = tmp
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) {
	double tmp;
	if ((z * z) <= 2e+303) {
		tmp = Math.sqrt((((z * z) + ((x * x) + (y * y))) / 3.0));
	} else {
		tmp = 1.0 / (Math.sqrt(3.0) / z);
	}
	return tmp;
}

Python

x = abs(x)
y = abs(y)
z = abs(z)
[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if (z * z) <= 2e+303:
		tmp = math.sqrt((((z * z) + ((x * x) + (y * y))) / 3.0))
	else:
		tmp = 1.0 / (math.sqrt(3.0) / z)
	return tmp

Julia

x = abs(x)
y = abs(y)
z = abs(z)
x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 2e+303)
		tmp = sqrt(Float64(Float64(Float64(z * z) + Float64(Float64(x * x) + Float64(y * y))) / 3.0));
	else
		tmp = Float64(1.0 / Float64(sqrt(3.0) / z));
	end
	return tmp
end

MATLAB

x = abs(x)
y = abs(y)
z = abs(z)
x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 2e+303)
		tmp = sqrt((((z * z) + ((x * x) + (y * y))) / 3.0));
	else
		tmp = 1.0 / (sqrt(3.0) / z);
	end
	tmp_2 = tmp;
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_] := If[LessEqual[N[(z * z), $MachinePrecision], 2e+303], N[Sqrt[N[(N[(N[(z * z), $MachinePrecision] + N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision]], $MachinePrecision], N[(1.0 / N[(N[Sqrt[3.0], $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 2e303

   1. Initial program 60.5%

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

   if 2e303 < (*.f64 z z)

   1. Initial program 7.7%

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

   2. Taylor expanded in z around inf 35.5%

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

      1. add-sqr-sqrt 35.3%

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

      2. sqrt-prod 7.6%

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

      3. sqrt-prod 7.6%

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

      4. metadata-eval 7.6%

         \[\leadsto \sqrt{\left(z \cdot z\right) \cdot \color{blue}{\frac{1}{3}}} \]

      5. div-inv 7.6%

         \[\leadsto \sqrt{\color{blue}{\frac{z \cdot z}{3}}} \]

      6. sqrt-div 7.6%

         \[\leadsto \color{blue}{\frac{\sqrt{z \cdot z}}{\sqrt{3}}} \]

      7. sqrt-prod 35.3%

         \[\leadsto \frac{\color{blue}{\sqrt{z} \cdot \sqrt{z}}}{\sqrt{3}} \]

      8. add-sqr-sqrt 35.5%

         \[\leadsto \frac{\color{blue}{z}}{\sqrt{3}} \]

      9. clear-num 35.5%

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{3}}{z}}} \]

   4. Applied egg-rr 35.5%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{3}}{z}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+303}:\\ \;\;\;\;\sqrt{\frac{z \cdot z + \left(x \cdot x + y \cdot y\right)}{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{3}}{z}}\\ \end{array} \]

Alternative 3: 97.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} x = |x|\\ y = |y|\\ z = |z|\\ [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \frac{1}{\frac{\sqrt{3}}{z}} \end{array} \]

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 (/ 1.0 (/ (sqrt 3.0) 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 1.0 / (sqrt(3.0) / 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 = 1.0d0 / (sqrt(3.0d0) / 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 1.0 / (Math.sqrt(3.0) / z);
}

Python

x = abs(x)
y = abs(y)
z = abs(z)
[x, y, z] = sort([x, y, z])
def code(x, y, z):
	return 1.0 / (math.sqrt(3.0) / 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(1.0 / Float64(sqrt(3.0) / 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 = 1.0 / (sqrt(3.0) / 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[(1.0 / N[(N[Sqrt[3.0], $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 47.7%

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

2. Taylor expanded in z around inf 16.1%

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

   1. add-sqr-sqrt 15.5%

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

   2. sqrt-prod 15.6%

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

   3. sqrt-prod 15.6%

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

   4. metadata-eval 15.6%

      \[\leadsto \sqrt{\left(z \cdot z\right) \cdot \color{blue}{\frac{1}{3}}} \]

   5. div-inv 15.6%

      \[\leadsto \sqrt{\color{blue}{\frac{z \cdot z}{3}}} \]

   6. sqrt-div 15.6%

      \[\leadsto \color{blue}{\frac{\sqrt{z \cdot z}}{\sqrt{3}}} \]

   7. sqrt-prod 15.5%

      \[\leadsto \frac{\color{blue}{\sqrt{z} \cdot \sqrt{z}}}{\sqrt{3}} \]

