bug366 (missed optimization)

Percentage Accurate: 43.9% → 100.0%

Time: 7.2s

Alternatives: 7

Speedup: 1.4×

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: 43.9% 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, 0.3× speedup

Math

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

FPCore

z_m = (fabs.f64 z)
y_m = (fabs.f64 y)
x_m = (fabs.f64 x)
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
(FPCore (x_m y_m z_m) :precision binary64 (hypot z_m y_m))

C

z_m = fabs(z);
y_m = fabs(y);
x_m = fabs(x);
assert(x_m < y_m && y_m < z_m);
double code(double x_m, double y_m, double z_m) {
	return hypot(z_m, y_m);
}

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

z_m = N[Abs[z], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
code[x$95$m_, y$95$m_, z$95$m_] := N[Sqrt[z$95$m ^ 2 + y$95$m ^ 2], $MachinePrecision]

Derivation

1. Initial program 39.8%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. unpow2 N/A

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

   4. lower-hypot.f64 65.3

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

5. Applied rewrites 65.3%

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

Alternative 2: 99.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ y_m = \left|y\right| \\ x_m = \left|x\right| \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ \mathsf{fma}\left(1, \mathsf{fma}\left(\frac{x\_m}{z\_m}, x\_m, \frac{y\_m}{z\_m} \cdot y\_m\right) \cdot 0.5, z\_m\right) \end{array} \]

FPCore

z_m = (fabs.f64 z)
y_m = (fabs.f64 y)
x_m = (fabs.f64 x)
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
(FPCore (x_m y_m z_m)
 :precision binary64
 (fma 1.0 (* (fma (/ x_m z_m) x_m (* (/ y_m z_m) y_m)) 0.5) z_m))

C

z_m = fabs(z);
y_m = fabs(y);
x_m = fabs(x);
assert(x_m < y_m && y_m < z_m);
double code(double x_m, double y_m, double z_m) {
	return fma(1.0, (fma((x_m / z_m), x_m, ((y_m / z_m) * y_m)) * 0.5), z_m);
}

Julia

z_m = abs(z)
y_m = abs(y)
x_m = abs(x)
x_m, y_m, z_m = sort([x_m, y_m, z_m])
function code(x_m, y_m, z_m)
	return fma(1.0, Float64(fma(Float64(x_m / z_m), x_m, Float64(Float64(y_m / z_m) * y_m)) * 0.5), z_m)
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
code[x$95$m_, y$95$m_, z$95$m_] := N[(1.0 * N[(N[(N[(x$95$m / z$95$m), $MachinePrecision] * x$95$m + N[(N[(y$95$m / z$95$m), $MachinePrecision] * y$95$m), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] + z$95$m), $MachinePrecision]

Derivation

1. Initial program 39.8%

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

3. Taylor expanded in z around inf

   \[\leadsto \color{blue}{z \cdot \left(1 + \frac{1}{2} \cdot \frac{{x}^{2} + {y}^{2}}{{z}^{2}}\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto z \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{{x}^{2} + {y}^{2}}{{z}^{2}} + 1\right)} \]

   2. distribute-lft-in N/A

      \[\leadsto \color{blue}{z \cdot \left(\frac{1}{2} \cdot \frac{{x}^{2} + {y}^{2}}{{z}^{2}}\right) + z \cdot 1} \]

   3. associate-*r/ N/A

      \[\leadsto z \cdot \color{blue}{\frac{\frac{1}{2} \cdot \left({x}^{2} + {y}^{2}\right)}{{z}^{2}}} + z \cdot 1 \]

   4. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{z \cdot \left(\frac{1}{2} \cdot \left({x}^{2} + {y}^{2}\right)\right)}{{z}^{2}}} + z \cdot 1 \]

   5. unpow2 N/A

      \[\leadsto \frac{z \cdot \left(\frac{1}{2} \cdot \left({x}^{2} + {y}^{2}\right)\right)}{\color{blue}{z \cdot z}} + z \cdot 1 \]

   6. times-frac N/A

      \[\leadsto \color{blue}{\frac{z}{z} \cdot \frac{\frac{1}{2} \cdot \left({x}^{2} + {y}^{2}\right)}{z}} + z \cdot 1 \]

   7. *-rgt-identity N/A

      \[\leadsto \frac{z}{z} \cdot \frac{\frac{1}{2} \cdot \left({x}^{2} + {y}^{2}\right)}{z} + \color{blue}{z} \]

