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

Percentage Accurate: 45.0% → 99.4%

Time: 6.3s

Alternatives: 6

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z)
use fmin_fmax_functions
    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: 45.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z)
use fmin_fmax_functions
    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, 0.3× 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 41.3%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. sqrt-div N/A

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

   4. lower-/.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. +-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. lower-fma.f64 N/A

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lift-*.f64 N/A

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

   13. lower-fma.f64 N/A

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

   14. lower-sqrt.f64 41.2

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

4. Applied rewrites 41.2%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 68.0%

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

Alternative 2: 99.4% accurate, 0.4× speedup

Math

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

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
 (* (hypot z_m y_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 hypot(z_m, y_m) * sqrt(0.3333333333333333);
}

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(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 math.hypot(z_m, y_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(hypot(z_m, y_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 = hypot(z_m, y_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[Sqrt[z$95$m ^ 2 + y$95$m ^ 2], $MachinePrecision] * N[Sqrt[0.3333333333333333], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 41.3%

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

5. Applied rewrites 67.9%

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

Alternative 3: 98.6% accurate, 0.5× speedup

Math

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

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

Derivation

1. Initial program 41.3%

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

5. Applied rewrites 14.5%

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

7. Applied rewrites 17.0%

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

9. Applied rewrites 17.0%

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

Alternative 4: 98.6% accurate, 1.1× speedup

Math

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

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

Derivation

1. Initial program 41.3%

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

5. Applied rewrites 67.9%

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

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 15.3%

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

10. Applied rewrites 17.1%

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

Alternative 5: 97.8% 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 =     private
y_m =     private
x_m =     private
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x_m, y_m, z_m)
use fmin_fmax_functions
    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 41.3%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. sqrt-div N/A

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

   4. lower-/.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. +-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. lower-fma.f64 N/A

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lift-*.f64 N/A

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

   13. lower-fma.f64 N/A

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

   14. lower-sqrt.f64 41.2

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

4. Applied rewrites 41.2%

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

5. Taylor expanded in z around inf

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

7. Applied rewrites 16.8%

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

Alternative 6: 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 =     private
y_m =     private
x_m =     private
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x_m, y_m, z_m)
use fmin_fmax_functions
    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 41.3%

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

3. Taylor expanded in z around inf

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

5. Applied rewrites 16.7%

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

Developer Target 1: 97.1% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z)
use fmin_fmax_functions
    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 2025019 
(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)))