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

Percentage Accurate: 44.7% → 99.4%

Time: 6.4s

Alternatives: 11

Speedup: 0.6×

• 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: 44.7% 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 44.7%

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

   1. sqrt-div 44.6%

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

   2. div-inv 44.3%

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

   3. associate-+l+ 44.3%

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

   4. add-sqr-sqrt 44.3%

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

   5. hypot-def 59.7%

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

   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/ 99.4%

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

   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 60.2%

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

   4. +-commutative 60.2%

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

   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: 64.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{0.3333333333333333} \cdot \mathsf{hypot}\left(y, x\right)\\ \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-21}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \cdot z \leq 5 \cdot 10^{+177}:\\ \;\;\;\;\sqrt{\frac{z \cdot z + x \cdot x}{3}}\\ \mathbf{elif}\;z \cdot z \leq 10^{+224}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;z \cdot \sqrt{0.3333333333333333}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (sqrt 0.3333333333333333) (hypot y x))))
   (if (<= (* z z) 5e-21)
     t_0
     (if (<= (* z z) 5e+177)
       (sqrt (/ (+ (* z z) (* x x)) 3.0))
       (if (<= (* z z) 1e+224) t_0 (* z (sqrt 0.3333333333333333)))))))

C

double code(double x, double y, double z) {
	double t_0 = sqrt(0.3333333333333333) * hypot(y, x);
	double tmp;
	if ((z * z) <= 5e-21) {
		tmp = t_0;
	} else if ((z * z) <= 5e+177) {
		tmp = sqrt((((z * z) + (x * x)) / 3.0));
	} else if ((z * z) <= 1e+224) {
		tmp = t_0;
	} else {
		tmp = z * sqrt(0.3333333333333333);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.sqrt(0.3333333333333333) * Math.hypot(y, x);
	double tmp;
	if ((z * z) <= 5e-21) {
		tmp = t_0;
	} else if ((z * z) <= 5e+177) {
		tmp = Math.sqrt((((z * z) + (x * x)) / 3.0));
	} else if ((z * z) <= 1e+224) {
		tmp = t_0;
	} else {
		tmp = z * Math.sqrt(0.3333333333333333);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.sqrt(0.3333333333333333) * math.hypot(y, x)
	tmp = 0
	if (z * z) <= 5e-21:
		tmp = t_0
	elif (z * z) <= 5e+177:
		tmp = math.sqrt((((z * z) + (x * x)) / 3.0))
	elif (z * z) <= 1e+224:
		tmp = t_0
	else:
		tmp = z * math.sqrt(0.3333333333333333)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(sqrt(0.3333333333333333) * hypot(y, x))
	tmp = 0.0
	if (Float64(z * z) <= 5e-21)
		tmp = t_0;
	elseif (Float64(z * z) <= 5e+177)
		tmp = sqrt(Float64(Float64(Float64(z * z) + Float64(x * x)) / 3.0));
	elseif (Float64(z * z) <= 1e+224)
		tmp = t_0;
	else
		tmp = Float64(z * sqrt(0.3333333333333333));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = sqrt(0.3333333333333333) * hypot(y, x);
	tmp = 0.0;
	if ((z * z) <= 5e-21)
		tmp = t_0;
	elseif ((z * z) <= 5e+177)
		tmp = sqrt((((z * z) + (x * x)) / 3.0));
	elseif ((z * z) <= 1e+224)
		tmp = t_0;
	else
		tmp = z * sqrt(0.3333333333333333);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Sqrt[0.3333333333333333], $MachinePrecision] * N[Sqrt[y ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(z * z), $MachinePrecision], 5e-21], t$95$0, If[LessEqual[N[(z * z), $MachinePrecision], 5e+177], N[Sqrt[N[(N[(N[(z * z), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(z * z), $MachinePrecision], 1e+224], t$95$0, N[(z * N[Sqrt[0.3333333333333333], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z z) < 4.99999999999999973e-21 or 5.0000000000000003e177 < (*.f64 z z) < 9.9999999999999997e223