   8. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z}{z}, \frac{\frac{1}{2} \cdot \left({x}^{2} + {y}^{2}\right)}{z}, z\right)} \]

5. Applied rewrites 12.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z}{z}, \frac{\mathsf{fma}\left(y, y, x \cdot x\right)}{z} \cdot 0.5, z\right)} \]
6. Step-by-step derivation

7. Applied rewrites 14.3%

   \[\leadsto \mathsf{fma}\left(\frac{z}{z}, \mathsf{fma}\left(y, \frac{y}{z}, \frac{x \cdot x}{z}\right) \cdot 0.5, z\right) \]

8. Taylor expanded in z around 0

   \[\leadsto \mathsf{fma}\left(1, \color{blue}{\mathsf{fma}\left(y, \frac{y}{z}, \frac{x \cdot x}{z}\right)} \cdot \frac{1}{2}, z\right) \]
9. Step-by-step derivation

10. Applied rewrites 14.3%

    \[\leadsto \mathsf{fma}\left(1, \color{blue}{\mathsf{fma}\left(y, \frac{y}{z}, \frac{x \cdot x}{z}\right)} \cdot 0.5, z\right) \]
11. Step-by-step derivation

12. Applied rewrites 15.0%

    \[\leadsto \mathsf{fma}\left(1, \mathsf{fma}\left(\frac{x}{z}, x, \frac{y}{z} \cdot y\right) \cdot 0.5, z\right) \]
13. Add Preprocessing

Alternative 3: 98.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ y_m = \left|y\right| \\ x_m = \left|x\right| \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ \mathsf{fma}\left(\frac{0.5}{z\_m} \cdot x\_m, x\_m, z\_m\right) \end{array} \]

FPCore

z_m = (fabs.f64 z)
y_m = (fabs.f64 y)
x_m = (fabs.f64 x)
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
(FPCore (x_m y_m z_m) :precision binary64 (fma (* (/ 0.5 z_m) x_m) x_m z_m))

C

z_m = fabs(z);
y_m = fabs(y);
x_m = fabs(x);
assert(x_m < y_m && y_m < z_m);
double code(double x_m, double y_m, double z_m) {
	return fma(((0.5 / z_m) * x_m), x_m, z_m);
}

Julia

z_m = abs(z)
y_m = abs(y)
x_m = abs(x)
x_m, y_m, z_m = sort([x_m, y_m, z_m])
function code(x_m, y_m, z_m)
	return fma(Float64(Float64(0.5 / z_m) * x_m), x_m, z_m)
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
code[x$95$m_, y$95$m_, z$95$m_] := N[(N[(N[(0.5 / z$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m + z$95$m), $MachinePrecision]

Derivation

1. Initial program 39.8%

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

3. Taylor expanded in z around inf

   \[\leadsto \color{blue}{z \cdot \left(1 + \frac{1}{2} \cdot \frac{{x}^{2} + {y}^{2}}{{z}^{2}}\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto z \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{{x}^{2} + {y}^{2}}{{z}^{2}} + 1\right)} \]

   2. distribute-lft-in N/A

      \[\leadsto \color{blue}{z \cdot \left(\frac{1}{2} \cdot \frac{{x}^{2} + {y}^{2}}{{z}^{2}}\right) + z \cdot 1} \]

   3. associate-*r/ N/A

      \[\leadsto z \cdot \color{blue}{\frac{\frac{1}{2} \cdot \left({x}^{2} + {y}^{2}\right)}{{z}^{2}}} + z \cdot 1 \]

   4. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{z \cdot \left(\frac{1}{2} \cdot \left({x}^{2} + {y}^{2}\right)\right)}{{z}^{2}}} + z \cdot 1 \]

   5. unpow2 N/A

      \[\leadsto \frac{z \cdot \left(\frac{1}{2} \cdot \left({x}^{2} + {y}^{2}\right)\right)}{\color{blue}{z \cdot z}} + z \cdot 1 \]

   6. times-frac N/A

      \[\leadsto \color{blue}{\frac{z}{z} \cdot \frac{\frac{1}{2} \cdot \left({x}^{2} + {y}^{2}\right)}{z}} + z \cdot 1 \]

   7. *-rgt-identity N/A

      \[\leadsto \frac{z}{z} \cdot \frac{\frac{1}{2} \cdot \left({x}^{2} + {y}^{2}\right)}{z} + \color{blue}{z} \]

   8. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z}{z}, \frac{\frac{1}{2} \cdot \left({x}^{2} + {y}^{2}\right)}{z}, z\right)} \]

