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

Percentage Accurate: 45.5% → 98.8%

Time: 12.3s

Alternatives: 3

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: 45.5% 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: 98.8% accurate, 0.2× speedup

Math

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

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
  y_m
  (* y_m (exp (- (log (/ z_m (* (sqrt 0.3333333333333333) 0.5))))))
  (* 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(y_m, (y_m * exp(-log((z_m / (sqrt(0.3333333333333333) * 0.5))))), (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 fma(y_m, Float64(y_m * exp(Float64(-log(Float64(z_m / Float64(sqrt(0.3333333333333333) * 0.5)))))), Float64(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[(y$95$m * N[(y$95$m * N[Exp[(-N[Log[N[(z$95$m / N[(N[Sqrt[0.3333333333333333], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])], $MachinePrecision]), $MachinePrecision] + N[(z$95$m * N[Sqrt[0.3333333333333333], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 45.9%

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

3. Taylor expanded in z around inf

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

   1. distribute-rgt-in N/A

      \[\leadsto \color{blue}{\sqrt{\frac{1}{3}} \cdot z + \left(\frac{1}{2} \cdot \frac{\sqrt{\frac{1}{3}} \cdot \left({x}^{2} + {y}^{2}\right)}{{z}^{2}}\right) \cdot z} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{z \cdot \sqrt{\frac{1}{3}}} + \left(\frac{1}{2} \cdot \frac{\sqrt{\frac{1}{3}} \cdot \left({x}^{2} + {y}^{2}\right)}{{z}^{2}}\right) \cdot z \]

   3. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \sqrt{\frac{1}{3}}, \left(\frac{1}{2} \cdot \frac{\sqrt{\frac{1}{3}} \cdot \left({x}^{2} + {y}^{2}\right)}{{z}^{2}}\right) \cdot z\right)} \]

   4. lower-sqrt.f64 N/A

      \[\leadsto \mathsf{fma}\left(z, \color{blue}{\sqrt{\frac{1}{3}}}, \left(\frac{1}{2} \cdot \frac{\sqrt{\frac{1}{3}} \cdot \left({x}^{2} + {y}^{2}\right)}{{z}^{2}}\right) \cdot z\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(z, \sqrt{\frac{1}{3}}, \color{blue}{z \cdot \left(\frac{1}{2} \cdot \frac{\sqrt{\frac{1}{3}} \cdot \left({x}^{2} + {y}^{2}\right)}{{z}^{2}}\right)}\right) \]

   6. associate-*r/ N/A

      \[\leadsto \mathsf{fma}\left(z, \sqrt{\frac{1}{3}}, z \cdot \color{blue}{\frac{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{3}} \cdot \left({x}^{2} + {y}^{2}\right)\right)}{{z}^{2}}}\right) \]

   7. associate-*r/ N/A

      \[\leadsto \mathsf{fma}\left(z, \sqrt{\frac{1}{3}}, \color{blue}{\frac{z \cdot \left(\frac{1}{2} \cdot \left(\sqrt{\frac{1}{3}} \cdot \left({x}^{2} + {y}^{2}\right)\right)\right)}{{z}^{2}}}\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(z, \sqrt{\frac{1}{3}}, \frac{z \cdot \left(\frac{1}{2} \cdot \left(\sqrt{\frac{1}{3}} \cdot \left({x}^{2} + {y}^{2}\right)\right)\right)}{\color{blue}{z \cdot z}}\right) \]

   9. times-frac N/A

      \[\leadsto \mathsf{fma}\left(z, \sqrt{\frac{1}{3}}, \color{blue}{\frac{z}{z} \cdot \frac{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{3}} \cdot \left({x}^{2} + {y}^{2}\right)\right)}{z}}\right) \]

   10. *-inverses N/A

       \[\leadsto \mathsf{fma}\left(z, \sqrt{\frac{1}{3}}, \color{blue}{1} \cdot \frac{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{3}} \cdot \left({x}^{2} + {y}^{2}\right)\right)}{z}\right) \]

   11. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(z, \sqrt{\frac{1}{3}}, \color{blue}{1 \cdot \frac{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{3}} \cdot \left({x}^{2} + {y}^{2}\right)\right)}{z}}\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(z, \sqrt{\frac{1}{3}}, 1 \cdot \frac{\color{blue}{\left(\sqrt{\frac{1}{3}} \cdot \left({x}^{2} + {y}^{2}\right)\right) \cdot \frac{1}{2}}}{z}\right) \]

