bug366 (missed optimization)

Percentage Accurate: 45.1% → 83.5%

Time: 2.3s

Alternatives: 5

Speedup: 111.0×

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

Specification

Math

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

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(x * x) + Float64(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[(x * x), $MachinePrecision] + N[(N[(y * y), $MachinePrecision] + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Initial Program: 45.1% accurate, 1.0× speedup

Math

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

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(x * x) + Float64(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[(x * x), $MachinePrecision] + N[(N[(y * y), $MachinePrecision] + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Alternative 1: 83.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 1.85 \cdot 10^{-96}:\\ \;\;\;\;\mathsf{hypot}\left(y, x\right)\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{+148}:\\ \;\;\;\;\sqrt{x \cdot x + \left(y \cdot y + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(z, y\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= z 1.85e-96)
   (hypot y x)
   (if (<= z 1.4e+148) (sqrt (+ (* x x) (+ (* y y) (* z z)))) (hypot z y))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.85e-96) {
		tmp = hypot(y, x);
	} else if (z <= 1.4e+148) {
		tmp = sqrt(((x * x) + ((y * y) + (z * z))));
	} else {
		tmp = hypot(z, y);
	}
	return tmp;
}

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.85e-96) {
		tmp = Math.hypot(y, x);
	} else if (z <= 1.4e+148) {
		tmp = Math.sqrt(((x * x) + ((y * y) + (z * z))));
	} else {
		tmp = Math.hypot(z, y);
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if z <= 1.85e-96:
		tmp = math.hypot(y, x)
	elif z <= 1.4e+148:
		tmp = math.sqrt(((x * x) + ((y * y) + (z * z))))
	else:
		tmp = math.hypot(z, y)
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (z <= 1.85e-96)
		tmp = hypot(y, x);
	elseif (z <= 1.4e+148)
		tmp = sqrt(Float64(Float64(x * x) + Float64(Float64(y * y) + Float64(z * z))));
	else
		tmp = hypot(z, y);
	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 <= 1.85e-96)
		tmp = hypot(y, x);
	elseif (z <= 1.4e+148)
		tmp = sqrt(((x * x) + ((y * y) + (z * z))));
	else
		tmp = hypot(z, y);
	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, 1.85e-96], N[Sqrt[y ^ 2 + x ^ 2], $MachinePrecision], If[LessEqual[z, 1.4e+148], N[Sqrt[N[(N[(x * x), $MachinePrecision] + N[(N[(y * y), $MachinePrecision] + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[z ^ 2 + y ^ 2], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < 1.84999999999999993e-96

   1. Initial program 41.9%

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

   2. Taylor expanded in z around 0 36.5%

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

      1. unpow2 36.5%

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

      2. unpow2 36.5%

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

      3. hypot-def 77.5%

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

   4. Simplified 77.5%

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

   if 1.84999999999999993e-96 < z < 1.3999999999999999e148

   1. Initial program 62.6%

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

   if 1.3999999999999999e148 < z

   1. Initial program 10.2%

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

   2. Taylor expanded in x around 0 7.4%

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

      1. unpow2 7.4%

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

      2. unpow2 7.4%

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

      3. hypot-def 88.2%

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

   4. Simplified 88.2%

      \[\leadsto \color{blue}{\mathsf{hypot}\left(z, y\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.85 \cdot 10^{-96}:\\ \;\;\;\;\mathsf{hypot}\left(y, x\right)\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{+148}:\\ \;\;\;\;\sqrt{x \cdot x + \left(y \cdot y + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(z, y\right)\\ \end{array} \]

Alternative 2: 79.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 9.2 \cdot 10^{-23}:\\ \;\;\;\;\mathsf{hypot}\left(y, x\right)\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z) :precision binary64 (if (<= z 9.2e-23) (hypot y x) z))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (z <= 9.2e-23) {
		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 <= 9.2e-23) {
		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 <= 9.2e-23:
		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 <= 9.2e-23)
		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 <= 9.2e-23)
		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, 9.2e-23], N[Sqrt[y ^ 2 + x ^ 2], $MachinePrecision], z]

