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

Percentage Accurate: 45.3% → 99.6%

Time: 8.8s

Alternatives: 7

Speedup: 2.7×

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: 45.3% 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: 99.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ y_m = \left|y\right| \\ x_m = \left|x\right| \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ \left(\mathsf{hypot}\left(y\_m, z\_m\right) \cdot {0.3333333333333333}^{0.25}\right) \cdot {0.3333333333333333}^{0.25} \end{array} \]

FPCore

z_m = (fabs.f64 z)
y_m = (fabs.f64 y)
x_m = (fabs.f64 x)
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 y_m z_m) (pow 0.3333333333333333 0.25))
  (pow 0.3333333333333333 0.25)))

C

z_m = fabs(z);
y_m = fabs(y);
x_m = fabs(x);
assert(x_m < y_m && y_m < z_m);
double code(double x_m, double y_m, double z_m) {
	return (hypot(y_m, z_m) * pow(0.3333333333333333, 0.25)) * pow(0.3333333333333333, 0.25);
}

Java

z_m = Math.abs(z);
y_m = Math.abs(y);
x_m = Math.abs(x);
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(y_m, z_m) * Math.pow(0.3333333333333333, 0.25)) * Math.pow(0.3333333333333333, 0.25);
}

Python

z_m = math.fabs(z)
y_m = math.fabs(y)
x_m = math.fabs(x)
[x_m, y_m, z_m] = sort([x_m, y_m, z_m])
def code(x_m, y_m, z_m):
	return (math.hypot(y_m, z_m) * math.pow(0.3333333333333333, 0.25)) * math.pow(0.3333333333333333, 0.25)

Julia

z_m = abs(z)
y_m = abs(y)
x_m = abs(x)
x_m, y_m, z_m = sort([x_m, y_m, z_m])
function code(x_m, y_m, z_m)
	return Float64(Float64(hypot(y_m, z_m) * (0.3333333333333333 ^ 0.25)) * (0.3333333333333333 ^ 0.25))
end

MATLAB

z_m = abs(z);
y_m = abs(y);
x_m = abs(x);
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(y_m, z_m) * (0.3333333333333333 ^ 0.25)) * (0.3333333333333333 ^ 0.25);
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
x_m = N[Abs[x], $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[(N[Sqrt[y$95$m ^ 2 + z$95$m ^ 2], $MachinePrecision] * N[Power[0.3333333333333333, 0.25], $MachinePrecision]), $MachinePrecision] * N[Power[0.3333333333333333, 0.25], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 42.1%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. unpow2 N/A

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

   5. unpow2 N/A

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

   6. lower-hypot.f64 N/A

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

   7. lower-sqrt.f64 69.4

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

5. Applied rewrites 69.4%

   \[\leadsto \color{blue}{\mathsf{hypot}\left(z, y\right) \cdot \sqrt{0.3333333333333333}} \]
6. Step-by-step derivation

7. Applied rewrites 69.4%

   \[\leadsto {0.3333333333333333}^{0.25} \cdot \color{blue}{\left({0.3333333333333333}^{0.25} \cdot \mathsf{hypot}\left(y, z\right)\right)} \]

8. Final simplification 69.4%

   \[\leadsto \left(\mathsf{hypot}\left(y, z\right) \cdot {0.3333333333333333}^{0.25}\right) \cdot {0.3333333333333333}^{0.25} \]
9. Add Preprocessing

Alternative 2: 99.4% accurate, 0.3× speedup

Math

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

FPCore

z_m = (fabs.f64 z)
y_m = (fabs.f64 y)
x_m = (fabs.f64 x)
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

z_m = fabs(z);
y_m = fabs(y);
x_m = fabs(x);
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

z_m = Math.abs(z);
y_m = Math.abs(y);
x_m = Math.abs(x);
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

z_m = math.fabs(z)
y_m = math.fabs(y)
x_m = math.fabs(x)
[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

z_m = abs(z)
y_m = abs(y)
x_m = abs(x)
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

z_m = abs(z);
y_m = abs(y);
x_m = abs(x);
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

z_m = N[Abs[z], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
x_m = N[Abs[x], $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 42.1%

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

3. Taylor expanded in z around inf

   \[\leadsto \sqrt{\frac{\color{blue}{{z}^{2}}}{3}} \]
4. Step-by-step derivation

   1. unpow2 N/A

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

   2. lower-*.f64 15.4

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

5. Applied rewrites 15.4%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. sqrt-div N/A

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

   4. lower-/.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-sqrt.f64 15.3

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

7. Applied rewrites 15.3%

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

8. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. unpow2 N/A

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

   4. lower-hypot.f64 69.4

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

10. Applied rewrites 69.4%

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

Alternative 3: 99.4% accurate, 0.4× speedup

Math

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

FPCore

z_m = (fabs.f64 z)
y_m = (fabs.f64 y)
x_m = (fabs.f64 x)
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 z_m y_m)))