   1. Initial program 56.4%

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

   2. Taylor expanded in z around 0 50.6%

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

      1. *-commutative 50.6%

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

      2. unpow2 50.6%

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

      3. unpow2 50.6%

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

      4. hypot-def 92.6%

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

   4. Simplified 92.6%

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

   if 4.99999999999999973e-21 < (*.f64 z z) < 5.0000000000000003e177

   1. Initial program 55.0%

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

   2. Taylor expanded in x around inf 34.4%

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

      1. unpow2 34.4%

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

   4. Simplified 34.4%

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

   if 9.9999999999999997e223 < (*.f64 z z)

   1. Initial program 17.1%

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

   2. Taylor expanded in z around inf 40.5%

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

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-21}:\\ \;\;\;\;\sqrt{0.3333333333333333} \cdot \mathsf{hypot}\left(y, x\right)\\ \mathbf{elif}\;z \cdot z \leq 5 \cdot 10^{+177}:\\ \;\;\;\;\sqrt{\frac{z \cdot z + x \cdot x}{3}}\\ \mathbf{elif}\;z \cdot z \leq 10^{+224}:\\ \;\;\;\;\sqrt{0.3333333333333333} \cdot \mathsf{hypot}\left(y, x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \sqrt{0.3333333333333333}\\ \end{array} \]

Alternative 3: 67.4% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 44.7%

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

2. Taylor expanded in y around 0 29.6%

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

   1. *-commutative 29.6%

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

   2. unpow2 29.6%

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

   3. unpow2 29.6%

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

   4. hypot-def 69.1%

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

4. Simplified 69.1%

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

5. Final simplification 69.1%

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

Alternative 4: 67.4% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 44.7%

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

   1. sqrt-div 44.6%

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

   2. div-inv 44.3%

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

   3. associate-+l+ 44.3%

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

   4. add-sqr-sqrt 44.3%

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

   5. hypot-def 59.7%

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

   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/ 99.4%

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

   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 60.2%

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

   4. +-commutative 60.2%

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

   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. Taylor expanded in y around 0 29.4%

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

   1. associate-*r/ 29.6%

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

   2. unpow2 29.6%

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

   3. unpow2 29.6%

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

   4. +-commutative 29.6%

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

   5. hypot-def 69.1%

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

   6. *-rgt-identity 69.1%

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

   7. hypot-def 29.6%

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

   8. +-commutative 29.6%

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

   9. hypot-def 69.1%

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

8. Simplified 69.1%

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

9. Final simplification 69.1%

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

Alternative 5: 31.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(-\sqrt{0.3333333333333333}\right)\\ \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-21}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \cdot z \leq 5 \cdot 10^{+177}:\\ \;\;\;\;\sqrt{\frac{z \cdot z + x \cdot x}{3}}\\ \mathbf{elif}\;z \cdot z \leq 10^{+224}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;z \cdot \sqrt{0.3333333333333333}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (- (sqrt 0.3333333333333333)))))
   (if (<= (* z z) 5e-21)
     t_0
     (if (<= (* z z) 5e+177)
       (sqrt (/ (+ (* z z) (* x x)) 3.0))
       (if (<= (* z z) 1e+224) t_0 (* z (sqrt 0.3333333333333333)))))))

