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

Percentage Accurate: 44.8% → 99.4%

Time: 6.2s

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: 44.8% 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

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ z_m = \left|z\right| \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ \frac{\mathsf{hypot}\left(z_m, y_m\right)}{\sqrt{3}} \end{array} \]

FPCore

x_m = (fabs.f64 x)
y_m = (fabs.f64 y)
z_m = (fabs.f64 z)
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
(FPCore (x_m y_m z_m) :precision binary64 (/ (hypot z_m y_m) (sqrt 3.0)))

C

x_m = fabs(x);
y_m = fabs(y);
z_m = fabs(z);
assert(x_m < y_m && y_m < z_m);
double code(double x_m, double y_m, double z_m) {
	return hypot(z_m, y_m) / sqrt(3.0);
}

Java

x_m = Math.abs(x);
y_m = Math.abs(y);
z_m = Math.abs(z);
assert x_m < y_m && y_m < z_m;
public static double code(double x_m, double y_m, double z_m) {
	return Math.hypot(z_m, y_m) / Math.sqrt(3.0);
}

Python

x_m = math.fabs(x)
y_m = math.fabs(y)
z_m = math.fabs(z)
[x_m, y_m, z_m] = sort([x_m, y_m, z_m])
def code(x_m, y_m, z_m):
	return math.hypot(z_m, y_m) / math.sqrt(3.0)

Julia

x_m = abs(x)
y_m = abs(y)
z_m = abs(z)
x_m, y_m, z_m = sort([x_m, y_m, z_m])
function code(x_m, y_m, z_m)
	return Float64(hypot(z_m, y_m) / sqrt(3.0))
end

MATLAB

x_m = abs(x);
y_m = abs(y);
z_m = abs(z);
x_m, y_m, z_m = num2cell(sort([x_m, y_m, z_m])){:}
function tmp = code(x_m, y_m, z_m)
	tmp = hypot(z_m, y_m) / sqrt(3.0);
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
code[x$95$m_, y$95$m_, z$95$m_] := N[(N[Sqrt[z$95$m ^ 2 + y$95$m ^ 2], $MachinePrecision] / N[Sqrt[3.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 43.1%

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

   1. sqr-neg 43.1%

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

   2. sqr-neg 43.1%

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

   3. sqr-neg 43.1%

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

   4. sqr-neg 43.1%

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

   5. sqr-neg 43.1%

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

3. Simplified 43.1%

   \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(x, x, \mathsf{fma}\left(y, y, z \cdot z\right)\right) \cdot 0.3333333333333333}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. fma-udef 43.1%

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

   2. fma-udef 43.1%

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

   3. associate-+l+ 43.1%

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

   4. metadata-eval 43.1%

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

   5. div-inv 43.1%

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

   6. sqrt-div 43.0%

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

   7. div-inv 42.7%

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

   8. +-commutative 42.7%

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

   9. add-sqr-sqrt 42.7%

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

   10. hypot-def 57.8%

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

   11. hypot-def 98.6%

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

6. Applied egg-rr 98.6%

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

   1. associate-*r/ 99.4%

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

   2. *-rgt-identity 99.4%

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

   3. hypot-def 58.2%

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

   4. unpow2 58.2%

      \[\leadsto \frac{\mathsf{hypot}\left(z, \sqrt{\color{blue}{{x}^{2}} + y \cdot y}\right)}{\sqrt{3}} \]

   5. unpow2 58.2%

      \[\leadsto \frac{\mathsf{hypot}\left(z, \sqrt{{x}^{2} + \color{blue}{{y}^{2}}}\right)}{\sqrt{3}} \]

   6. +-commutative 58.2%

      \[\leadsto \frac{\mathsf{hypot}\left(z, \sqrt{\color{blue}{{y}^{2} + {x}^{2}}}\right)}{\sqrt{3}} \]

   7. unpow2 58.2%

      \[\leadsto \frac{\mathsf{hypot}\left(z, \sqrt{\color{blue}{y \cdot y} + {x}^{2}}\right)}{\sqrt{3}} \]

