bug366 (missed optimization)

Percentage Accurate: 45.3% → 100.0%

Time: 2.6s

Alternatives: 3

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: 45.3% 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, 1.1× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
NOTE: z should be positive before calling this function
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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
NOTE: z should be positive before calling this function
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 46.0%

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

2. Taylor expanded in x around 0 31.5%

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

   1. unpow2 31.5%

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

   2. unpow2 31.5%

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

   3. hypot-def 66.7%

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

4. Simplified 66.7%

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

5. Final simplification 66.7%

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

Alternative 2: 99.3% accurate, 12.3× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
NOTE: z should be positive before calling this function
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

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

Fortran

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
NOTE: z should be positive before calling this function
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

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

Python

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

Julia

x = abs(x)
y = abs(y)
z = abs(z)
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 = abs(x)
y = abs(y)
z = abs(z)
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 should be positive before calling this function
NOTE: y should be positive before calling this function
NOTE: z should be positive before calling this function
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 46.0%

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

2. Taylor expanded in z around inf 16.3%

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

   1. unpow2 16.3%

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

   2. unpow2 16.3%

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

4. Simplified 16.3%

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

5. Taylor expanded in x around 0 17.3%

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

   1. *-commutative 17.3%

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

   2. unpow2 17.3%

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

   3. associate-/l* 17.7%

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

7. Simplified 17.7%

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

8. Final simplification 17.7%

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

Alternative 3: 98.6% accurate, 111.0× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
NOTE: z should be positive before calling this function
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

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 46.0%

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

2. Taylor expanded in z around inf 17.1%

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

3. Final simplification 17.1%

   \[\leadsto z \]

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 2023279 
(FPCore (x y z)
  :name "bug366 (missed optimization)"
  :precision binary64

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

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