   13. associate-/l* N/A

       \[\leadsto \mathsf{fma}\left(z, \sqrt{\frac{1}{3}}, 1 \cdot \color{blue}{\left(\left(\sqrt{\frac{1}{3}} \cdot \left({x}^{2} + {y}^{2}\right)\right) \cdot \frac{\frac{1}{2}}{z}\right)}\right) \]

   14. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(z, \sqrt{\frac{1}{3}}, 1 \cdot \color{blue}{\left(\left(\sqrt{\frac{1}{3}} \cdot \left({x}^{2} + {y}^{2}\right)\right) \cdot \frac{\frac{1}{2}}{z}\right)}\right) \]

5. Applied rewrites 17.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(z, \sqrt{0.3333333333333333}, 1 \cdot \left(\left(\mathsf{fma}\left(y, y, x \cdot x\right) \cdot \sqrt{0.3333333333333333}\right) \cdot \frac{0.5}{z}\right)\right)} \]
6. Step-by-step derivation

7. Applied rewrites 17.3%

   \[\leadsto \frac{1}{\color{blue}{\frac{1}{\mathsf{fma}\left(z, \sqrt{0.3333333333333333}, \frac{\sqrt{0.3333333333333333} \cdot \mathsf{fma}\left(x, x, y \cdot y\right)}{z \cdot 2}\right)}}} \]

8. Taylor expanded in x around 0

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

10. Applied rewrites 19.2%

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

12. Applied rewrites 18.9%

    \[\leadsto \mathsf{fma}\left(y, y \cdot e^{\log \left(\frac{z}{\sqrt{0.3333333333333333} \cdot 0.5}\right) \cdot -1}, \sqrt{0.3333333333333333} \cdot z\right) \]

13. Final simplification 18.9%

    \[\leadsto \mathsf{fma}\left(y, y \cdot e^{-\log \left(\frac{z}{\sqrt{0.3333333333333333} \cdot 0.5}\right)}, z \cdot \sqrt{0.3333333333333333}\right) \]
14. Add Preprocessing

Alternative 2: 98.8% accurate, 0.8× speedup

Math

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

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
  y_m
  (* y_m (/ (* (sqrt 0.3333333333333333) 0.5) z_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(y_m, (y_m * ((sqrt(0.3333333333333333) * 0.5) / z_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 fma(y_m, Float64(y_m * Float64(Float64(sqrt(0.3333333333333333) * 0.5) / z_m)), Float64(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[(y$95$m * N[(y$95$m * N[(N[(N[Sqrt[0.3333333333333333], $MachinePrecision] * 0.5), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision] + N[(z$95$m * N[Sqrt[0.3333333333333333], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 45.9%

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

3. Taylor expanded in z around inf

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

   1. distribute-rgt-in N/A

      \[\leadsto \color{blue}{\sqrt{\frac{1}{3}} \cdot z + \left(\frac{1}{2} \cdot \frac{\sqrt{\frac{1}{3}} \cdot \left({x}^{2} + {y}^{2}\right)}{{z}^{2}}\right) \cdot z} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{z \cdot \sqrt{\frac{1}{3}}} + \left(\frac{1}{2} \cdot \frac{\sqrt{\frac{1}{3}} \cdot \left({x}^{2} + {y}^{2}\right)}{{z}^{2}}\right) \cdot z \]

   3. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \sqrt{\frac{1}{3}}, \left(\frac{1}{2} \cdot \frac{\sqrt{\frac{1}{3}} \cdot \left({x}^{2} + {y}^{2}\right)}{{z}^{2}}\right) \cdot z\right)} \]

   4. lower-sqrt.f64 N/A

      \[\leadsto \mathsf{fma}\left(z, \color{blue}{\sqrt{\frac{1}{3}}}, \left(\frac{1}{2} \cdot \frac{\sqrt{\frac{1}{3}} \cdot \left({x}^{2} + {y}^{2}\right)}{{z}^{2}}\right) \cdot z\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(z, \sqrt{\frac{1}{3}}, \color{blue}{z \cdot \left(\frac{1}{2} \cdot \frac{\sqrt{\frac{1}{3}} \cdot \left({x}^{2} + {y}^{2}\right)}{{z}^{2}}\right)}\right) \]