   8. unpow2 58.2%

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

   9. hypot-def 99.4%

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

8. Simplified 99.4%

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

9. Taylor expanded in x around 0 27.2%

   \[\leadsto \color{blue}{\frac{1}{\sqrt{3}} \cdot \sqrt{{y}^{2} + {z}^{2}}} \]
10. Step-by-step derivation

    1. associate-*l/ 27.4%

       \[\leadsto \color{blue}{\frac{1 \cdot \sqrt{{y}^{2} + {z}^{2}}}{\sqrt{3}}} \]

    2. +-commutative 27.4%

       \[\leadsto \frac{1 \cdot \sqrt{\color{blue}{{z}^{2} + {y}^{2}}}}{\sqrt{3}} \]

    3. unpow2 27.4%

       \[\leadsto \frac{1 \cdot \sqrt{\color{blue}{z \cdot z} + {y}^{2}}}{\sqrt{3}} \]

    4. unpow2 27.4%

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

    5. hypot-def 63.8%

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

    6. *-lft-identity 63.8%

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

11. Simplified 63.8%

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

12. Final simplification 63.8%

    \[\leadsto \frac{\mathsf{hypot}\left(z, y\right)}{\sqrt{3}} \]
13. Add Preprocessing

Alternative 2: 99.4% accurate, 0.6× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
y_m = (fabs.f64 y)
z_m = (fabs.f64 z)
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
(FPCore (x_m y_m z_m)
 :precision binary64
 (* (sqrt 0.3333333333333333) (hypot y_m z_m)))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
code[x$95$m_, y$95$m_, z$95$m_] := N[(N[Sqrt[0.3333333333333333], $MachinePrecision] * N[Sqrt[y$95$m ^ 2 + z$95$m ^ 2], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 43.1%

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

   1. sqr-neg 43.1%

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

   2. sqr-neg 43.1%

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

   3. sqr-neg 43.1%

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

   4. sqr-neg 43.1%

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

   5. sqr-neg 43.1%

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

3. Simplified 43.1%

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

5. Taylor expanded in x around 0 27.4%

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

   1. unpow2 27.4%

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

   2. unpow2 27.4%

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

   3. hypot-def 63.8%

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

7. Simplified 63.8%

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

8. Final simplification 63.8%

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

Alternative 3: 97.8% accurate, 1.1× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
y_m = (fabs.f64 y)
z_m = (fabs.f64 z)
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
(FPCore (x_m y_m z_m) :precision binary64 (/ 1.0 (/ (sqrt 3.0) z_m)))

C

x_m = fabs(x);
y_m = fabs(y);
z_m = fabs(z);
assert(x_m < y_m && y_m < z_m);
double code(double x_m, double y_m, double z_m) {
	return 1.0 / (sqrt(3.0) / z_m);
}

Fortran

x_m = abs(x)
y_m = abs(y)
z_m = abs(z)
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, y_m, z_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    code = 1.0d0 / (sqrt(3.0d0) / z_m)
end function

Java

x_m = Math.abs(x);
y_m = Math.abs(y);
z_m = Math.abs(z);
assert x_m < y_m && y_m < z_m;
public static double code(double x_m, double y_m, double z_m) {
	return 1.0 / (Math.sqrt(3.0) / z_m);
}

Python

x_m = math.fabs(x)
y_m = math.fabs(y)
z_m = math.fabs(z)
[x_m, y_m, z_m] = sort([x_m, y_m, z_m])
def code(x_m, y_m, z_m):
	return 1.0 / (math.sqrt(3.0) / z_m)

Julia

x_m = abs(x)
y_m = abs(y)
z_m = abs(z)
x_m, y_m, z_m = sort([x_m, y_m, z_m])
function code(x_m, y_m, z_m)
	return Float64(1.0 / Float64(sqrt(3.0) / z_m))
end

MATLAB

x_m = abs(x);
y_m = abs(y);
z_m = abs(z);
x_m, y_m, z_m = num2cell(sort([x_m, y_m, z_m])){:}
function tmp = code(x_m, y_m, z_m)
	tmp = 1.0 / (sqrt(3.0) / z_m);
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
code[x$95$m_, y$95$m_, z$95$m_] := N[(1.0 / N[(N[Sqrt[3.0], $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 43.1%

