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

Percentage Accurate: 44.8% → 99.4%

Time: 6.6s

Alternatives: 6

Speedup: 0.6×

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = sqrt(((((x * x) + (y * y)) + (z * z)) / 3.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[Sqrt[N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] + N[(z * z), $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision]], $MachinePrecision]

Initial Program: 44.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = sqrt(((((x * x) + (y * y)) + (z * z)) / 3.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[Sqrt[N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] + N[(z * z), $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision]], $MachinePrecision]

Alternative 1: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{hypot}\left(x, \mathsf{hypot}\left(z, y\right)\right)}{\sqrt{3}} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(N[Sqrt[x ^ 2 + N[Sqrt[z ^ 2 + y ^ 2], $MachinePrecision] ^ 2], $MachinePrecision] / N[Sqrt[3.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.0%

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

   1. sqrt-div 49.9%

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

   2. div-inv 49.5%

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

   3. associate-+l+ 49.5%

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

   4. add-sqr-sqrt 49.5%

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

   5. hypot-def 59.5%

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

   6. hypot-def 98.7%

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

3. Applied egg-rr 98.7%

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

   1. associate-*r/ 99.4%

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

   2. *-rgt-identity 99.4%

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

   3. hypot-def 59.9%

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

   4. +-commutative 59.9%

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

   5. hypot-def 99.4%

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

5. Simplified 99.4%

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

6. Final simplification 99.4%

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

Alternative 2: 35.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.7 \cdot 10^{+138}:\\ \;\;\;\;\sqrt{0.3333333333333333} \cdot \mathsf{hypot}\left(y, x\right)\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-59}:\\ \;\;\;\;\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{\sqrt{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -5.7e+138)
   (* (sqrt 0.3333333333333333) (hypot y x))
   (if (<= x -5e-59)
     (sqrt (/ (+ (+ (* x x) (* y y)) (* z z)) 3.0))
     (/ z (sqrt 3.0)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -5.7e+138) {
		tmp = sqrt(0.3333333333333333) * hypot(y, x);
	} else if (x <= -5e-59) {
		tmp = sqrt(((((x * x) + (y * y)) + (z * z)) / 3.0));
	} else {
		tmp = z / sqrt(3.0);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -5.7e+138) {
		tmp = Math.sqrt(0.3333333333333333) * Math.hypot(y, x);
	} else if (x <= -5e-59) {
		tmp = Math.sqrt(((((x * x) + (y * y)) + (z * z)) / 3.0));
	} else {
		tmp = z / Math.sqrt(3.0);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -5.7e+138:
		tmp = math.sqrt(0.3333333333333333) * math.hypot(y, x)
	elif x <= -5e-59:
		tmp = math.sqrt(((((x * x) + (y * y)) + (z * z)) / 3.0))
	else:
		tmp = z / math.sqrt(3.0)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -5.7e+138)
		tmp = Float64(sqrt(0.3333333333333333) * hypot(y, x));
	elseif (x <= -5e-59)
		tmp = sqrt(Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) + Float64(z * z)) / 3.0));
	else
		tmp = Float64(z / sqrt(3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -5.7e+138)
		tmp = sqrt(0.3333333333333333) * hypot(y, x);
	elseif (x <= -5e-59)
		tmp = sqrt(((((x * x) + (y * y)) + (z * z)) / 3.0));
	else
		tmp = z / sqrt(3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -5.7e+138], N[(N[Sqrt[0.3333333333333333], $MachinePrecision] * N[Sqrt[y ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -5e-59], N[Sqrt[N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] + N[(z * z), $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision]], $MachinePrecision], N[(z / N[Sqrt[3.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.69999999999999986e138