C

double code(double x, double y, double z) {
	double t_0 = x * -sqrt(0.3333333333333333);
	double tmp;
	if ((z * z) <= 5e-21) {
		tmp = t_0;
	} else if ((z * z) <= 5e+177) {
		tmp = sqrt((((z * z) + (x * x)) / 3.0));
	} else if ((z * z) <= 1e+224) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = x * -sqrt(0.3333333333333333d0)
    if ((z * z) <= 5d-21) then
        tmp = t_0
    else if ((z * z) <= 5d+177) then
        tmp = sqrt((((z * z) + (x * x)) / 3.0d0))
    else if ((z * z) <= 1d+224) then
        tmp = t_0
    else
        tmp = z * sqrt(0.3333333333333333d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x * -Math.sqrt(0.3333333333333333);
	double tmp;
	if ((z * z) <= 5e-21) {
		tmp = t_0;
	} else if ((z * z) <= 5e+177) {
		tmp = Math.sqrt((((z * z) + (x * x)) / 3.0));
	} else if ((z * z) <= 1e+224) {
		tmp = t_0;
	} else {
		tmp = z * Math.sqrt(0.3333333333333333);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * -math.sqrt(0.3333333333333333)
	tmp = 0
	if (z * z) <= 5e-21:
		tmp = t_0
	elif (z * z) <= 5e+177:
		tmp = math.sqrt((((z * z) + (x * x)) / 3.0))
	elif (z * z) <= 1e+224:
		tmp = t_0
	else:
		tmp = z * math.sqrt(0.3333333333333333)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(-sqrt(0.3333333333333333)))
	tmp = 0.0
	if (Float64(z * z) <= 5e-21)
		tmp = t_0;
	elseif (Float64(z * z) <= 5e+177)
		tmp = sqrt(Float64(Float64(Float64(z * z) + Float64(x * x)) / 3.0));
	elseif (Float64(z * z) <= 1e+224)
		tmp = t_0;
	else
		tmp = Float64(z * sqrt(0.3333333333333333));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * -sqrt(0.3333333333333333);
	tmp = 0.0;
	if ((z * z) <= 5e-21)
		tmp = t_0;
	elseif ((z * z) <= 5e+177)
		tmp = sqrt((((z * z) + (x * x)) / 3.0));
	elseif ((z * z) <= 1e+224)
		tmp = t_0;
	else
		tmp = z * sqrt(0.3333333333333333);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * (-N[Sqrt[0.3333333333333333], $MachinePrecision])), $MachinePrecision]}, If[LessEqual[N[(z * z), $MachinePrecision], 5e-21], t$95$0, If[LessEqual[N[(z * z), $MachinePrecision], 5e+177], N[Sqrt[N[(N[(N[(z * z), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(z * z), $MachinePrecision], 1e+224], t$95$0, N[(z * N[Sqrt[0.3333333333333333], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z z) < 4.99999999999999973e-21 or 5.0000000000000003e177 < (*.f64 z z) < 9.9999999999999997e223

   1. Initial program 56.4%

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

   2. Taylor expanded in x around -inf 29.5%

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

      1. mul-1-neg 29.5%

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

      2. distribute-rgt-neg-in 29.5%

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

   4. Simplified 29.5%

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

   if 4.99999999999999973e-21 < (*.f64 z z) < 5.0000000000000003e177

   1. Initial program 55.0%

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

   2. Taylor expanded in x around inf 34.4%

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

      1. unpow2 34.4%

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

   4. Simplified 34.4%

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

   if 9.9999999999999997e223 < (*.f64 z z)

   1. Initial program 17.1%

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

   2. Taylor expanded in z around inf 40.5%

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

4. Final simplification 33.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-21}:\\ \;\;\;\;x \cdot \left(-\sqrt{0.3333333333333333}\right)\\ \mathbf{elif}\;z \cdot z \leq 5 \cdot 10^{+177}:\\ \;\;\;\;\sqrt{\frac{z \cdot z + x \cdot x}{3}}\\ \mathbf{elif}\;z \cdot z \leq 10^{+224}:\\ \;\;\;\;x \cdot \left(-\sqrt{0.3333333333333333}\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \sqrt{0.3333333333333333}\\ \end{array} \]

Alternative 6: 28.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(-\sqrt{0.3333333333333333}\right)\\ \mathbf{if}\;z \leq 2050000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 7.4 \cdot 10^{+22}:\\ \;\;\;\;\frac{z}{\sqrt{3}}\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{+113}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;z \cdot \sqrt{0.3333333333333333}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (- (sqrt 0.3333333333333333)))))
   (if (<= z 2050000.0)
     t_0
     (if (<= z 7.4e+22)
       (/ z (sqrt 3.0))
       (if (<= z 2.1e+113) t_0 (* z (sqrt 0.3333333333333333)))))))

