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

Percentage Accurate: 44.9% → 97.9%
Time: 2.5s
Alternatives: 3
Speedup: 2.3×

Specification

?
\[\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}} \]
(FPCore (x y z)
  :precision binary64
  :pre TRUE
  (sqrt (/ (+ (+ (* x x) (* y y)) (* z z)) 3.0)))
double code(double x, double y, double z) {
	return sqrt(((((x * x) + (y * y)) + (z * z)) / 3.0));
}

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

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

def code(x, y, z):
	return math.sqrt(((((x * x) + (y * y)) + (z * z)) / 3.0))

function code(x, y, z)
	return sqrt(Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) + Float64(z * z)) / 3.0))
end

function tmp = code(x, y, z)
	tmp = sqrt(((((x * x) + (y * y)) + (z * z)) / 3.0));
end

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]

f(x, y, z):
	x in [-inf, +inf],
	y in [-inf, +inf],
	z in [-inf, +inf]
code: THEORY
BEGIN
f(x, y, z: real): real =
	sqrt(((((x * x) + (y * y)) + (z * z)) / (3)))
END code
\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}

Local Percentage Accuracy vs ?

The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples.

Accuracy vs Speed?

Herbie found 3 alternatives:

AlternativeAccuracySpeedup
The accuracy (vertical axis) and speed (horizontal axis) of each alternatives. Up and to the right is better. The red square shows the initial program, and each blue circle shows an alternative.The line shows the best available speed-accuracy tradeoffs.

Initial Program: 44.9% accurate, 1.0× speedup?

\[\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}} \]
(FPCore (x y z)
  :precision binary64
  :pre TRUE
  (sqrt (/ (+ (+ (* x x) (* y y)) (* z z)) 3.0)))
double code(double x, double y, double z) {
	return sqrt(((((x * x) + (y * y)) + (z * z)) / 3.0));
}

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

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

def code(x, y, z):
	return math.sqrt(((((x * x) + (y * y)) + (z * z)) / 3.0))

function code(x, y, z)
	return sqrt(Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) + Float64(z * z)) / 3.0))
end

function tmp = code(x, y, z)
	tmp = sqrt(((((x * x) + (y * y)) + (z * z)) / 3.0));
end

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]

f(x, y, z):
	x in [-inf, +inf],
	y in [-inf, +inf],
	z in [-inf, +inf]
code: THEORY
BEGIN
f(x, y, z: real): real =
	sqrt(((((x * x) + (y * y)) + (z * z)) / (3)))
END code
\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}

Alternative 1: 97.9% accurate, 1.6× speedup?

\[\mathsf{max}\left(\left|y\right|, \mathsf{max}\left(\left|x\right|, \left|z\right|\right)\right) \cdot 0.5773502691896257 \]
(FPCore (x y z)
  :precision binary64
  :pre TRUE
  (* (fmax (fabs y) (fmax (fabs x) (fabs z))) 0.5773502691896257))
double code(double x, double y, double z) {
	return fmax(fabs(y), fmax(fabs(x), fabs(z))) * 0.5773502691896257;
}

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 = fmax(abs(y), fmax(abs(x), abs(z))) * 0.5773502691896257d0
end function

public static double code(double x, double y, double z) {
	return fmax(Math.abs(y), fmax(Math.abs(x), Math.abs(z))) * 0.5773502691896257;
}

def code(x, y, z):
	return fmax(math.fabs(y), fmax(math.fabs(x), math.fabs(z))) * 0.5773502691896257

function code(x, y, z)
	return Float64(fmax(abs(y), fmax(abs(x), abs(z))) * 0.5773502691896257)
end

function tmp = code(x, y, z)
	tmp = max(abs(y), max(abs(x), abs(z))) * 0.5773502691896257;
end

code[x_, y_, z_] := N[(N[Max[N[Abs[y], $MachinePrecision], N[Max[N[Abs[x], $MachinePrecision], N[Abs[z], $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 0.5773502691896257), $MachinePrecision]

f(x, y, z):
	x in [-inf, +inf],
	y in [-inf, +inf],
	z in [-inf, +inf]
code: THEORY
BEGIN
f(x, y, z: real): real =
	LET tmp_1 = IF ((abs(x)) > (abs(z))) THEN (abs(x)) ELSE (abs(z)) ENDIF IN
	LET tmp_2 = IF ((abs(x)) > (abs(z))) THEN (abs(x)) ELSE (abs(z)) ENDIF IN
	LET tmp = IF ((abs(y)) > tmp_1) THEN (abs(y)) ELSE tmp_2 ENDIF IN
	tmp * (577350269189625731058868041145615279674530029296875e-51)
END code
\mathsf{max}\left(\left|y\right|, \mathsf{max}\left(\left|x\right|, \left|z\right|\right)\right) \cdot 0.5773502691896257
Derivation

