bug366 (missed optimization)

Percentage Accurate: 44.7% → 99.3%

Time: 7.0s

Alternatives: 2

Speedup: 32.0×

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: 44.7% 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: 99.3% accurate, 1.4× speedup

Math

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

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 y_m (/ (* y_m 0.5) 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 fma(y_m, ((y_m * 0.5) / 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 fma(y_m, Float64(Float64(y_m * 0.5) / 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[(y$95$m * N[(N[(y$95$m * 0.5), $MachinePrecision] / z$95$m), $MachinePrecision] + z$95$m), $MachinePrecision]

Derivation

1. Initial program 43.9%

   \[\sqrt{x \cdot x + \left(y \cdot y + z \cdot z\right)} \]
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. distribute-rgt-in N/A

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

   2. *-lft-identity N/A

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

   3. +-commutative N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. associate-*r/ N/A

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

   7. unpow2 N/A

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

   8. times-frac N/A

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

   9. *-inverses N/A

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

   10. associate-/l* N/A

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

   11. *-lft-identity N/A

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

   12. lower-fma.f64 N/A

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

5. Applied rewrites 13.9%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 16.7%

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

Alternative 2: 98.4% accurate, 32.0× speedup

Math

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

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 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 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 = 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 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 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 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 = 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_] := z$95$m

Derivation

1. Initial program 43.9%

   \[\sqrt{x \cdot x + \left(y \cdot y + z \cdot z\right)} \]
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. distribute-rgt-in N/A

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

   2. *-lft-identity N/A

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

   3. +-commutative N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. associate-*r/ N/A

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

   7. unpow2 N/A

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

   8. times-frac N/A

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

   9. *-inverses N/A

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

   10. associate-/l* N/A

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

   11. *-lft-identity N/A

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

   12. lower-fma.f64 N/A

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

5. Applied rewrites 13.9%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 16.7%

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

10. Applied rewrites 16.2%

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

11. Taylor expanded in z around -inf

    \[\leadsto -1 \cdot \left(z \cdot \color{blue}{{\left(\sqrt{-1}\right)}^{2}}\right) \]
12. Step-by-step derivation

13. Applied rewrites 16.2%

    \[\leadsto z \]
14. 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 2024234 
(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)))))