   1. Initial program 13.6%

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

   2. Taylor expanded in z around 0 13.8%

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

      1. *-commutative 13.8%

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

      2. unpow2 13.8%

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

      3. unpow2 13.8%

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

      4. hypot-def 82.1%

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

   4. Simplified 82.1%

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

   if -5.69999999999999986e138 < x < -5.0000000000000001e-59

   1. Initial program 72.6%

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

   if -5.0000000000000001e-59 < x

   1. Initial program 49.5%

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

      1. sqrt-div 49.5%

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

      2. div-inv 49.0%

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

      3. associate-+l+ 49.0%

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

      4. add-sqr-sqrt 49.0%

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

      5. hypot-def 56.1%

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

      6. hypot-def 98.6%

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

   3. Applied egg-rr 98.6%

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

      1. associate-*r/ 99.5%

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

      2. *-rgt-identity 99.5%

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

      3. hypot-def 56.5%

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

      4. +-commutative 56.5%

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

      5. hypot-def 99.5%

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

   5. Simplified 99.5%

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

   6. Taylor expanded in z around inf 23.6%

      \[\leadsto \color{blue}{\frac{z}{\sqrt{3}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 41.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.7 \cdot 10^{+138}:\\ \;\;\;\;\sqrt{0.3333333333333333} \cdot \mathsf{hypot}\left(y, x\right)\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-59}:\\ \;\;\;\;\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{\sqrt{3}}\\ \end{array} \]

Alternative 3: 68.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \sqrt{0.3333333333333333} \cdot \mathsf{hypot}\left(z, x\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* (sqrt 0.3333333333333333) (hypot z x)))

C

double code(double x, double y, double z) {
	return sqrt(0.3333333333333333) * hypot(z, x);
}

Java

public static double code(double x, double y, double z) {
	return Math.sqrt(0.3333333333333333) * Math.hypot(z, x);
}

Python

def code(x, y, z):
	return math.sqrt(0.3333333333333333) * math.hypot(z, x)

Julia

function code(x, y, z)
	return Float64(sqrt(0.3333333333333333) * hypot(z, x))
end

MATLAB

function tmp = code(x, y, z)
	tmp = sqrt(0.3333333333333333) * hypot(z, x);
end

Wolfram

code[x_, y_, z_] := N[(N[Sqrt[0.3333333333333333], $MachinePrecision] * N[Sqrt[z ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.0%

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

2. Taylor expanded in y around 0 37.1%

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

   1. *-commutative 37.1%

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

   2. unpow2 37.1%

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

   3. unpow2 37.1%

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

   4. hypot-def 71.1%

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

4. Simplified 71.1%

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

5. Final simplification 71.1%

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

Alternative 4: 34.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.42 \cdot 10^{+141}:\\ \;\;\;\;x \cdot \left(-\sqrt{0.3333333333333333}\right)\\ \mathbf{elif}\;x \leq -2.5 \cdot 10^{-58}:\\ \;\;\;\;\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{\sqrt{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.42e+141)
   (* x (- (sqrt 0.3333333333333333)))
   (if (<= x -2.5e-58)
     (sqrt (/ (+ (+ (* x x) (* y y)) (* z z)) 3.0))
     (/ z (sqrt 3.0)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.42e+141) {
		tmp = x * -sqrt(0.3333333333333333);
	} else if (x <= -2.5e-58) {
		tmp = sqrt(((((x * x) + (y * y)) + (z * z)) / 3.0));
	} else {
		tmp = z / sqrt(3.0);
	}
	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 (x <= (-1.42d+141)) then
        tmp = x * -sqrt(0.3333333333333333d0)
    else if (x <= (-2.5d-58)) then
        tmp = sqrt(((((x * x) + (y * y)) + (z * z)) / 3.0d0))
    else
        tmp = z / sqrt(3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.42e+141) {
		tmp = x * -Math.sqrt(0.3333333333333333);
	} else if (x <= -2.5e-58) {
		tmp = Math.sqrt(((((x * x) + (y * y)) + (z * z)) / 3.0));
	} else {
		tmp = z / Math.sqrt(3.0);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.42e+141:
		tmp = x * -math.sqrt(0.3333333333333333)
	elif x <= -2.5e-58:
		tmp = math.sqrt(((((x * x) + (y * y)) + (z * z)) / 3.0))
	else:
		tmp = z / math.sqrt(3.0)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.42e+141)
		tmp = Float64(x * Float64(-sqrt(0.3333333333333333)));
	elseif (x <= -2.5e-58)
		tmp = sqrt(Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) + Float64(z * z)) / 3.0));
	else
		tmp = Float64(z / sqrt(3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.42e+141)
		tmp = x * -sqrt(0.3333333333333333);
	elseif (x <= -2.5e-58)
		tmp = sqrt(((((x * x) + (y * y)) + (z * z)) / 3.0));
	else
		tmp = z / sqrt(3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.42e+141], N[(x * (-N[Sqrt[0.3333333333333333], $MachinePrecision])), $MachinePrecision], If[LessEqual[x, -2.5e-58], N[Sqrt[N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] + N[(z * z), $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision]], $MachinePrecision], N[(z / N[Sqrt[3.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.42000000000000005e141

