FRP.Yampa.Vector3:vector3Rho from Yampa-0.10.2

Percentage Accurate: 44.8% → 99.3%

Time: 2.5s

Alternatives: 4

Speedup: 111.0×

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

Specification

Math

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

FPCore

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

C

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

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)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 44.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \mathsf{hypot}\left(z, x\right) \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z) :precision binary64 (hypot z x))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	return hypot(z, x);
}

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	return Math.hypot(z, x);
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	return math.hypot(z, x)

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	return hypot(z, x)
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp = code(x, y, z)
	tmp = hypot(z, x);
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := N[Sqrt[z ^ 2 + x ^ 2], $MachinePrecision]

Derivation

1. Initial program 44.8%

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

2. Taylor expanded in y around 0 29.8%

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

   1. unpow2 29.8%

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

   2. unpow2 29.8%

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

   3. hypot-def 69.5%

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

4. Simplified 69.5%

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

5. Final simplification 69.5%

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

Alternative 2: 78.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 1780000:\\ \;\;\;\;\mathsf{hypot}\left(y, x\right)\\ \mathbf{elif}\;z \leq 7.4 \cdot 10^{+22}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 7.2 \cdot 10^{+111}:\\ \;\;\;\;\mathsf{hypot}\left(y, x\right)\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= z 1780000.0)
   (hypot y x)
   (if (<= z 7.4e+22) z (if (<= z 7.2e+111) (hypot y x) z))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (z <= 1780000.0) {
		tmp = hypot(y, x);
	} else if (z <= 7.4e+22) {
		tmp = z;
	} else if (z <= 7.2e+111) {
		tmp = hypot(y, x);
	} else {
		tmp = z;
	}
	return tmp;
}

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (z <= 1780000.0) {
		tmp = Math.hypot(y, x);
	} else if (z <= 7.4e+22) {
		tmp = z;
	} else if (z <= 7.2e+111) {
		tmp = Math.hypot(y, x);
	} else {
		tmp = z;
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if z <= 1780000.0:
		tmp = math.hypot(y, x)
	elif z <= 7.4e+22:
		tmp = z
	elif z <= 7.2e+111:
		tmp = math.hypot(y, x)
	else:
		tmp = z
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (z <= 1780000.0)
		tmp = hypot(y, x);
	elseif (z <= 7.4e+22)
		tmp = z;
	elseif (z <= 7.2e+111)
		tmp = hypot(y, x);
	else
		tmp = z;
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 1780000.0)
		tmp = hypot(y, x);
	elseif (z <= 7.4e+22)
		tmp = z;
	elseif (z <= 7.2e+111)
		tmp = hypot(y, x);
	else
		tmp = z;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[z, 1780000.0], N[Sqrt[y ^ 2 + x ^ 2], $MachinePrecision], If[LessEqual[z, 7.4e+22], z, If[LessEqual[z, 7.2e+111], N[Sqrt[y ^ 2 + x ^ 2], $MachinePrecision], z]]]

Derivation

1. Split input into 2 regimes

2. if z < 1.78e6 or 7.3999999999999996e22 < z < 7.2000000000000004e111

   1. Initial program 48.8%

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

   2. Taylor expanded in z around 0 40.1%

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

      1. unpow2 40.1%

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

      2. unpow2 40.1%

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

      3. hypot-def 78.4%

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

   4. Simplified 78.4%

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

   if 1.78e6 < z < 7.3999999999999996e22 or 7.2000000000000004e111 < z

   1. Initial program 23.6%

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

   2. Taylor expanded in z around inf 81.3%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1780000:\\ \;\;\;\;\mathsf{hypot}\left(y, x\right)\\ \mathbf{elif}\;z \leq 7.4 \cdot 10^{+22}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 7.2 \cdot 10^{+111}:\\ \;\;\;\;\mathsf{hypot}\left(y, x\right)\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]

Alternative 3: 77.8% accurate, 13.6× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 2050000:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{+23}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{+112}:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= z 2050000.0)
   (- x)
   (if (<= z 2.6e+23) z (if (<= z 3.5e+112) (- x) z))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (z <= 2050000.0) {
		tmp = -x;
	} else if (z <= 2.6e+23) {
		tmp = z;
	} else if (z <= 3.5e+112) {
		tmp = -x;
	} else {
		tmp = z;
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
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
    else if (z <= 2.6d+23) then
        tmp = z
    else if (z <= 3.5d+112) then
        tmp = -x
    else
        tmp = z
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (z <= 2050000.0) {
		tmp = -x;
	} else if (z <= 2.6e+23) {
		tmp = z;
	} else if (z <= 3.5e+112) {
		tmp = -x;
	} else {
		tmp = z;
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if z <= 2050000.0:
		tmp = -x
	elif z <= 2.6e+23:
		tmp = z
	elif z <= 3.5e+112:
		tmp = -x
	else:
		tmp = z
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (z <= 2050000.0)
		tmp = Float64(-x);
	elseif (z <= 2.6e+23)
		tmp = z;
	elseif (z <= 3.5e+112)
		tmp = Float64(-x);
	else
		tmp = z;
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 2050000.0)
		tmp = -x;
	elseif (z <= 2.6e+23)
		tmp = z;
	elseif (z <= 3.5e+112)
		tmp = -x;
	else
		tmp = z;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[z, 2050000.0], (-x), If[LessEqual[z, 2.6e+23], z, If[LessEqual[z, 3.5e+112], (-x), z]]]

Derivation

1. Split input into 2 regimes

2. if z < 2.05e6 or 2.59999999999999992e23 < z < 3.49999999999999997e112

   1. Initial program 48.8%

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

   2. Taylor expanded in x around -inf 24.7%

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

      1. mul-1-neg 24.7%

         \[\leadsto \color{blue}{-x} \]

   4. Simplified 24.7%

      \[\leadsto \color{blue}{-x} \]

   if 2.05e6 < z < 2.59999999999999992e23 or 3.49999999999999997e112 < z

   1. Initial program 23.6%

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

   2. Taylor expanded in z around inf 81.3%

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

4. Final simplification 33.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2050000:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{+23}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{+112}:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]

Alternative 4: 50.9% accurate, 111.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ z \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z) :precision binary64 z)

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	return z;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = z
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	return z;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	return z

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	return z
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp = code(x, y, z)
	tmp = z;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := z

Derivation

1. Initial program 44.8%

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

2. Taylor expanded in z around inf 18.3%

   \[\leadsto \color{blue}{z} \]

3. Final simplification 18.3%

   \[\leadsto z \]

Developer target: 70.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023171 
(FPCore (x y z)
  :name "FRP.Yampa.Vector3:vector3Rho from Yampa-0.10.2"
  :precision binary64

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

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