C

double code(double x, double y, double z) {
	double t_0 = x * -sqrt(0.3333333333333333);
	double tmp;
	if (z <= 2050000.0) {
		tmp = t_0;
	} else if (z <= 7.4e+22) {
		tmp = z / sqrt(3.0);
	} else if (z <= 2.1e+113) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = x * -sqrt(0.3333333333333333d0)
    if (z <= 2050000.0d0) then
        tmp = t_0
    else if (z <= 7.4d+22) then
        tmp = z / sqrt(3.0d0)
    else if (z <= 2.1d+113) then
        tmp = t_0
    else
        tmp = z * sqrt(0.3333333333333333d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x * -Math.sqrt(0.3333333333333333);
	double tmp;
	if (z <= 2050000.0) {
		tmp = t_0;
	} else if (z <= 7.4e+22) {
		tmp = z / Math.sqrt(3.0);
	} else if (z <= 2.1e+113) {
		tmp = t_0;
	} else {
		tmp = z * Math.sqrt(0.3333333333333333);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * -math.sqrt(0.3333333333333333)
	tmp = 0
	if z <= 2050000.0:
		tmp = t_0
	elif z <= 7.4e+22:
		tmp = z / math.sqrt(3.0)
	elif z <= 2.1e+113:
		tmp = t_0
	else:
		tmp = z * math.sqrt(0.3333333333333333)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(-sqrt(0.3333333333333333)))
	tmp = 0.0
	if (z <= 2050000.0)
		tmp = t_0;
	elseif (z <= 7.4e+22)
		tmp = Float64(z / sqrt(3.0));
	elseif (z <= 2.1e+113)
		tmp = t_0;
	else
		tmp = Float64(z * sqrt(0.3333333333333333));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * -sqrt(0.3333333333333333);
	tmp = 0.0;
	if (z <= 2050000.0)
		tmp = t_0;
	elseif (z <= 7.4e+22)
		tmp = z / sqrt(3.0);
	elseif (z <= 2.1e+113)
		tmp = t_0;
	else
		tmp = z * sqrt(0.3333333333333333);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * (-N[Sqrt[0.3333333333333333], $MachinePrecision])), $MachinePrecision]}, If[LessEqual[z, 2050000.0], t$95$0, If[LessEqual[z, 7.4e+22], N[(z / N[Sqrt[3.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.1e+113], t$95$0, N[(z * N[Sqrt[0.3333333333333333], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if z < 2.05e6 or 7.3999999999999996e22 < z < 2.0999999999999999e113

   1. Initial program 48.6%

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

   2. Taylor expanded in x around -inf 24.6%

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

      1. mul-1-neg 24.6%

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

      2. distribute-rgt-neg-in 24.6%

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

   4. Simplified 24.6%

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

   if 2.05e6 < z < 7.3999999999999996e22

   1. Initial program 77.1%

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

      1. sqrt-div 77.1%

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

      2. div-inv 76.3%

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

      3. associate-+l+ 76.3%

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

      4. add-sqr-sqrt 76.3%

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

      5. hypot-def 98.8%

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

      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/ 100.0%

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

      2. *-rgt-identity 100.0%

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

      3. hypot-def 100.0%

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

      4. +-commutative 100.0%

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

      5. hypot-def 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in z around inf 52.2%

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

   if 2.0999999999999999e113 < z

   1. Initial program 17.6%

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

   2. Taylor expanded in z around inf 83.9%

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

4. Final simplification 33.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2050000:\\ \;\;\;\;x \cdot \left(-\sqrt{0.3333333333333333}\right)\\ \mathbf{elif}\;z \leq 7.4 \cdot 10^{+22}:\\ \;\;\;\;\frac{z}{\sqrt{3}}\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{+113}:\\ \;\;\;\;x \cdot \left(-\sqrt{0.3333333333333333}\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \sqrt{0.3333333333333333}\\ \end{array} \]