   1. Initial program 13.6%

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

   2. Taylor expanded in x around -inf 69.5%

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{0.3333333333333333} \cdot x\right)} \]
   3. Step-by-step derivation

      1. mul-1-neg 69.5%

         \[\leadsto \color{blue}{-\sqrt{0.3333333333333333} \cdot x} \]

      2. distribute-rgt-neg-in 69.5%

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

   4. Simplified 69.5%

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

   if -1.42000000000000005e141 < x < -2.49999999999999989e-58

   1. Initial program 72.6%

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

   if -2.49999999999999989e-58 < x

   1. Initial program 49.5%

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

      1. sqrt-div 49.5%

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

      2. div-inv 49.0%

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

      3. associate-+l+ 49.0%

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

      4. add-sqr-sqrt 49.0%

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

      5. hypot-def 56.1%

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

      6. hypot-def 98.6%

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

   3. Applied egg-rr 98.6%

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

      1. associate-*r/ 99.5%

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

      2. *-rgt-identity 99.5%

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

      3. hypot-def 56.5%

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

      4. +-commutative 56.5%

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

      5. hypot-def 99.5%

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

   5. Simplified 99.5%

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

   6. Taylor expanded in z around inf 23.6%

      \[\leadsto \color{blue}{\frac{z}{\sqrt{3}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 39.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.42 \cdot 10^{+141}:\\ \;\;\;\;x \cdot \left(-\sqrt{0.3333333333333333}\right)\\ \mathbf{elif}\;x \leq -2.5 \cdot 10^{-58}:\\ \;\;\;\;\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{\sqrt{3}}\\ \end{array} \]

Alternative 5: 30.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.00085:\\ \;\;\;\;x \cdot \left(-\sqrt{0.3333333333333333}\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \sqrt{0.3333333333333333}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -0.00085)
   (* x (- (sqrt 0.3333333333333333)))
   (* z (sqrt 0.3333333333333333))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -0.00085) {
		tmp = x * -sqrt(0.3333333333333333);
	} else {
		tmp = z * sqrt(0.3333333333333333);
	}
	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 (x <= (-0.00085d0)) then
        tmp = x * -sqrt(0.3333333333333333d0)
    else
        tmp = z * sqrt(0.3333333333333333d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -0.00085) {
		tmp = x * -Math.sqrt(0.3333333333333333);
	} else {
		tmp = z * Math.sqrt(0.3333333333333333);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -0.00085:
		tmp = x * -math.sqrt(0.3333333333333333)
	else:
		tmp = z * math.sqrt(0.3333333333333333)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -0.00085)
		tmp = Float64(x * Float64(-sqrt(0.3333333333333333)));
	else
		tmp = Float64(z * sqrt(0.3333333333333333));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -0.00085)
		tmp = x * -sqrt(0.3333333333333333);
	else
		tmp = z * sqrt(0.3333333333333333);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -0.00085], N[(x * (-N[Sqrt[0.3333333333333333], $MachinePrecision])), $MachinePrecision], N[(z * N[Sqrt[0.3333333333333333], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -8.49999999999999953e-4

   1. Initial program 49.1%

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

   2. Taylor expanded in x around -inf 65.1%

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{0.3333333333333333} \cdot x\right)} \]
   3. Step-by-step derivation

      1. mul-1-neg 65.1%

         \[\leadsto \color{blue}{-\sqrt{0.3333333333333333} \cdot x} \]

      2. distribute-rgt-neg-in 65.1%

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

   4. Simplified 65.1%

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

   if -8.49999999999999953e-4 < x

   1. Initial program 50.4%

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

   2. Taylor expanded in z around inf 25.5%

      \[\leadsto \color{blue}{z \cdot \sqrt{0.3333333333333333}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00085:\\ \;\;\;\;x \cdot \left(-\sqrt{0.3333333333333333}\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \sqrt{0.3333333333333333}\\ \end{array} \]

Alternative 6: 18.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ z \cdot \sqrt{0.3333333333333333} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* z (sqrt 0.3333333333333333)))

