bug366 (missed optimization)

Percentage Accurate: 45.5% → 100.0%

Time: 3.1s

Alternatives: 4

Speedup: 111.0×

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: 45.5% 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: 100.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 46.5%

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

   1. associate-+r+ 46.5%

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

   2. add-sqr-sqrt 46.5%

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

   3. hypot-def 60.4%

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

   4. hypot-def 100.0%

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

3. Applied egg-rr 100.0%

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

   1. hypot-def 60.4%

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

   2. unpow2 60.4%

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

   3. unpow2 60.4%

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

   4. unpow2 60.4%

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

   5. +-commutative 60.4%

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

   6. unpow2 60.4%

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

   7. hypot-def 100.0%

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

5. Simplified 100.0%

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

6. Final simplification 100.0%

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

Alternative 2: 52.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 46.5%

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

2. Taylor expanded in x around 0 34.6%

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

   1. unpow2 34.6%

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

   2. unpow2 34.6%

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

   3. hypot-def 66.8%

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

4. Simplified 66.8%

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

5. Final simplification 66.8%

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

Alternative 3: 51.3% accurate, 12.3× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ z + \frac{y}{\frac{z}{y}} \cdot 0.5 \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z) :precision binary64 (+ z (* (/ y (/ z y)) 0.5)))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	return z + ((y / (z / y)) * 0.5);
}

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 + ((y / (z / y)) * 0.5d0)
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	return z + ((y / (z / y)) * 0.5);
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	return z + ((y / (z / y)) * 0.5)

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	return Float64(z + Float64(Float64(y / Float64(z / y)) * 0.5))
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp = code(x, y, z)
	tmp = z + ((y / (z / y)) * 0.5);
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := N[(z + N[(N[(y / N[(z / y), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 46.5%

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

2. Taylor expanded in z around inf 14.0%

   \[\leadsto \color{blue}{z + 0.5 \cdot \frac{{x}^{2} + {y}^{2}}{z}} \]
3. Step-by-step derivation

   1. unpow2 14.0%

      \[\leadsto z + 0.5 \cdot \frac{\color{blue}{x \cdot x} + {y}^{2}}{z} \]

   2. unpow2 14.0%

      \[\leadsto z + 0.5 \cdot \frac{x \cdot x + \color{blue}{y \cdot y}}{z} \]

4. Simplified 14.0%

   \[\leadsto \color{blue}{z + 0.5 \cdot \frac{x \cdot x + y \cdot y}{z}} \]

5. Taylor expanded in x around 0 15.0%

   \[\leadsto z + \color{blue}{0.5 \cdot \frac{{y}^{2}}{z}} \]
6. Step-by-step derivation

   1. *-commutative 15.0%

      \[\leadsto z + \color{blue}{\frac{{y}^{2}}{z} \cdot 0.5} \]

   2. unpow2 15.0%

      \[\leadsto z + \frac{\color{blue}{y \cdot y}}{z} \cdot 0.5 \]

   3. associate-/l* 16.1%

      \[\leadsto z + \color{blue}{\frac{y}{\frac{z}{y}}} \cdot 0.5 \]

7. Simplified 16.1%

   \[\leadsto z + \color{blue}{\frac{y}{\frac{z}{y}} \cdot 0.5} \]

8. Final simplification 16.1%

   \[\leadsto z + \frac{y}{\frac{z}{y}} \cdot 0.5 \]

Alternative 4: 50.8% 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 46.5%

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

2. Taylor expanded in z around inf 15.7%

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

3. Final simplification 15.7%

   \[\leadsto z \]

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 2023293 
(FPCore (x y z)
  :name "bug366 (missed optimization)"
  :precision binary64

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

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