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

   1. sqr-neg 43.1%

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

   2. sqr-neg 43.1%

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

   3. sqr-neg 43.1%

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

   4. sqr-neg 43.1%

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

   5. sqr-neg 43.1%

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

3. Simplified 43.1%

   \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(x, x, \mathsf{fma}\left(y, y, z \cdot z\right)\right) \cdot 0.3333333333333333}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. fma-udef 43.1%

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

   2. fma-udef 43.1%

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

   3. associate-+l+ 43.1%

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

   4. metadata-eval 43.1%

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

   5. div-inv 43.1%

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

   6. sqrt-div 43.0%

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

   7. div-inv 42.7%

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

   8. +-commutative 42.7%

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

   9. add-sqr-sqrt 42.7%

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

   10. hypot-def 57.8%

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

   11. hypot-def 98.6%

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

6. Applied egg-rr 98.6%

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

   1. associate-*r/ 99.4%

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

   2. *-rgt-identity 99.4%

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

   3. hypot-def 58.2%

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

   4. unpow2 58.2%

      \[\leadsto \frac{\mathsf{hypot}\left(z, \sqrt{\color{blue}{{x}^{2}} + y \cdot y}\right)}{\sqrt{3}} \]

   5. unpow2 58.2%

      \[\leadsto \frac{\mathsf{hypot}\left(z, \sqrt{{x}^{2} + \color{blue}{{y}^{2}}}\right)}{\sqrt{3}} \]

   6. +-commutative 58.2%

      \[\leadsto \frac{\mathsf{hypot}\left(z, \sqrt{\color{blue}{{y}^{2} + {x}^{2}}}\right)}{\sqrt{3}} \]

   7. unpow2 58.2%

      \[\leadsto \frac{\mathsf{hypot}\left(z, \sqrt{\color{blue}{y \cdot y} + {x}^{2}}\right)}{\sqrt{3}} \]

   8. unpow2 58.2%

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

   9. hypot-def 99.4%

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

8. Simplified 99.4%

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

9. Taylor expanded in z around inf 14.6%

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

    1. frac-2neg 14.6%

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

    2. div-inv 14.6%

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

11. Applied egg-rr 14.6%

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

13. Applied egg-rr 14.7%

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

14. Final simplification 14.7%

    \[\leadsto \frac{1}{\frac{\sqrt{3}}{z}} \]
15. Add Preprocessing

Alternative 4: 97.9% accurate, 1.1× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
y_m = (fabs.f64 y)
z_m = (fabs.f64 z)
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
(FPCore (x_m y_m z_m) :precision binary64 (* z_m (sqrt 0.3333333333333333)))

C

x_m = fabs(x);
y_m = fabs(y);
z_m = fabs(z);
assert(x_m < y_m && y_m < z_m);
double code(double x_m, double y_m, double z_m) {
	return z_m * sqrt(0.3333333333333333);
}

Fortran

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

Java

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

Python

x_m = math.fabs(x)
y_m = math.fabs(y)
z_m = math.fabs(z)
[x_m, y_m, z_m] = sort([x_m, y_m, z_m])
def code(x_m, y_m, z_m):
	return z_m * math.sqrt(0.3333333333333333)

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
code[x$95$m_, y$95$m_, z$95$m_] := N[(z$95$m * N[Sqrt[0.3333333333333333], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 43.1%

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

   1. sqr-neg 43.1%

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

   2. sqr-neg 43.1%

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

   3. sqr-neg 43.1%

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

   4. sqr-neg 43.1%

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

   5. sqr-neg 43.1%

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

3. Simplified 43.1%

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

5. Taylor expanded in z around inf 14.6%

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

   1. *-commutative 14.6%

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

7. Simplified 14.6%

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

8. Final simplification 14.6%

   \[\leadsto z \cdot \sqrt{0.3333333333333333} \]
9. Add Preprocessing

Alternative 5: 97.8% accurate, 1.1× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
y_m = (fabs.f64 y)
z_m = (fabs.f64 z)
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
(FPCore (x_m y_m z_m) :precision binary64 (/ z_m (sqrt 3.0)))