C

z_m = fabs(z);
y_m = fabs(y);
x_m = fabs(x);
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(z_m, y_m);
}

Java

z_m = Math.abs(z);
y_m = Math.abs(y);
x_m = Math.abs(x);
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(z_m, y_m);
}

Python

z_m = math.fabs(z)
y_m = math.fabs(y)
x_m = math.fabs(x)
[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(z_m, y_m)

Julia

z_m = abs(z)
y_m = abs(y)
x_m = abs(x)
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(z_m, y_m))
end

MATLAB

z_m = abs(z);
y_m = abs(y);
x_m = abs(x);
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(z_m, y_m);
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
x_m = N[Abs[x], $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[z$95$m ^ 2 + y$95$m ^ 2], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 42.1%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. unpow2 N/A

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

   5. unpow2 N/A

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

   6. lower-hypot.f64 N/A

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

   7. lower-sqrt.f64 69.4

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

5. Applied rewrites 69.4%

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

6. Final simplification 69.4%

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

Alternative 4: 98.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ y_m = \left|y\right| \\ x_m = \left|x\right| \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ \mathsf{fma}\left(\frac{0.16666666666666666 \cdot y\_m}{z\_m}, \frac{y\_m}{\sqrt{0.3333333333333333}}, \sqrt{0.3333333333333333} \cdot z\_m\right) \end{array} \]

FPCore

z_m = (fabs.f64 z)
y_m = (fabs.f64 y)
x_m = (fabs.f64 x)
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
 (fma
  (/ (* 0.16666666666666666 y_m) z_m)
  (/ y_m (sqrt 0.3333333333333333))
  (* (sqrt 0.3333333333333333) z_m)))

C

z_m = fabs(z);
y_m = fabs(y);
x_m = fabs(x);
assert(x_m < y_m && y_m < z_m);
double code(double x_m, double y_m, double z_m) {
	return fma(((0.16666666666666666 * y_m) / z_m), (y_m / sqrt(0.3333333333333333)), (sqrt(0.3333333333333333) * z_m));
}

Julia

z_m = abs(z)
y_m = abs(y)
x_m = abs(x)
x_m, y_m, z_m = sort([x_m, y_m, z_m])
function code(x_m, y_m, z_m)
	return fma(Float64(Float64(0.16666666666666666 * y_m) / z_m), Float64(y_m / sqrt(0.3333333333333333)), Float64(sqrt(0.3333333333333333) * z_m))
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
x_m = N[Abs[x], $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[(N[(0.16666666666666666 * y$95$m), $MachinePrecision] / z$95$m), $MachinePrecision] * N[(y$95$m / N[Sqrt[0.3333333333333333], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[0.3333333333333333], $MachinePrecision] * z$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 42.1%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. unpow2 N/A

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

   5. unpow2 N/A

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

   6. lower-hypot.f64 N/A

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

   7. lower-sqrt.f64 69.4

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

5. Applied rewrites 69.4%

   \[\leadsto \color{blue}{\mathsf{hypot}\left(z, y\right) \cdot \sqrt{0.3333333333333333}} \]
6. Step-by-step derivation

7. Applied rewrites 69.4%

   \[\leadsto {0.3333333333333333}^{0.25} \cdot \color{blue}{\left({0.3333333333333333}^{0.25} \cdot \mathsf{hypot}\left(y, z\right)\right)} \]

8. Taylor expanded in z around inf

   \[\leadsto z \cdot \color{blue}{\left(\sqrt{\frac{1}{3}} + \frac{1}{6} \cdot \frac{{y}^{2}}{{z}^{2} \cdot \sqrt{\frac{1}{3}}}\right)} \]
9. Step-by-step derivation

10. Applied rewrites 14.8%

    \[\leadsto \mathsf{fma}\left(\frac{y}{\sqrt{0.3333333333333333}} \cdot \frac{y}{z \cdot z}, 0.16666666666666666, \sqrt{0.3333333333333333}\right) \cdot \color{blue}{z} \]

11. Taylor expanded in y around 0

    \[\leadsto \frac{1}{6} \cdot \frac{{y}^{2}}{z \cdot \sqrt{\frac{1}{3}}} + \color{blue}{z \cdot \sqrt{\frac{1}{3}}} \]
12. Step-by-step derivation

13. Applied rewrites 15.1%

    \[\leadsto \mathsf{fma}\left(\frac{0.16666666666666666 \cdot y}{z}, \color{blue}{\frac{y}{\sqrt{0.3333333333333333}}}, \sqrt{0.3333333333333333} \cdot z\right) \]
14. Add Preprocessing

Alternative 5: 98.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ y_m = \left|y\right| \\ x_m = \left|x\right| \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ \mathsf{fma}\left(\frac{0.5}{z\_m} \cdot y\_m, y\_m, z\_m\right) \cdot \sqrt{0.3333333333333333} \end{array} \]

FPCore

z_m = (fabs.f64 z)
y_m = (fabs.f64 y)
x_m = (fabs.f64 x)
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
 (* (fma (* (/ 0.5 z_m) y_m) y_m z_m) (sqrt 0.3333333333333333)))

