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

Percentage Accurate: 44.9% → 99.4%

Time: 4.7s

Alternatives: 7

Speedup: 0.5×

• Unsound rule application detected in e-graph. Results may not be sound.

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: 44.9% 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 42.2%

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

   1. sqrt-div 42.0%

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

   2. div-inv 41.8%

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

   3. associate-+l+ 41.8%

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

   4. add-sqr-sqrt 41.8%

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

   5. hypot-def 55.8%

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

   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/ 99.5%

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

   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 56.2%

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

   4. unpow2 56.2%

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

   5. unpow2 56.2%

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

   6. +-commutative 56.2%

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

   7. unpow2 56.2%

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

   8. unpow2 56.2%

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

   9. 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. Final simplification 99.5%

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

Alternative 2: 72.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.85 \cdot 10^{-96}:\\ \;\;\;\;\sqrt{0.3333333333333333} \cdot \mathsf{hypot}\left(y, x\right)\\ \mathbf{elif}\;z \leq 8.3 \cdot 10^{+152}:\\ \;\;\;\;\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{\sqrt{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 1.85e-96)
   (* (sqrt 0.3333333333333333) (hypot y x))
   (if (<= z 8.3e+152)
     (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 (z <= 1.85e-96) {
		tmp = sqrt(0.3333333333333333) * hypot(y, x);
	} else if (z <= 8.3e+152) {
		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 (z <= 1.85e-96) {
		tmp = Math.sqrt(0.3333333333333333) * Math.hypot(y, x);
	} else if (z <= 8.3e+152) {
		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 z <= 1.85e-96:
		tmp = math.sqrt(0.3333333333333333) * math.hypot(y, x)
	elif z <= 8.3e+152:
		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 (z <= 1.85e-96)
		tmp = Float64(sqrt(0.3333333333333333) * hypot(y, x));
	elseif (z <= 8.3e+152)
		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 (z <= 1.85e-96)
		tmp = sqrt(0.3333333333333333) * hypot(y, x);
	elseif (z <= 8.3e+152)
		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[z, 1.85e-96], N[(N[Sqrt[0.3333333333333333], $MachinePrecision] * N[Sqrt[y ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 8.3e+152], 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 z < 1.84999999999999993e-96

   1. Initial program 41.8%

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

   2. Taylor expanded in z around 0 36.4%

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

      1. *-commutative 36.4%

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

      2. unpow2 36.4%

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

      3. unpow2 36.4%

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

      4. hypot-def 77.1%

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

   4. Simplified 77.1%

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

   if 1.84999999999999993e-96 < z < 8.3000000000000001e152

   1. Initial program 63.2%

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

   if 8.3000000000000001e152 < z

   1. Initial program 7.2%

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

      1. sqrt-div 7.2%

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

      2. div-inv 7.2%

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

      3. associate-+l+ 7.2%

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

      4. add-sqr-sqrt 7.2%

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

      5. hypot-def 7.2%

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

      6. hypot-def 98.9%

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

   3. Applied egg-rr 98.9%

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

      1. associate-*r/ 99.6%

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

      2. *-rgt-identity 99.6%

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

      3. hypot-def 7.2%

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

      4. unpow2 7.2%

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

      5. unpow2 7.2%

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

      6. +-commutative 7.2%

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

      7. unpow2 7.2%

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

      8. unpow2 7.2%

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

      9. hypot-def 99.6%

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

   5. Simplified 99.6%

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

   6. Taylor expanded in z around inf 81.0%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.85 \cdot 10^{-96}:\\ \;\;\;\;\sqrt{0.3333333333333333} \cdot \mathsf{hypot}\left(y, x\right)\\ \mathbf{elif}\;z \leq 8.3 \cdot 10^{+152}:\\ \;\;\;\;\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{\sqrt{3}}\\ \end{array} \]

