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

Percentage Accurate: 45.3% → 99.4%

Time: 6.6s

Alternatives: 5

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: 45.3% 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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 48.8%

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

   1. sqr-neg 48.8%

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

   2. sqr-neg 48.8%

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

   3. sqr-neg 48.8%

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

   4. sqr-neg 48.8%

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

   5. sqr-neg 48.8%

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

3. Simplified 48.8%

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

   1. fma-udef 48.8%

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

   2. fma-udef 48.8%

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

   3. associate-+l+ 48.8%

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

   4. metadata-eval 48.8%

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

   5. div-inv 48.8%

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

   6. sqrt-div 48.7%

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

   7. div-inv 48.3%

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

   8. +-commutative 48.3%

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

   9. add-sqr-sqrt 48.3%

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

   10. hypot-def 63.7%

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

   11. hypot-def 98.6%

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

5. Applied egg-rr 98.6%

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

   1. associate-*r/ 99.4%

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

   2. *-rgt-identity 99.4%

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

   3. hypot-def 64.2%

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

   4. unpow2 64.2%

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

   5. unpow2 64.2%

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

   6. +-commutative 64.2%

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

   7. unpow2 64.2%

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

   8. unpow2 64.2%

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

   9. hypot-def 99.4%

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

7. Simplified 99.4%

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

8. Final simplification 99.4%

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

Alternative 2: 67.8% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 48.8%

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

   1. sqr-neg 48.8%

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

   2. sqr-neg 48.8%

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

   3. sqr-neg 48.8%

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

   4. sqr-neg 48.8%

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

   5. sqr-neg 48.8%

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

3. Simplified 48.8%

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

4. Taylor expanded in x around 0 34.9%

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

   1. unpow2 34.9%

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

   2. unpow2 34.9%

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

   3. hypot-def 71.9%

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

6. Simplified 71.9%

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

7. Final simplification 71.9%

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

Alternative 3: 67.8% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 48.8%

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

   1. sqr-neg 48.8%

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

   2. sqr-neg 48.8%

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

   3. sqr-neg 48.8%

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

   4. sqr-neg 48.8%

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

   5. sqr-neg 48.8%

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

3. Simplified 48.8%

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

   1. fma-udef 48.8%

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

   2. fma-udef 48.8%

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

   3. associate-+l+ 48.8%

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

   4. metadata-eval 48.8%

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

   5. div-inv 48.8%

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

   6. sqrt-div 48.7%

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

   7. div-inv 48.3%

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

   8. +-commutative 48.3%

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

   9. add-sqr-sqrt 48.3%

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

   10. hypot-def 63.7%

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

   11. hypot-def 98.6%

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

5. Applied egg-rr 98.6%

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

   1. associate-*r/ 99.4%

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

   2. *-rgt-identity 99.4%

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

   3. hypot-def 64.2%

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

   4. unpow2 64.2%

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

   5. unpow2 64.2%

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

   6. +-commutative 64.2%

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

   7. unpow2 64.2%

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

   8. unpow2 64.2%

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

   9. hypot-def 99.4%

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

7. Simplified 99.4%

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

8. Taylor expanded in x around 0 34.7%

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

   1. associate-*l/ 35.0%

      \[\leadsto \color{blue}{\frac{1 \cdot \sqrt{{y}^{2} + {z}^{2}}}{\sqrt{3}}} \]

   2. +-commutative 35.0%

      \[\leadsto \frac{1 \cdot \sqrt{\color{blue}{{z}^{2} + {y}^{2}}}}{\sqrt{3}} \]

   3. unpow2 35.0%

      \[\leadsto \frac{1 \cdot \sqrt{\color{blue}{z \cdot z} + {y}^{2}}}{\sqrt{3}} \]

   4. unpow2 35.0%

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

   5. hypot-def 71.9%

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

   6. *-lft-identity 71.9%

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

10. Simplified 71.9%

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

11. Final simplification 71.9%

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

Alternative 4: 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 48.8%

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

   1. sqr-neg 48.8%

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

   2. sqr-neg 48.8%

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

   3. sqr-neg 48.8%

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

   4. sqr-neg 48.8%

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

   5. sqr-neg 48.8%

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

3. Simplified 48.8%

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

4. Taylor expanded in z around inf 25.2%

   \[\leadsto \color{blue}{z \cdot \sqrt{0.3333333333333333}} \]
5. Step-by-step derivation

   1. *-commutative 25.2%

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

6. Simplified 25.2%

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

7. Final simplification 25.2%

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

Alternative 5: 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 48.8%

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

   1. sqr-neg 48.8%

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

   2. sqr-neg 48.8%

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

   3. sqr-neg 48.8%

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

   4. sqr-neg 48.8%

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

   5. sqr-neg 48.8%

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

3. Simplified 48.8%

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

   1. fma-udef 48.8%

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

   2. fma-udef 48.8%

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

   3. associate-+l+ 48.8%

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

   4. metadata-eval 48.8%

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

   5. div-inv 48.8%

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

   6. sqrt-div 48.7%

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

   7. div-inv 48.3%

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

   8. +-commutative 48.3%

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

   9. add-sqr-sqrt 48.3%

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

   10. hypot-def 63.7%

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

   11. hypot-def 98.6%

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

5. Applied egg-rr 98.6%

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

   1. associate-*r/ 99.4%

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

   2. *-rgt-identity 99.4%

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

   3. hypot-def 64.2%

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

   4. unpow2 64.2%

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

   5. unpow2 64.2%

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

   6. +-commutative 64.2%

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

   7. unpow2 64.2%

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

   8. unpow2 64.2%

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

   9. hypot-def 99.4%

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

7. Simplified 99.4%

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

8. Taylor expanded in z around inf 25.3%

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

9. Final simplification 25.3%

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

Developer target: 62.4% 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 2023320 
(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)))