C

z_m = fabs(z);
y_m = fabs(y);
x_m = fabs(x);
assert(x_m < y_m && y_m < z_m);
double code(double x_m, double y_m, double z_m) {
	return fma(((0.5 / z_m) * y_m), y_m, z_m) * sqrt(0.3333333333333333);
}

Julia

z_m = abs(z)
y_m = abs(y)
x_m = abs(x)
x_m, y_m, z_m = sort([x_m, y_m, z_m])
function code(x_m, y_m, z_m)
	return Float64(fma(Float64(Float64(0.5 / z_m) * y_m), y_m, z_m) * sqrt(0.3333333333333333))
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
x_m = N[Abs[x], $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[(N[(N[(0.5 / z$95$m), $MachinePrecision] * y$95$m), $MachinePrecision] * y$95$m + z$95$m), $MachinePrecision] * N[Sqrt[0.3333333333333333], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 42.1%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. unpow2 N/A

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

   5. unpow2 N/A

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

   6. lower-hypot.f64 N/A

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

   7. lower-sqrt.f64 69.4

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

5. Applied rewrites 69.4%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 14.5%

   \[\leadsto \mathsf{fma}\left(\frac{0.5}{z}, y \cdot y, z\right) \cdot \sqrt{\color{blue}{0.3333333333333333}} \]
9. Step-by-step derivation

10. Applied rewrites 15.1%

    \[\leadsto \mathsf{fma}\left(\frac{0.5}{z} \cdot y, y, z\right) \cdot \sqrt{0.3333333333333333} \]
11. Add Preprocessing

Alternative 6: 97.7% accurate, 2.0× speedup

Math

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

FPCore

z_m = (fabs.f64 z)
y_m = (fabs.f64 y)
x_m = (fabs.f64 x)
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

z_m = fabs(z);
y_m = fabs(y);
x_m = fabs(x);
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

z_m = abs(z)
y_m = abs(y)
x_m = abs(x)
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

z_m = Math.abs(z);
y_m = Math.abs(y);
x_m = Math.abs(x);
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

z_m = math.fabs(z)
y_m = math.fabs(y)
x_m = math.fabs(x)
[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

z_m = abs(z)
y_m = abs(y)
x_m = abs(x)
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

z_m = abs(z);
y_m = abs(y);
x_m = abs(x);
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

z_m = N[Abs[z], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
x_m = N[Abs[x], $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 42.1%

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

3. Taylor expanded in z around inf

   \[\leadsto \sqrt{\frac{\color{blue}{{z}^{2}}}{3}} \]
4. Step-by-step derivation

   1. unpow2 N/A

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

   2. lower-*.f64 15.4

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

5. Applied rewrites 15.4%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. sqrt-div N/A

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

   4. lower-/.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-sqrt.f64 15.3

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

7. Applied rewrites 15.3%

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

8. Taylor expanded in z around inf

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

   1. lower-/.f64 N/A

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

   2. lower-sqrt.f64 14.6

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

10. Applied rewrites 14.6%

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

Alternative 7: 97.8% accurate, 2.7× speedup

Math

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

FPCore

z_m = (fabs.f64 z)
y_m = (fabs.f64 y)
x_m = (fabs.f64 x)
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) z_m))

C

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

Fortran

z_m = abs(z)
y_m = abs(y)
x_m = abs(x)
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 = sqrt(0.3333333333333333d0) * z_m
end function

Java

z_m = Math.abs(z);
y_m = Math.abs(y);
x_m = Math.abs(x);
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) * z_m;
}

Python

z_m = math.fabs(z)
y_m = math.fabs(y)
x_m = math.fabs(x)
[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) * z_m

Julia

z_m = abs(z)
y_m = abs(y)
x_m = abs(x)
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) * z_m)
end

MATLAB

z_m = abs(z);
y_m = abs(y);
x_m = abs(x);
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) * z_m;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
x_m = N[Abs[x], $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] * z$95$m), $MachinePrecision]

Derivation

1. Initial program 42.1%

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

3. Taylor expanded in z around inf

   \[\leadsto \color{blue}{z \cdot \sqrt{\frac{1}{3}}} \]
4. Step-by-step derivation

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 14.6

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

5. Applied rewrites 14.6%

   \[\leadsto \color{blue}{\sqrt{0.3333333333333333} \cdot z} \]
6. Add Preprocessing

Developer Target 1: 97.2% accurate, 0.7× 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 2024279 
(FPCore (x y z)
  :name "Data.Array.Repa.Algorithms.Pixel:doubleRmsOfRGB8 from repa-algorithms-3.4.0.1"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< z -63964793941097760000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (/ (- z) (sqrt 3)) (if (< z 7320293694404182000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (/ (sqrt (+ (+ (* z z) (* x x)) (* y y))) (sqrt 3)) (* (sqrt 3333333333333333/10000000000000000) z))))

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