Alternative 3: 34.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 7.5 \cdot 10^{-100}:\\ \;\;\;\;x \cdot \left(-\sqrt{0.3333333333333333}\right)\\ \mathbf{elif}\;z \leq 8.3 \cdot 10^{+152}:\\ \;\;\;\;\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{\sqrt{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 7.5e-100)
   (* x (- (sqrt 0.3333333333333333)))
   (if (<= z 8.3e+152)
     (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 (z <= 7.5e-100) {
		tmp = x * -sqrt(0.3333333333333333);
	} else if (z <= 8.3e+152) {
		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 (z <= 7.5d-100) then
        tmp = x * -sqrt(0.3333333333333333d0)
    else if (z <= 8.3d+152) 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 (z <= 7.5e-100) {
		tmp = x * -Math.sqrt(0.3333333333333333);
	} else if (z <= 8.3e+152) {
		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 z <= 7.5e-100:
		tmp = x * -math.sqrt(0.3333333333333333)
	elif z <= 8.3e+152:
		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 (z <= 7.5e-100)
		tmp = Float64(x * Float64(-sqrt(0.3333333333333333)));
	elseif (z <= 8.3e+152)
		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 (z <= 7.5e-100)
		tmp = x * -sqrt(0.3333333333333333);
	elseif (z <= 8.3e+152)
		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[z, 7.5e-100], N[(x * (-N[Sqrt[0.3333333333333333], $MachinePrecision])), $MachinePrecision], If[LessEqual[z, 8.3e+152], 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 z < 7.50000000000000015e-100

   1. Initial program 41.8%

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

   2. Taylor expanded in x around -inf 18.9%

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

      1. mul-1-neg 18.9%

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

      2. distribute-rgt-neg-in 18.9%

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

   4. Simplified 18.9%

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

   if 7.50000000000000015e-100 < z < 8.3000000000000001e152

   1. Initial program 63.2%

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

   if 8.3000000000000001e152 < z

   1. Initial program 7.2%

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

      1. sqrt-div 7.2%

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

      2. div-inv 7.2%

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

      3. associate-+l+ 7.2%

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

      4. add-sqr-sqrt 7.2%

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

      5. hypot-def 7.2%

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

      6. hypot-def 98.9%

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

   3. Applied egg-rr 98.9%

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

      1. associate-*r/ 99.6%

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

      2. *-rgt-identity 99.6%

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

      3. hypot-def 7.2%

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

      4. unpow2 7.2%

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

      5. unpow2 7.2%

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

      6. +-commutative 7.2%

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

      7. unpow2 7.2%

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

      8. unpow2 7.2%

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

      9. hypot-def 99.6%

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

   5. Simplified 99.6%

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

   6. Taylor expanded in z around inf 81.0%

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

4. Final simplification 35.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 7.5 \cdot 10^{-100}:\\ \;\;\;\;x \cdot \left(-\sqrt{0.3333333333333333}\right)\\ \mathbf{elif}\;z \leq 8.3 \cdot 10^{+152}:\\ \;\;\;\;\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{\sqrt{3}}\\ \end{array} \]

Alternative 4: 30.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 9.2 \cdot 10^{-23}:\\ \;\;\;\;x \cdot \left(-\sqrt{0.3333333333333333}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{\sqrt{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 9.2e-23) (* x (- (sqrt 0.3333333333333333))) (/ z (sqrt 3.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 9.2e-23) {
		tmp = x * -sqrt(0.3333333333333333);
	} 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 (z <= 9.2d-23) then
        tmp = x * -sqrt(0.3333333333333333d0)
    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 (z <= 9.2e-23) {
		tmp = x * -Math.sqrt(0.3333333333333333);
	} else {
		tmp = z / Math.sqrt(3.0);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 9.2e-23:
		tmp = x * -math.sqrt(0.3333333333333333)
	else:
		tmp = z / math.sqrt(3.0)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 9.2e-23)
		tmp = Float64(x * Float64(-sqrt(0.3333333333333333)));
	else
		tmp = Float64(z / sqrt(3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 9.2e-23)
		tmp = x * -sqrt(0.3333333333333333);
	else
		tmp = z / sqrt(3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 9.2e-23], N[(x * (-N[Sqrt[0.3333333333333333], $MachinePrecision])), $MachinePrecision], N[(z / N[Sqrt[3.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 9.2000000000000004e-23

   1. Initial program 45.0%

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

   2. Taylor expanded in x around -inf 20.2%

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

      1. mul-1-neg 20.2%

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

      2. distribute-rgt-neg-in 20.2%

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

   4. Simplified 20.2%

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

   if 9.2000000000000004e-23 < z

   1. Initial program 32.7%

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

      1. sqrt-div 32.6%

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

      2. div-inv 32.4%

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

      3. associate-+l+ 32.4%

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

      4. add-sqr-sqrt 32.4%

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

      5. hypot-def 38.8%

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

      6. hypot-def 98.8%

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

   3. Applied egg-rr 98.8%

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

      1. associate-*r/ 99.6%

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

      2. *-rgt-identity 99.6%

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

      3. hypot-def 39.0%

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

      4. unpow2 39.0%

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

      5. unpow2 39.0%

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

      6. +-commutative 39.0%

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

      7. unpow2 39.0%

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

      8. unpow2 39.0%

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

      9. hypot-def 99.6%

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

   5. Simplified 99.6%

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

   6. Taylor expanded in z around inf 59.4%

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

4. Final simplification 29.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 9.2 \cdot 10^{-23}:\\ \;\;\;\;x \cdot \left(-\sqrt{0.3333333333333333}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{\sqrt{3}}\\ \end{array} \]