  1. Initial program 44.9%

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

  2. Taylor expanded in z around inf

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

    1. Applied rewrites18.1%

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

    2. Evaluated real constant18.1%

      \[\leadsto z \cdot 0.5773502691896257 \]
    3. Add Preprocessing

    Alternative 2: 67.7% accurate, 2.3× speedup?

    \[\mathsf{max}\left(\left|x\right|, \left|y\right|\right) \cdot 0.5773502691896257 \]
    (FPCore (x y z)
  :precision binary64
  :pre TRUE
  (* (fmax (fabs x) (fabs y)) 0.5773502691896257))
    double code(double x, double y, double z) {
	return fmax(fabs(x), fabs(y)) * 0.5773502691896257;
}

    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 = fmax(abs(x), abs(y)) * 0.5773502691896257d0
end function

    public static double code(double x, double y, double z) {
	return fmax(Math.abs(x), Math.abs(y)) * 0.5773502691896257;
}

    def code(x, y, z):
	return fmax(math.fabs(x), math.fabs(y)) * 0.5773502691896257

    function code(x, y, z)
	return Float64(fmax(abs(x), abs(y)) * 0.5773502691896257)
end

    function tmp = code(x, y, z)
	tmp = max(abs(x), abs(y)) * 0.5773502691896257;
end

    code[x_, y_, z_] := N[(N[Max[N[Abs[x], $MachinePrecision], N[Abs[y], $MachinePrecision]], $MachinePrecision] * 0.5773502691896257), $MachinePrecision]

    f(x, y, z):
	x in [-inf, +inf],
	y in [-inf, +inf],
	z in [-inf, +inf]
code: THEORY
BEGIN
f(x, y, z: real): real =
	LET tmp = IF ((abs(x)) > (abs(y))) THEN (abs(x)) ELSE (abs(y)) ENDIF IN
	tmp * (577350269189625731058868041145615279674530029296875e-51)
END code
    \mathsf{max}\left(\left|x\right|, \left|y\right|\right) \cdot 0.5773502691896257
    Derivation

    1. Initial program 44.9%

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

    2. Taylor expanded in y around inf

      \[\leadsto y \cdot \sqrt{\frac{1}{3}} \]
    3. Step-by-step derivation

      1. Applied rewrites18.9%

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

      2. Evaluated real constant18.9%

        \[\leadsto y \cdot 0.5773502691896257 \]
      3. Add Preprocessing

      Alternative 3: 36.2% accurate, 4.2× speedup?

      \[\left|x\right| \cdot 0.5773502691896257 \]
      (FPCore (x y z)
  :precision binary64
  :pre TRUE
  (* (fabs x) 0.5773502691896257))
      double code(double x, double y, double z) {
	return fabs(x) * 0.5773502691896257;
}

      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 = abs(x) * 0.5773502691896257d0
end function

      public static double code(double x, double y, double z) {
	return Math.abs(x) * 0.5773502691896257;
}

      def code(x, y, z):
	return math.fabs(x) * 0.5773502691896257

      function code(x, y, z)
	return Float64(abs(x) * 0.5773502691896257)
end

      function tmp = code(x, y, z)
	tmp = abs(x) * 0.5773502691896257;
end

      code[x_, y_, z_] := N[(N[Abs[x], $MachinePrecision] * 0.5773502691896257), $MachinePrecision]

      f(x, y, z):
	x in [-inf, +inf],
	y in [-inf, +inf],
	z in [-inf, +inf]
code: THEORY
BEGIN
f(x, y, z: real): real =
	(abs(x)) * (577350269189625731058868041145615279674530029296875e-51)
END code
      \left|x\right| \cdot 0.5773502691896257
      Derivation

      1. Initial program 44.9%

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

      2. Taylor expanded in x around inf

        \[\leadsto x \cdot \sqrt{\frac{1}{3}} \]
      3. Step-by-step derivation

        1. Applied rewrites18.6%

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

        2. Evaluated real constant18.6%

          \[\leadsto x \cdot 0.5773502691896257 \]
        3. Add Preprocessing

        Reproduce

        ?
        herbie shell --seed 2026092 
(FPCore (x y z)
  :name "Data.Array.Repa.Algorithms.Pixel:doubleRmsOfRGB8 from repa-algorithms-3.4.0.1"
  :precision binary64
  (sqrt (/ (+ (+ (* x x) (* y y)) (* z z)) 3.0)))