Derivation

1. Split input into 2 regimes

2. if z < 9.2000000000000004e-23

   1. Initial program 45.0%

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

   2. Taylor expanded in z around 0 38.9%

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

      1. unpow2 38.9%

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

      2. unpow2 38.9%

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

      3. hypot-def 78.9%

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

   4. Simplified 78.9%

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

   if 9.2000000000000004e-23 < z

   1. Initial program 32.8%

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

   2. Taylor expanded in z around inf 59.6%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 9.2 \cdot 10^{-23}:\\ \;\;\;\;\mathsf{hypot}\left(y, x\right)\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]

Alternative 3: 80.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 1.1 \cdot 10^{-22}:\\ \;\;\;\;\mathsf{hypot}\left(y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(z, y\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= z 1.1e-22) (hypot y x) (hypot z y)))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.1e-22) {
		tmp = hypot(y, x);
	} else {
		tmp = hypot(z, y);
	}
	return tmp;
}

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.1e-22) {
		tmp = Math.hypot(y, x);
	} else {
		tmp = Math.hypot(z, y);
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if z <= 1.1e-22:
		tmp = math.hypot(y, x)
	else:
		tmp = math.hypot(z, y)
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (z <= 1.1e-22)
		tmp = hypot(y, x);
	else
		tmp = hypot(z, y);
	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 <= 1.1e-22)
		tmp = hypot(y, x);
	else
		tmp = hypot(z, y);
	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, 1.1e-22], N[Sqrt[y ^ 2 + x ^ 2], $MachinePrecision], N[Sqrt[z ^ 2 + y ^ 2], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.1e-22

   1. Initial program 45.0%

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

   2. Taylor expanded in z around 0 38.9%

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

      1. unpow2 38.9%

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

      2. unpow2 38.9%

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

      3. hypot-def 78.9%

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

   4. Simplified 78.9%

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

   if 1.1e-22 < z

   1. Initial program 32.8%

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

   2. Taylor expanded in x around 0 24.8%

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

      1. unpow2 24.8%

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

      2. unpow2 24.8%

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

      3. hypot-def 80.8%

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

   4. Simplified 80.8%

      \[\leadsto \color{blue}{\mathsf{hypot}\left(z, y\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.1 \cdot 10^{-22}:\\ \;\;\;\;\mathsf{hypot}\left(y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(z, y\right)\\ \end{array} \]

Alternative 4: 79.4% accurate, 27.3× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 1.1 \cdot 10^{-22}:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z) :precision binary64 (if (<= z 1.1e-22) (- x) z))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.1e-22) {
		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 <= 1.1d-22) 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 <= 1.1e-22) {
		tmp = -x;
	} else {
		tmp = z;
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if z <= 1.1e-22:
		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 <= 1.1e-22)
		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 <= 1.1e-22)
		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, 1.1e-22], (-x), z]

Derivation

1. Split input into 2 regimes

2. if z < 1.1e-22

   1. Initial program 45.0%

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

   2. Taylor expanded in x around -inf 20.3%

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

      1. mul-1-neg 20.3%

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

   4. Simplified 20.3%

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

   if 1.1e-22 < z

   1. Initial program 32.8%

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

   2. Taylor expanded in z around inf 59.6%

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

4. Final simplification 29.2%

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

Alternative 5: 51.5% 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 42.2%

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

2. Taylor expanded in z around inf 16.2%

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

3. Final simplification 16.2%

   \[\leadsto z \]

Developer target: 100.0% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x y z) :precision binary64 (hypot x (hypot y z)))

C

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

Java

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

Python

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

Julia

function code(x, y, z)
	return hypot(x, hypot(y, z))
end

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023178 
(FPCore (x y z)
  :name "bug366 (missed optimization)"
  :precision binary64

  :herbie-target
  (hypot x (hypot y z))

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