   8. add-sqr-sqrt 16.1%

      \[\leadsto \frac{\color{blue}{z}}{\sqrt{3}} \]

   9. clear-num 16.1%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{3}}{z}}} \]

4. Applied egg-rr 16.1%

   \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{3}}{z}}} \]

5. Final simplification 16.1%

   \[\leadsto \frac{1}{\frac{\sqrt{3}}{z}} \]

Alternative 4: 97.9% 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 47.7%

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

2. Taylor expanded in z around inf 16.1%

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

3. Final simplification 16.1%

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

Alternative 5: 97.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} x = |x|\\ y = |y|\\ z = |z|\\ [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \frac{z}{\sqrt{3}} \end{array} \]

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 (/ z (sqrt 3.0)))

C

x = abs(x);
y = abs(y);
z = abs(z);
assert(x < y && y < z);
double code(double x, double y, double z) {
	return z / sqrt(3.0);
}

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 = z / sqrt(3.0d0)
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 z / Math.sqrt(3.0);
}

Python

x = abs(x)
y = abs(y)
z = abs(z)
[x, y, z] = sort([x, y, z])
def code(x, y, z):
	return z / math.sqrt(3.0)

Julia

x = abs(x)
y = abs(y)
z = abs(z)
x, y, z = sort([x, y, z])
function code(x, y, z)
	return Float64(z / sqrt(3.0))
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 = z / sqrt(3.0);
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[(z / N[Sqrt[3.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 47.7%

   \[\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}} \]
2. Step-by-step derivation

   1. sqrt-div 47.5%

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

   2. div-inv 47.3%

      \[\leadsto \color{blue}{\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z} \cdot \frac{1}{\sqrt{3}}} \]

   3. associate-+l+ 47.3%

      \[\leadsto \sqrt{\color{blue}{x \cdot x + \left(y \cdot y + z \cdot z\right)}} \cdot \frac{1}{\sqrt{3}} \]

   4. add-sqr-sqrt 47.3%

      \[\leadsto \sqrt{x \cdot x + \color{blue}{\sqrt{y \cdot y + z \cdot z} \cdot \sqrt{y \cdot y + z \cdot z}}} \cdot \frac{1}{\sqrt{3}} \]

   5. hypot-def 60.4%

      \[\leadsto \color{blue}{\mathsf{hypot}\left(x, \sqrt{y \cdot y + z \cdot z}\right)} \cdot \frac{1}{\sqrt{3}} \]

   6. hypot-def 98.7%

      \[\leadsto \mathsf{hypot}\left(x, \color{blue}{\mathsf{hypot}\left(y, z\right)}\right) \cdot \frac{1}{\sqrt{3}} \]

3. Applied egg-rr 98.7%

   \[\leadsto \color{blue}{\mathsf{hypot}\left(x, \mathsf{hypot}\left(y, z\right)\right) \cdot \frac{1}{\sqrt{3}}} \]
4. Step-by-step derivation

   1. associate-*r/ 99.4%

      \[\leadsto \color{blue}{\frac{\mathsf{hypot}\left(x, \mathsf{hypot}\left(y, z\right)\right) \cdot 1}{\sqrt{3}}} \]

   2. *-rgt-identity 99.4%

      \[\leadsto \frac{\color{blue}{\mathsf{hypot}\left(x, \mathsf{hypot}\left(y, z\right)\right)}}{\sqrt{3}} \]

   3. hypot-def 60.8%

      \[\leadsto \frac{\mathsf{hypot}\left(x, \color{blue}{\sqrt{y \cdot y + z \cdot z}}\right)}{\sqrt{3}} \]

   4. +-commutative 60.8%

      \[\leadsto \frac{\mathsf{hypot}\left(x, \sqrt{\color{blue}{z \cdot z + y \cdot y}}\right)}{\sqrt{3}} \]

   5. hypot-def 99.4%

      \[\leadsto \frac{\mathsf{hypot}\left(x, \color{blue}{\mathsf{hypot}\left(z, y\right)}\right)}{\sqrt{3}} \]

5. Simplified 99.4%

   \[\leadsto \color{blue}{\frac{\mathsf{hypot}\left(x, \mathsf{hypot}\left(z, y\right)\right)}{\sqrt{3}}} \]

6. Taylor expanded in z around inf 16.1%

   \[\leadsto \color{blue}{\frac{z}{\sqrt{3}}} \]

7. Final simplification 16.1%

   \[\leadsto \frac{z}{\sqrt{3}} \]

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 2023240 
(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)))