Alternative 7: 28.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 2600000:\\ \;\;\;\;x \cdot \left(-\sqrt{0.3333333333333333}\right)\\ \mathbf{elif}\;z \leq 8 \cdot 10^{+22}:\\ \;\;\;\;\frac{z}{\sqrt{3}}\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{+112}:\\ \;\;\;\;\frac{-x}{\sqrt{3}}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \sqrt{0.3333333333333333}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 2600000.0)
   (* x (- (sqrt 0.3333333333333333)))
   (if (<= z 8e+22)
     (/ z (sqrt 3.0))
     (if (<= z 2.3e+112)
       (/ (- x) (sqrt 3.0))
       (* z (sqrt 0.3333333333333333))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 2600000.0) {
		tmp = x * -sqrt(0.3333333333333333);
	} else if (z <= 8e+22) {
		tmp = z / sqrt(3.0);
	} else if (z <= 2.3e+112) {
		tmp = -x / sqrt(3.0);
	} 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 (z <= 2600000.0d0) then
        tmp = x * -sqrt(0.3333333333333333d0)
    else if (z <= 8d+22) then
        tmp = z / sqrt(3.0d0)
    else if (z <= 2.3d+112) then
        tmp = -x / sqrt(3.0d0)
    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 (z <= 2600000.0) {
		tmp = x * -Math.sqrt(0.3333333333333333);
	} else if (z <= 8e+22) {
		tmp = z / Math.sqrt(3.0);
	} else if (z <= 2.3e+112) {
		tmp = -x / Math.sqrt(3.0);
	} else {
		tmp = z * Math.sqrt(0.3333333333333333);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 2600000.0:
		tmp = x * -math.sqrt(0.3333333333333333)
	elif z <= 8e+22:
		tmp = z / math.sqrt(3.0)
	elif z <= 2.3e+112:
		tmp = -x / math.sqrt(3.0)
	else:
		tmp = z * math.sqrt(0.3333333333333333)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 2600000.0)
		tmp = Float64(x * Float64(-sqrt(0.3333333333333333)));
	elseif (z <= 8e+22)
		tmp = Float64(z / sqrt(3.0));
	elseif (z <= 2.3e+112)
		tmp = Float64(Float64(-x) / sqrt(3.0));
	else
		tmp = Float64(z * sqrt(0.3333333333333333));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 2600000.0)
		tmp = x * -sqrt(0.3333333333333333);
	elseif (z <= 8e+22)
		tmp = z / sqrt(3.0);
	elseif (z <= 2.3e+112)
		tmp = -x / sqrt(3.0);
	else
		tmp = z * sqrt(0.3333333333333333);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 2600000.0], N[(x * (-N[Sqrt[0.3333333333333333], $MachinePrecision])), $MachinePrecision], If[LessEqual[z, 8e+22], N[(z / N[Sqrt[3.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.3e+112], N[((-x) / N[Sqrt[3.0], $MachinePrecision]), $MachinePrecision], N[(z * N[Sqrt[0.3333333333333333], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if z < 2.6e6

   1. Initial program 49.3%

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

   2. Taylor expanded in x around -inf 23.4%

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

      1. mul-1-neg 23.4%

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

      2. distribute-rgt-neg-in 23.4%

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

   4. Simplified 23.4%

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

   if 2.6e6 < z < 8e22

   1. Initial program 77.1%

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

      1. sqrt-div 77.1%

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

      2. div-inv 76.3%

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

      3. associate-+l+ 76.3%

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

      4. add-sqr-sqrt 76.3%

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

      5. hypot-def 98.8%

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

      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/ 100.0%

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

      2. *-rgt-identity 100.0%

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

      3. hypot-def 100.0%

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

      4. +-commutative 100.0%

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

      5. hypot-def 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in z around inf 52.2%

