bug366 (missed optimization)

Percentage Accurate: 29.6% → 29.6%
Time: 1.2s
Alternatives: 1
Speedup: 1.0×

Specification

?
\[\begin{array}{l} \\ \sqrt{x \cdot x + \left(y \cdot y + z \cdot z\right)} \end{array} \]
(FPCore (x y z) :precision binary64 (sqrt (+ (* x x) (+ (* y y) (* z z)))))
double code(double x, double y, double z) {
	return sqrt(((x * x) + ((y * y) + (z * z))));
}

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

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

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

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

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

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

\begin{array}{l}

\\
\sqrt{x \cdot x + \left(y \cdot y + z \cdot z\right)}
\end{array}

Sampling outcomes in binary64 precision:

Local Percentage Accuracy vs ?

The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples.

Accuracy vs Speed?

Herbie found 1 alternatives:

AlternativeAccuracySpeedup
The accuracy (vertical axis) and speed (horizontal axis) of each alternatives. Up and to the right is better. The red square shows the initial program, and each blue circle shows an alternative.The line shows the best available speed-accuracy tradeoffs.

Initial Program: 29.6% accurate, 1.0× speedup?

\[\begin{array}{l} \\ \sqrt{x \cdot x + \left(y \cdot y + z \cdot z\right)} \end{array} \]
(FPCore (x y z) :precision binary64 (sqrt (+ (* x x) (+ (* y y) (* z z)))))
double code(double x, double y, double z) {
	return sqrt(((x * x) + ((y * y) + (z * z))));
}

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

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

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

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

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

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

\begin{array}{l}

\\
\sqrt{x \cdot x + \left(y \cdot y + z \cdot z\right)}
\end{array}

Alternative 1: 29.6% accurate, 1.0× speedup?

\[\begin{array}{l} x_m = \left|x\right| \\ [x_m, y, z] = \mathsf{sort}([x_m, y, z])\\ \\ \sqrt{\left(z \cdot z + y \cdot y\right) + x\_m \cdot x\_m} \end{array} \]
x_m = (fabs.f64 x)
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
(FPCore (x_m y z)
 :precision binary64
 (sqrt (+ (+ (* z z) (* y y)) (* x_m x_m))))
x_m = fabs(x);
assert(x_m < y && y < z);
double code(double x_m, double y, double z) {
	return sqrt((((z * z) + (y * y)) + (x_m * x_m)));
}

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

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

x_m = math.fabs(x)
[x_m, y, z] = sort([x_m, y, z])
def code(x_m, y, z):
	return math.sqrt((((z * z) + (y * y)) + (x_m * x_m)))

x_m = abs(x)
x_m, y, z = sort([x_m, y, z])
function code(x_m, y, z)
	return sqrt(Float64(Float64(Float64(z * z) + Float64(y * y)) + Float64(x_m * x_m)))
end

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

x_m = N[Abs[x], $MachinePrecision]
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
code[x$95$m_, y_, z_] := N[Sqrt[N[(N[(N[(z * z), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] + N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

\begin{array}{l}
x_m = \left|x\right|
\\
[x_m, y, z] = \mathsf{sort}([x_m, y, z])\\
\\
\sqrt{\left(z \cdot z + y \cdot y\right) + x\_m \cdot x\_m}
\end{array}
Derivation

  1. Initial program 48.5%

    \[\sqrt{x \cdot x + \left(y \cdot y + z \cdot z\right)} \]
  2. Add Preprocessing

  3. Final simplification48.5%

    \[\leadsto \sqrt{\left(z \cdot z + y \cdot y\right) + x \cdot x} \]
  4. Add Preprocessing

Developer Target 1: 100.0% accurate, 0.2× speedup?

\[\begin{array}{l} \\ \mathsf{hypot}\left(x, \mathsf{hypot}\left(y, z\right)\right) \end{array} \]
(FPCore (x y z) :precision binary64 (hypot x (hypot y z)))
double code(double x, double y, double z) {
	return hypot(x, hypot(y, z));
}

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

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

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

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

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

\begin{array}{l}

\\
\mathsf{hypot}\left(x, \mathsf{hypot}\left(y, z\right)\right)
\end{array}

Reproduce

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herbie shell --seed 2024343 
(FPCore (x y z)
  :name "bug366 (missed optimization)"
  :precision binary64

  :alt
  (! :herbie-platform default (hypot x (hypot y z)))

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