C

x_m = fabs(x);
y_m = fabs(y);
z_m = fabs(z);
assert(x_m < y_m && y_m < z_m);
double code(double x_m, double y_m, double z_m) {
	return z_m / sqrt(3.0);
}

Fortran

x_m = abs(x)
y_m = abs(y)
z_m = abs(z)
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, y_m, z_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    code = z_m / sqrt(3.0d0)
end function

Java

x_m = Math.abs(x);
y_m = Math.abs(y);
z_m = Math.abs(z);
assert x_m < y_m && y_m < z_m;
public static double code(double x_m, double y_m, double z_m) {
	return z_m / Math.sqrt(3.0);
}

Python

x_m = math.fabs(x)
y_m = math.fabs(y)
z_m = math.fabs(z)
[x_m, y_m, z_m] = sort([x_m, y_m, z_m])
def code(x_m, y_m, z_m):
	return z_m / math.sqrt(3.0)

Julia

x_m = abs(x)
y_m = abs(y)
z_m = abs(z)
x_m, y_m, z_m = sort([x_m, y_m, z_m])
function code(x_m, y_m, z_m)
	return Float64(z_m / sqrt(3.0))
end

MATLAB

x_m = abs(x);
y_m = abs(y);
z_m = abs(z);
x_m, y_m, z_m = num2cell(sort([x_m, y_m, z_m])){:}
function tmp = code(x_m, y_m, z_m)
	tmp = z_m / sqrt(3.0);
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
code[x$95$m_, y$95$m_, z$95$m_] := N[(z$95$m / N[Sqrt[3.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 43.1%

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

   1. sqr-neg 43.1%

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

   2. sqr-neg 43.1%

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

   3. sqr-neg 43.1%

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

   4. sqr-neg 43.1%

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

   5. sqr-neg 43.1%

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

3. Simplified 43.1%

   \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(x, x, \mathsf{fma}\left(y, y, z \cdot z\right)\right) \cdot 0.3333333333333333}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. fma-udef 43.1%

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

   2. fma-udef 43.1%

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

   3. associate-+l+ 43.1%

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

   4. metadata-eval 43.1%

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

   5. div-inv 43.1%

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

   6. sqrt-div 43.0%

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

   7. div-inv 42.7%

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

   8. +-commutative 42.7%

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

   9. add-sqr-sqrt 42.7%

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

   10. hypot-def 57.8%

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

   11. hypot-def 98.6%

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

6. Applied egg-rr 98.6%

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

   1. associate-*r/ 99.4%

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

   2. *-rgt-identity 99.4%

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

   3. hypot-def 58.2%

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

   4. unpow2 58.2%

      \[\leadsto \frac{\mathsf{hypot}\left(z, \sqrt{\color{blue}{{x}^{2}} + y \cdot y}\right)}{\sqrt{3}} \]

   5. unpow2 58.2%

      \[\leadsto \frac{\mathsf{hypot}\left(z, \sqrt{{x}^{2} + \color{blue}{{y}^{2}}}\right)}{\sqrt{3}} \]

   6. +-commutative 58.2%

      \[\leadsto \frac{\mathsf{hypot}\left(z, \sqrt{\color{blue}{{y}^{2} + {x}^{2}}}\right)}{\sqrt{3}} \]

   7. unpow2 58.2%

      \[\leadsto \frac{\mathsf{hypot}\left(z, \sqrt{\color{blue}{y \cdot y} + {x}^{2}}\right)}{\sqrt{3}} \]

   8. unpow2 58.2%

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

   9. hypot-def 99.4%

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

8. Simplified 99.4%

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

9. Taylor expanded in z around inf 14.6%

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

10. Final simplification 14.6%

    \[\leadsto \frac{z}{\sqrt{3}} \]
11. Add Preprocessing

Developer target: 97.2% 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 2024017 
(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)))