C

double code(double x, double y, double z) {
	return z * sqrt(0.3333333333333333);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = z * sqrt(0.3333333333333333d0)
end function

Java

public static double code(double x, double y, double z) {
	return z * Math.sqrt(0.3333333333333333);
}

Python

def code(x, y, z):
	return z * math.sqrt(0.3333333333333333)

Julia

function code(x, y, z)
	return Float64(z * sqrt(0.3333333333333333))
end

MATLAB

function tmp = code(x, y, z)
	tmp = z * sqrt(0.3333333333333333);
end

Wolfram

code[x_, y_, z_] := N[(z * N[Sqrt[0.3333333333333333], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.0%

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

2. Taylor expanded in z around inf 22.3%

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

3. Final simplification 22.3%

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

Developer target: 62.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z < -6.396479394109776 \cdot 10^{+136}:\\ \;\;\;\;\frac{-z}{\sqrt{3}}\\ \mathbf{elif}\;z < 7.320293694404182 \cdot 10^{+117}:\\ \;\;\;\;\frac{\sqrt{\left(z \cdot z + x \cdot x\right) + y \cdot y}}{\sqrt{3}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.3333333333333333} \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (< z -6.396479394109776e+136)
   (/ (- z) (sqrt 3.0))
   (if (< z 7.320293694404182e+117)
     (/ (sqrt (+ (+ (* z z) (* x x)) (* y y))) (sqrt 3.0))
     (* (sqrt 0.3333333333333333) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z < -6.396479394109776e+136) {
		tmp = -z / sqrt(3.0);
	} else if (z < 7.320293694404182e+117) {
		tmp = sqrt((((z * z) + (x * x)) + (y * y))) / sqrt(3.0);
	} else {
		tmp = sqrt(0.3333333333333333) * z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z < (-6.396479394109776d+136)) then
        tmp = -z / sqrt(3.0d0)
    else if (z < 7.320293694404182d+117) then
        tmp = sqrt((((z * z) + (x * x)) + (y * y))) / sqrt(3.0d0)
    else
        tmp = sqrt(0.3333333333333333d0) * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z < -6.396479394109776e+136) {
		tmp = -z / Math.sqrt(3.0);
	} else if (z < 7.320293694404182e+117) {
		tmp = Math.sqrt((((z * z) + (x * x)) + (y * y))) / Math.sqrt(3.0);
	} else {
		tmp = Math.sqrt(0.3333333333333333) * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z < -6.396479394109776e+136:
		tmp = -z / math.sqrt(3.0)
	elif z < 7.320293694404182e+117:
		tmp = math.sqrt((((z * z) + (x * x)) + (y * y))) / math.sqrt(3.0)
	else:
		tmp = math.sqrt(0.3333333333333333) * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z < -6.396479394109776e+136)
		tmp = Float64(Float64(-z) / sqrt(3.0));
	elseif (z < 7.320293694404182e+117)
		tmp = Float64(sqrt(Float64(Float64(Float64(z * z) + Float64(x * x)) + Float64(y * y))) / sqrt(3.0));
	else
		tmp = Float64(sqrt(0.3333333333333333) * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z < -6.396479394109776e+136)
		tmp = -z / sqrt(3.0);
	elseif (z < 7.320293694404182e+117)
		tmp = sqrt((((z * z) + (x * x)) + (y * y))) / sqrt(3.0);
	else
		tmp = sqrt(0.3333333333333333) * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Less[z, -6.396479394109776e+136], N[((-z) / N[Sqrt[3.0], $MachinePrecision]), $MachinePrecision], If[Less[z, 7.320293694404182e+117], N[(N[Sqrt[N[(N[(N[(z * z), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[3.0], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[0.3333333333333333], $MachinePrecision] * z), $MachinePrecision]]]

Reproduce

herbie shell --seed 2023185 
(FPCore (x y z)
  :name "Data.Array.Repa.Algorithms.Pixel:doubleRmsOfRGB8 from repa-algorithms-3.4.0.1"
  :precision binary64

  :herbie-target
  (if (< z -6.396479394109776e+136) (/ (- z) (sqrt 3.0)) (if (< z 7.320293694404182e+117) (/ (sqrt (+ (+ (* z z) (* x x)) (* y y))) (sqrt 3.0)) (* (sqrt 0.3333333333333333) z)))

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