Alternative 5: 30.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.1 \cdot 10^{-22}:\\ \;\;\;\;\frac{-x}{\sqrt{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{\sqrt{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 1.1e-22) (/ (- x) (sqrt 3.0)) (/ z (sqrt 3.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.1e-22) {
		tmp = -x / sqrt(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 (z <= 1.1d-22) then
        tmp = -x / sqrt(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 (z <= 1.1e-22) {
		tmp = -x / Math.sqrt(3.0);
	} else {
		tmp = z / Math.sqrt(3.0);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 1.1e-22:
		tmp = -x / math.sqrt(3.0)
	else:
		tmp = z / math.sqrt(3.0)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 1.1e-22)
		tmp = Float64(Float64(-x) / sqrt(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 (z <= 1.1e-22)
		tmp = -x / sqrt(3.0);
	else
		tmp = z / sqrt(3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 1.1e-22], N[((-x) / N[Sqrt[3.0], $MachinePrecision]), $MachinePrecision], N[(z / N[Sqrt[3.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.1e-22

   1. Initial program 45.0%

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

      1. sqrt-div 44.8%

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

      2. div-inv 44.5%

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

      3. associate-+l+ 44.5%

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

      4. add-sqr-sqrt 44.5%

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

      5. hypot-def 60.8%

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

      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/ 99.5%

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

      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 61.3%

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

      4. unpow2 61.3%

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

      5. unpow2 61.3%

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

      6. +-commutative 61.3%

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

      7. unpow2 61.3%

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

      8. unpow2 61.3%

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

      9. 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 x around -inf 20.3%

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

      1. mul-1-neg 20.3%

         \[\leadsto \color{blue}{-\frac{x}{\sqrt{3}}} \]

      2. distribute-neg-frac 20.3%

         \[\leadsto \color{blue}{\frac{-x}{\sqrt{3}}} \]

   8. Simplified 20.3%

      \[\leadsto \color{blue}{\frac{-x}{\sqrt{3}}} \]

   if 1.1e-22 < z

   1. Initial program 32.7%

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

      1. sqrt-div 32.6%

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

      2. div-inv 32.4%

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

      3. associate-+l+ 32.4%

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

      4. add-sqr-sqrt 32.4%

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

      5. hypot-def 38.8%

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

      6. hypot-def 98.8%

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

   3. Applied egg-rr 98.8%

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

      1. associate-*r/ 99.6%

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

      2. *-rgt-identity 99.6%

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

      3. hypot-def 39.0%

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

      4. unpow2 39.0%

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

      5. unpow2 39.0%

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

      6. +-commutative 39.0%

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

      7. unpow2 39.0%

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

      8. unpow2 39.0%

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

      9. hypot-def 99.6%

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

   5. Simplified 99.6%

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

   6. Taylor expanded in z around inf 59.4%

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

4. Final simplification 29.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.1 \cdot 10^{-22}:\\ \;\;\;\;\frac{-x}{\sqrt{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{\sqrt{3}}\\ \end{array} \]

Alternative 6: 18.2% 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 42.2%

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

2. Taylor expanded in z around inf 16.1%

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

3. Final simplification 16.1%

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

Alternative 7: 18.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{z}{\sqrt{3}} \end{array} \]

FPCore

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

C

double code(double x, double y, double z) {
	return z / sqrt(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 = z / sqrt(3.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 42.2%

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

   1. sqrt-div 42.0%

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

   2. div-inv 41.8%

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

   3. associate-+l+ 41.8%

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

   4. add-sqr-sqrt 41.8%

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

   5. hypot-def 55.8%

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

   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/ 99.5%

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

   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 56.2%

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

   4. unpow2 56.2%

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

   5. unpow2 56.2%

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

   6. +-commutative 56.2%

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

   7. unpow2 56.2%

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

   8. unpow2 56.2%

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

   9. 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 16.2%

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

7. Final simplification 16.2%

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

Developer target: 61.7% 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 2023178 
(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)))