   6. associate-*r/ N/A

      \[\leadsto \mathsf{fma}\left(z, \sqrt{\frac{1}{3}}, z \cdot \color{blue}{\frac{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{3}} \cdot \left({x}^{2} + {y}^{2}\right)\right)}{{z}^{2}}}\right) \]

   7. associate-*r/ N/A

      \[\leadsto \mathsf{fma}\left(z, \sqrt{\frac{1}{3}}, \color{blue}{\frac{z \cdot \left(\frac{1}{2} \cdot \left(\sqrt{\frac{1}{3}} \cdot \left({x}^{2} + {y}^{2}\right)\right)\right)}{{z}^{2}}}\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(z, \sqrt{\frac{1}{3}}, \frac{z \cdot \left(\frac{1}{2} \cdot \left(\sqrt{\frac{1}{3}} \cdot \left({x}^{2} + {y}^{2}\right)\right)\right)}{\color{blue}{z \cdot z}}\right) \]

   9. times-frac N/A

      \[\leadsto \mathsf{fma}\left(z, \sqrt{\frac{1}{3}}, \color{blue}{\frac{z}{z} \cdot \frac{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{3}} \cdot \left({x}^{2} + {y}^{2}\right)\right)}{z}}\right) \]

   10. *-inverses N/A

       \[\leadsto \mathsf{fma}\left(z, \sqrt{\frac{1}{3}}, \color{blue}{1} \cdot \frac{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{3}} \cdot \left({x}^{2} + {y}^{2}\right)\right)}{z}\right) \]

   11. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(z, \sqrt{\frac{1}{3}}, \color{blue}{1 \cdot \frac{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{3}} \cdot \left({x}^{2} + {y}^{2}\right)\right)}{z}}\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(z, \sqrt{\frac{1}{3}}, 1 \cdot \frac{\color{blue}{\left(\sqrt{\frac{1}{3}} \cdot \left({x}^{2} + {y}^{2}\right)\right) \cdot \frac{1}{2}}}{z}\right) \]

   13. associate-/l* N/A

       \[\leadsto \mathsf{fma}\left(z, \sqrt{\frac{1}{3}}, 1 \cdot \color{blue}{\left(\left(\sqrt{\frac{1}{3}} \cdot \left({x}^{2} + {y}^{2}\right)\right) \cdot \frac{\frac{1}{2}}{z}\right)}\right) \]

   14. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(z, \sqrt{\frac{1}{3}}, 1 \cdot \color{blue}{\left(\left(\sqrt{\frac{1}{3}} \cdot \left({x}^{2} + {y}^{2}\right)\right) \cdot \frac{\frac{1}{2}}{z}\right)}\right) \]

5. Applied rewrites 17.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(z, \sqrt{0.3333333333333333}, 1 \cdot \left(\left(\mathsf{fma}\left(y, y, x \cdot x\right) \cdot \sqrt{0.3333333333333333}\right) \cdot \frac{0.5}{z}\right)\right)} \]
6. Step-by-step derivation

7. Applied rewrites 17.3%

   \[\leadsto \frac{1}{\color{blue}{\frac{1}{\mathsf{fma}\left(z, \sqrt{0.3333333333333333}, \frac{\sqrt{0.3333333333333333} \cdot \mathsf{fma}\left(x, x, y \cdot y\right)}{z \cdot 2}\right)}}} \]

8. Taylor expanded in x around 0

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

10. Applied rewrites 19.2%

    \[\leadsto \mathsf{fma}\left(y, \color{blue}{y \cdot \frac{\sqrt{0.3333333333333333} \cdot 0.5}{z}}, \sqrt{0.3333333333333333} \cdot z\right) \]

11. Final simplification 19.2%

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

Alternative 3: 98.0% accurate, 2.7× speedup

Math

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

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 (* 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 z_m * sqrt(0.3333333333333333);
}

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(0.3333333333333333d0)
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(0.3333333333333333);
}

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(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(z_m * sqrt(0.3333333333333333))
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(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[(z$95$m * N[Sqrt[0.3333333333333333], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 45.9%

   \[\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}} \]
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. lower-*.f64 N/A

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

   2. lower-sqrt.f64 18.7

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

5. Applied rewrites 18.7%

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

Developer Target 1: 97.3% 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 2024238 
(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)))