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

   if 8e22 < z < 2.3e112

   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 42.3%

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

      2. div-inv 42.0%

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

      3. associate-+l+ 42.0%

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

      4. add-sqr-sqrt 42.0%

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

      5. hypot-def 68.3%

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

      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 68.9%

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

      4. +-commutative 68.9%

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

      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 x around -inf 35.0%

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

      1. mul-1-neg 35.0%

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

      2. distribute-neg-frac 35.0%

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

   8. Simplified 35.0%

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

   if 2.3e112 < z

   1. Initial program 17.6%

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

   2. Taylor expanded in z around inf 83.9%

      \[\leadsto \color{blue}{z \cdot \sqrt{0.3333333333333333}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 33.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2600000:\\ \;\;\;\;x \cdot \left(-\sqrt{0.3333333333333333}\right)\\ \mathbf{elif}\;z \leq 8 \cdot 10^{+22}:\\ \;\;\;\;\frac{z}{\sqrt{3}}\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{+112}:\\ \;\;\;\;\frac{-x}{\sqrt{3}}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \sqrt{0.3333333333333333}\\ \end{array} \]

Alternative 8: 28.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 2050000:\\ \;\;\;\;x \cdot \left(-\sqrt{0.3333333333333333}\right)\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{+23}:\\ \;\;\;\;\sqrt{\frac{z}{\frac{3}{z}}}\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{+112}:\\ \;\;\;\;\frac{-x}{\sqrt{3}}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \sqrt{0.3333333333333333}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 2050000.0)
   (* x (- (sqrt 0.3333333333333333)))
   (if (<= z 2.6e+23)
     (sqrt (/ z (/ 3.0 z)))
     (if (<= z 3.6e+112)
       (/ (- x) (sqrt 3.0))
       (* z (sqrt 0.3333333333333333))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 2050000.0) {
		tmp = x * -sqrt(0.3333333333333333);
	} else if (z <= 2.6e+23) {
		tmp = sqrt((z / (3.0 / z)));
	} else if (z <= 3.6e+112) {
		tmp = -x / sqrt(3.0);
	} 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 (z <= 2050000.0d0) then
        tmp = x * -sqrt(0.3333333333333333d0)
    else if (z <= 2.6d+23) then
        tmp = sqrt((z / (3.0d0 / z)))
    else if (z <= 3.6d+112) then
        tmp = -x / sqrt(3.0d0)
    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 (z <= 2050000.0) {
		tmp = x * -Math.sqrt(0.3333333333333333);
	} else if (z <= 2.6e+23) {
		tmp = Math.sqrt((z / (3.0 / z)));
	} else if (z <= 3.6e+112) {
		tmp = -x / Math.sqrt(3.0);
	} else {
		tmp = z * Math.sqrt(0.3333333333333333);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 2050000.0:
		tmp = x * -math.sqrt(0.3333333333333333)
	elif z <= 2.6e+23:
		tmp = math.sqrt((z / (3.0 / z)))
	elif z <= 3.6e+112:
		tmp = -x / math.sqrt(3.0)
	else:
		tmp = z * math.sqrt(0.3333333333333333)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 2050000.0)
		tmp = Float64(x * Float64(-sqrt(0.3333333333333333)));
	elseif (z <= 2.6e+23)
		tmp = sqrt(Float64(z / Float64(3.0 / z)));
	elseif (z <= 3.6e+112)
		tmp = Float64(Float64(-x) / sqrt(3.0));
	else
		tmp = Float64(z * sqrt(0.3333333333333333));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 2050000.0)
		tmp = x * -sqrt(0.3333333333333333);
	elseif (z <= 2.6e+23)
		tmp = sqrt((z / (3.0 / z)));
	elseif (z <= 3.6e+112)
		tmp = -x / sqrt(3.0);
	else
		tmp = z * sqrt(0.3333333333333333);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 2050000.0], N[(x * (-N[Sqrt[0.3333333333333333], $MachinePrecision])), $MachinePrecision], If[LessEqual[z, 2.6e+23], N[Sqrt[N[(z / N[(3.0 / z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 3.6e+112], N[((-x) / N[Sqrt[3.0], $MachinePrecision]), $MachinePrecision], N[(z * N[Sqrt[0.3333333333333333], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if z < 2.05e6