5. Applied rewrites 12.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z}{z}, \frac{\mathsf{fma}\left(y, y, x \cdot x\right)}{z} \cdot 0.5, z\right)} \]

6. Taylor expanded in y around 0

   \[\leadsto z + \color{blue}{\frac{1}{2} \cdot \frac{{x}^{2}}{z}} \]
7. Step-by-step derivation

8. Applied rewrites 14.3%

   \[\leadsto \mathsf{fma}\left(\frac{0.5}{z}, \color{blue}{x \cdot x}, z\right) \]
9. Step-by-step derivation

10. Applied rewrites 15.0%

    \[\leadsto \mathsf{fma}\left(\frac{0.5}{z} \cdot x, x, z\right) \]
11. Add Preprocessing

Alternative 4: 43.9% accurate, 1.5× speedup

Math

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

FPCore

z_m = (fabs.f64 z)
y_m = (fabs.f64 y)
x_m = (fabs.f64 x)
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
(FPCore (x_m y_m z_m) :precision binary64 (sqrt (fma z_m z_m (* y_m y_m))))

C

z_m = fabs(z);
y_m = fabs(y);
x_m = fabs(x);
assert(x_m < y_m && y_m < z_m);
double code(double x_m, double y_m, double z_m) {
	return sqrt(fma(z_m, z_m, (y_m * y_m)));
}

Julia

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

Wolfram

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

Derivation

1. Initial program 39.8%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. lower-fma.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 28.6

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

5. Applied rewrites 28.6%

   \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(z, z, y \cdot y\right)}} \]
6. Add Preprocessing

Alternative 5: 43.2% accurate, 2.0× speedup

Math

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

FPCore

z_m = (fabs.f64 z)
y_m = (fabs.f64 y)
x_m = (fabs.f64 x)
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
(FPCore (x_m y_m z_m) :precision binary64 (sqrt (* z_m z_m)))

C

z_m = fabs(z);
y_m = fabs(y);
x_m = fabs(x);
assert(x_m < y_m && y_m < z_m);
double code(double x_m, double y_m, double z_m) {
	return sqrt((z_m * z_m));
}

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.8%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. lower-fma.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 28.6

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

5. Applied rewrites 28.6%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 14.1%

   \[\leadsto \sqrt{z \cdot \color{blue}{z}} \]
9. Add Preprocessing

Alternative 6: 5.5% accurate, 2.0× speedup

Math

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

FPCore

z_m = (fabs.f64 z)
y_m = (fabs.f64 y)
x_m = (fabs.f64 x)
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
(FPCore (x_m y_m z_m) :precision binary64 (sqrt (* y_m y_m)))

C

z_m = fabs(z);
y_m = fabs(y);
x_m = fabs(x);
assert(x_m < y_m && y_m < z_m);
double code(double x_m, double y_m, double z_m) {
	return sqrt((y_m * y_m));
}

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.8%

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

3. Taylor expanded in z around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. lower-fma.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 30.1

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

5. Applied rewrites 30.1%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 18.3%

   \[\leadsto \sqrt{y \cdot \color{blue}{y}} \]
9. Add Preprocessing

Alternative 7: 1.7% accurate, 10.7× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ y_m = \left|y\right| \\ x_m = \left|x\right| \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ -x\_m \end{array} \]

FPCore

z_m = (fabs.f64 z)
y_m = (fabs.f64 y)
x_m = (fabs.f64 x)
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
(FPCore (x_m y_m z_m) :precision binary64 (- x_m))

C

z_m = fabs(z);
y_m = fabs(y);
x_m = fabs(x);
assert(x_m < y_m && y_m < z_m);
double code(double x_m, double y_m, double z_m) {
	return -x_m;
}

Fortran

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

Java

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

Python

z_m = math.fabs(z)
y_m = math.fabs(y)
x_m = math.fabs(x)
[x_m, y_m, z_m] = sort([x_m, y_m, z_m])
def code(x_m, y_m, z_m):
	return -x_m

Julia

z_m = abs(z)
y_m = abs(y)
x_m = abs(x)
x_m, y_m, z_m = sort([x_m, y_m, z_m])
function code(x_m, y_m, z_m)
	return Float64(-x_m)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.8%

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

3. Taylor expanded in x around -inf

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

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(x\right)} \]

   2. lower-neg.f64 19.6

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

5. Applied rewrites 19.6%

   \[\leadsto \color{blue}{-x} \]
6. Add Preprocessing

Developer Target 1: 100.0% accurate, 0.2× 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 2024337 
(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)))))