bug366 (missed optimization)

Percentage Accurate: 45.5% → 99.2%

Time: 8.5s

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: 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: 99.2% accurate, 1.4× speedup

Math

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

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

Derivation

1. Initial program 45.0%

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

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. associate-*r/ N/A

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

   8. unpow2 N/A

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

   9. times-frac N/A

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

   10. *-inverses N/A

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

   11. associate-/l* N/A

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

   12. *-lft-identity N/A

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

   13. lower-fma.f64 N/A

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

5. Applied rewrites 16.3%

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

6. Taylor expanded in y around 0

   \[\leadsto \mathsf{fma}\left(\frac{{x}^{2}}{z}, \frac{1}{2}, z\right) \]
7. Step-by-step derivation

8. Applied rewrites 16.3%

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

9. Taylor expanded in x around inf

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

11. Applied rewrites 9.3%

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

12. Taylor expanded in y around inf

    \[\leadsto \mathsf{fma}\left(\frac{{y}^{2}}{z}, \frac{1}{2}, z\right) \]
13. Step-by-step derivation

14. Applied rewrites 17.4%

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

15. Final simplification 17.4%

    \[\leadsto \mathsf{fma}\left(\frac{y}{z} \cdot y, 0.5, z\right) \]
16. 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 45.0%

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

   1. lift-+.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. associate-+r+ N/A

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

   4. +-commutative N/A

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

   5. associate-+l+ N/A

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

   6. lift-*.f64 N/A

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

   7. lower-fma.f64 N/A

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

   8. +-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lower-hypot.f64 19.9

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

4. Applied rewrites 19.9%

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

   1. lift-hypot.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. flip-+ N/A

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

   4. lower-/.f64 N/A

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

   5. lower--.f64 N/A

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

   6. pow2 N/A

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

   7. lift-*.f64 N/A

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

   8. pow-prod-down N/A

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

   9. pow-prod-up N/A

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

   10. lower-pow.f64 N/A

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

   11. metadata-eval N/A

       \[\leadsto \sqrt{\mathsf{fma}\left(y, y, \frac{{z}^{\color{blue}{4}} - \left(x \cdot x\right) \cdot \left(x \cdot x\right)}{z \cdot z - x \cdot x}\right)} \]

   12. pow2 N/A

       \[\leadsto \sqrt{\mathsf{fma}\left(y, y, \frac{{z}^{4} - \color{blue}{{x}^{2}} \cdot \left(x \cdot x\right)}{z \cdot z - x \cdot x}\right)} \]

   13. pow2 N/A

       \[\leadsto \sqrt{\mathsf{fma}\left(y, y, \frac{{z}^{4} - {x}^{2} \cdot \color{blue}{{x}^{2}}}{z \cdot z - x \cdot x}\right)} \]

   14. pow-prod-up N/A

       \[\leadsto \sqrt{\mathsf{fma}\left(y, y, \frac{{z}^{4} - \color{blue}{{x}^{\left(2 + 2\right)}}}{z \cdot z - x \cdot x}\right)} \]

   15. lower-pow.f64 N/A

       \[\leadsto \sqrt{\mathsf{fma}\left(y, y, \frac{{z}^{4} - \color{blue}{{x}^{\left(2 + 2\right)}}}{z \cdot z - x \cdot x}\right)} \]

   16. metadata-eval N/A

       \[\leadsto \sqrt{\mathsf{fma}\left(y, y, \frac{{z}^{4} - {x}^{\color{blue}{4}}}{z \cdot z - x \cdot x}\right)} \]

   17. lift-*.f64 N/A

       \[\leadsto \sqrt{\mathsf{fma}\left(y, y, \frac{{z}^{4} - {x}^{4}}{\color{blue}{z \cdot z} - x \cdot x}\right)} \]

   18. difference-of-squares N/A

       \[\leadsto \sqrt{\mathsf{fma}\left(y, y, \frac{{z}^{4} - {x}^{4}}{\color{blue}{\left(z + x\right) \cdot \left(z - x\right)}}\right)} \]

   19. lower-*.f64 N/A

       \[\leadsto \sqrt{\mathsf{fma}\left(y, y, \frac{{z}^{4} - {x}^{4}}{\color{blue}{\left(z + x\right) \cdot \left(z - x\right)}}\right)} \]

   20. lower-+.f64 N/A

       \[\leadsto \sqrt{\mathsf{fma}\left(y, y, \frac{{z}^{4} - {x}^{4}}{\color{blue}{\left(z + x\right)} \cdot \left(z - x\right)}\right)} \]

   21. lower--.f64 26.5

       \[\leadsto \sqrt{\mathsf{fma}\left(y, y, \frac{{z}^{4} - {x}^{4}}{\left(z + x\right) \cdot \color{blue}{\left(z - x\right)}}\right)} \]

6. Applied rewrites 26.5%

   \[\leadsto \sqrt{\mathsf{fma}\left(y, y, \color{blue}{\frac{{z}^{4} - {x}^{4}}{\left(z + x\right) \cdot \left(z - x\right)}}\right)} \]

7. Taylor expanded in z around inf

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

9. Applied rewrites 17.2%

   \[\leadsto \color{blue}{z} \]
10. 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 2024249 
(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)))))