   1. Initial program 49.3%

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

   2. Taylor expanded in x around -inf 23.4%

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

      1. mul-1-neg 23.4%

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

      2. distribute-rgt-neg-in 23.4%

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

   4. Simplified 23.4%

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

   if 2.05e6 < z < 2.59999999999999992e23

   1. Initial program 77.1%

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

      1. sqrt-div 77.1%

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

      2. div-inv 76.3%

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

      3. associate-+l+ 76.3%

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

      4. add-sqr-sqrt 76.3%

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

      5. hypot-def 98.8%

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

      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/ 100.0%

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

      2. *-rgt-identity 100.0%

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

      3. hypot-def 100.0%

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

      4. +-commutative 100.0%

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

      5. hypot-def 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in z around inf 52.2%

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

      1. add-sqr-sqrt 51.4%

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

      2. sqrt-unprod 52.2%

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

      3. frac-times 52.2%

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

      4. add-sqr-sqrt 52.2%

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

   8. Applied egg-rr 52.2%

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

      1. associate-/l* 52.2%

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

   10. Simplified 52.2%

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

   if 2.59999999999999992e23 < z < 3.6e112

   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 42.3%

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

      2. div-inv 42.0%

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

      3. associate-+l+ 42.0%

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

      4. add-sqr-sqrt 42.0%

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

      5. hypot-def 68.3%

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

      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 68.9%

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

      4. +-commutative 68.9%

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

      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 x around -inf 35.0%

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

      1. mul-1-neg 35.0%

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

      2. distribute-neg-frac 35.0%

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

   8. Simplified 35.0%

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

   if 3.6e112 < z

   1. Initial program 17.6%

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

   2. Taylor expanded in z around inf 83.9%

      \[\leadsto \color{blue}{z \cdot \sqrt{0.3333333333333333}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 33.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2050000:\\ \;\;\;\;x \cdot \left(-\sqrt{0.3333333333333333}\right)\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{+23}:\\ \;\;\;\;\sqrt{\frac{z}{\frac{3}{z}}}\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{+112}:\\ \;\;\;\;\frac{-x}{\sqrt{3}}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \sqrt{0.3333333333333333}\\ \end{array} \]

Alternative 9: 28.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1700000:\\ \;\;\;\;x \cdot \left(-\sqrt{0.3333333333333333}\right)\\ \mathbf{elif}\;z \leq 9.6 \cdot 10^{+22}:\\ \;\;\;\;\sqrt{\frac{z \cdot z}{3}}\\ \mathbf{elif}\;z \leq 7.2 \cdot 10^{+111}:\\ \;\;\;\;\frac{-x}{\sqrt{3}}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \sqrt{0.3333333333333333}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 1700000.0)
   (* x (- (sqrt 0.3333333333333333)))
   (if (<= z 9.6e+22)
     (sqrt (/ (* z z) 3.0))
     (if (<= z 7.2e+111)
       (/ (- x) (sqrt 3.0))
       (* z (sqrt 0.3333333333333333))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 1700000.0) {
		tmp = x * -sqrt(0.3333333333333333);
	} else if (z <= 9.6e+22) {
		tmp = sqrt(((z * z) / 3.0));
	} else if (z <= 7.2e+111) {
		tmp = -x / sqrt(3.0);
	} 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 (z <= 1700000.0d0) then
        tmp = x * -sqrt(0.3333333333333333d0)
    else if (z <= 9.6d+22) then
        tmp = sqrt(((z * z) / 3.0d0))
    else if (z <= 7.2d+111) then
        tmp = -x / sqrt(3.0d0)
    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 (z <= 1700000.0) {
		tmp = x * -Math.sqrt(0.3333333333333333);
	} else if (z <= 9.6e+22) {
		tmp = Math.sqrt(((z * z) / 3.0));
	} else if (z <= 7.2e+111) {
		tmp = -x / Math.sqrt(3.0);
	} else {
		tmp = z * Math.sqrt(0.3333333333333333);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 1700000.0:
		tmp = x * -math.sqrt(0.3333333333333333)
	elif z <= 9.6e+22:
		tmp = math.sqrt(((z * z) / 3.0))
	elif z <= 7.2e+111:
		tmp = -x / math.sqrt(3.0)
	else:
		tmp = z * math.sqrt(0.3333333333333333)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 1700000.0)
		tmp = Float64(x * Float64(-sqrt(0.3333333333333333)));
	elseif (z <= 9.6e+22)
		tmp = sqrt(Float64(Float64(z * z) / 3.0));
	elseif (z <= 7.2e+111)
		tmp = Float64(Float64(-x) / sqrt(3.0));
	else
		tmp = Float64(z * sqrt(0.3333333333333333));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 1700000.0)
		tmp = x * -sqrt(0.3333333333333333);
	elseif (z <= 9.6e+22)
		tmp = sqrt(((z * z) / 3.0));
	elseif (z <= 7.2e+111)
		tmp = -x / sqrt(3.0);
	else
		tmp = z * sqrt(0.3333333333333333);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 1700000.0], N[(x * (-N[Sqrt[0.3333333333333333], $MachinePrecision])), $MachinePrecision], If[LessEqual[z, 9.6e+22], N[Sqrt[N[(N[(z * z), $MachinePrecision] / 3.0), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 7.2e+111], N[((-x) / N[Sqrt[3.0], $MachinePrecision]), $MachinePrecision], N[(z * N[Sqrt[0.3333333333333333], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if z < 1.7e6

   1. Initial program 49.3%

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

   2. Taylor expanded in x around -inf 23.4%

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

      1. mul-1-neg 23.4%

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

      2. distribute-rgt-neg-in 23.4%

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

   4. Simplified 23.4%

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

   if 1.7e6 < z < 9.6e22

   1. Initial program 77.1%

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

   2. Taylor expanded in z around inf 52.2%

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

      1. unpow2 52.2%

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

   4. Simplified 52.2%

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

   if 9.6e22 < z < 7.2000000000000004e111

   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 42.3%

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

      2. div-inv 42.0%

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

      3. associate-+l+ 42.0%

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

      4. add-sqr-sqrt 42.0%

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

      5. hypot-def 68.3%

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

      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 68.9%

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

      4. +-commutative 68.9%

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

      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 x around -inf 35.0%

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

      1. mul-1-neg 35.0%

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

      2. distribute-neg-frac 35.0%

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

   8. Simplified 35.0%

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

   if 7.2000000000000004e111 < z

   1. Initial program 17.6%

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

   2. Taylor expanded in z around inf 83.9%

      \[\leadsto \color{blue}{z \cdot \sqrt{0.3333333333333333}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 33.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1700000:\\ \;\;\;\;x \cdot \left(-\sqrt{0.3333333333333333}\right)\\ \mathbf{elif}\;z \leq 9.6 \cdot 10^{+22}:\\ \;\;\;\;\sqrt{\frac{z \cdot z}{3}}\\ \mathbf{elif}\;z \leq 7.2 \cdot 10^{+111}:\\ \;\;\;\;\frac{-x}{\sqrt{3}}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \sqrt{0.3333333333333333}\\ \end{array} \]

Alternative 10: 18.7% 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 44.7%

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

2. Taylor expanded in z around inf 18.2%

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

3. Final simplification 18.2%

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

Alternative 11: 18.7% 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 44.7%

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

   1. sqrt-div 44.6%

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

   2. div-inv 44.3%

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

   3. associate-+l+ 44.3%

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

   4. add-sqr-sqrt 44.3%

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

   5. hypot-def 59.7%

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

   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/ 99.4%

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

   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 60.2%

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

   4. +-commutative 60.2%

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

   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. Taylor expanded in z around inf 18.2%

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

7. Final simplification 18.2%

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

Developer target: 62.1% 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 2023171 
(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)))