Numeric.LinearAlgebra.Util:formatSparse from hmatrix-0.16.1.5

Percentage Accurate: 100.0% → 100.0%

Time: 5.8s

Alternatives: 3

Speedup: 1.1×

Specification

Math

\[\begin{array}{l} \\ \frac{\left|x - y\right|}{\left|y\right|} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ (fabs (- x y)) (fabs y)))

C

double code(double x, double y) {
	return fabs((x - y)) / fabs(y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = abs((x - y)) / abs(y)
end function

Java

public static double code(double x, double y) {
	return Math.abs((x - y)) / Math.abs(y);
}

Python

def code(x, y):
	return math.fabs((x - y)) / math.fabs(y)

Julia

function code(x, y)
	return Float64(abs(Float64(x - y)) / abs(y))
end

MATLAB

function tmp = code(x, y)
	tmp = abs((x - y)) / abs(y);
end

Wolfram

code[x_, y_] := N[(N[Abs[N[(x - y), $MachinePrecision]], $MachinePrecision] / N[Abs[y], $MachinePrecision]), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left|x - y\right|}{\left|y\right|} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ (fabs (- x y)) (fabs y)))

C

double code(double x, double y) {
	return fabs((x - y)) / fabs(y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = abs((x - y)) / abs(y)
end function

Java

public static double code(double x, double y) {
	return Math.abs((x - y)) / Math.abs(y);
}

Python

def code(x, y):
	return math.fabs((x - y)) / math.fabs(y)

Julia

function code(x, y)
	return Float64(abs(Float64(x - y)) / abs(y))
end

MATLAB

function tmp = code(x, y)
	tmp = abs((x - y)) / abs(y);
end

Wolfram

code[x_, y_] := N[(N[Abs[N[(x - y), $MachinePrecision]], $MachinePrecision] / N[Abs[y], $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left|\frac{y - x}{y}\right| \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (fabs (/ (- y x) y)))

C

double code(double x, double y) {
	return fabs(((y - x) / y));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = abs(((y - x) / y))
end function

Java

public static double code(double x, double y) {
	return Math.abs(((y - x) / y));
}

Python

def code(x, y):
	return math.fabs(((y - x) / y))

Julia

function code(x, y)
	return abs(Float64(Float64(y - x) / y))
end

MATLAB

function tmp = code(x, y)
	tmp = abs(((y - x) / y));
end

Wolfram

code[x_, y_] := N[Abs[N[(N[(y - x), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{\left|x - y\right|}{\left|y\right|} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\left|x - y\right|}{\left|y\right|}} \]

   2. lift-fabs.f64 N/A

      \[\leadsto \frac{\color{blue}{\left|x - y\right|}}{\left|y\right|} \]

   3. neg-fabs N/A

      \[\leadsto \frac{\color{blue}{\left|\mathsf{neg}\left(\left(x - y\right)\right)\right|}}{\left|y\right|} \]

   4. lift-fabs.f64 N/A

      \[\leadsto \frac{\left|\mathsf{neg}\left(\left(x - y\right)\right)\right|}{\color{blue}{\left|y\right|}} \]

   5. div-fabs N/A

      \[\leadsto \color{blue}{\left|\frac{\mathsf{neg}\left(\left(x - y\right)\right)}{y}\right|} \]

   6. lower-fabs.f64 N/A

      \[\leadsto \color{blue}{\left|\frac{\mathsf{neg}\left(\left(x - y\right)\right)}{y}\right|} \]

   7. lift--.f64 N/A

      \[\leadsto \left|\frac{\mathsf{neg}\left(\color{blue}{\left(x - y\right)}\right)}{y}\right| \]

   8. sub-neg N/A

      \[\leadsto \left|\frac{\mathsf{neg}\left(\color{blue}{\left(x + \left(\mathsf{neg}\left(y\right)\right)\right)}\right)}{y}\right| \]

   9. +-commutative N/A

      \[\leadsto \left|\frac{\mathsf{neg}\left(\color{blue}{\left(\left(\mathsf{neg}\left(y\right)\right) + x\right)}\right)}{y}\right| \]

   10. distribute-neg-in N/A

       \[\leadsto \left|\frac{\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right) + \left(\mathsf{neg}\left(x\right)\right)}}{y}\right| \]

   11. remove-double-neg N/A

       \[\leadsto \left|\frac{\color{blue}{y} + \left(\mathsf{neg}\left(x\right)\right)}{y}\right| \]

   12. sub-neg N/A

       \[\leadsto \left|\frac{\color{blue}{y - x}}{y}\right| \]

   13. div-sub N/A

       \[\leadsto \left|\color{blue}{\frac{y}{y} - \frac{x}{y}}\right| \]

   14. *-inverses N/A

       \[\leadsto \left|\color{blue}{1} - \frac{x}{y}\right| \]

   15. lower--.f64 N/A

       \[\leadsto \left|\color{blue}{1 - \frac{x}{y}}\right| \]

   16. lower-/.f64 100.0

       \[\leadsto \left|1 - \color{blue}{\frac{x}{y}}\right| \]

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\left|1 - \frac{x}{y}\right|} \]

5. Taylor expanded in x around 0

   \[\leadsto \left|\color{blue}{1 + -1 \cdot \frac{x}{y}}\right| \]
6. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \left|1 + \color{blue}{\left(\mathsf{neg}\left(\frac{x}{y}\right)\right)}\right| \]

   2. *-inverses N/A

      \[\leadsto \left|\color{blue}{\frac{y}{y}} + \left(\mathsf{neg}\left(\frac{x}{y}\right)\right)\right| \]

   3. sub-neg N/A

      \[\leadsto \left|\color{blue}{\frac{y}{y} - \frac{x}{y}}\right| \]

   4. div-sub N/A

      \[\leadsto \left|\color{blue}{\frac{y - x}{y}}\right| \]

   5. lower-/.f64 N/A

      \[\leadsto \left|\color{blue}{\frac{y - x}{y}}\right| \]

   6. lower--.f64 100.0

      \[\leadsto \left|\frac{\color{blue}{y - x}}{y}\right| \]

7. Applied rewrites 100.0%

   \[\leadsto \left|\color{blue}{\frac{y - x}{y}}\right| \]
8. Add Preprocessing

Alternative 2: 97.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left|y - x\right|}{\left|y\right|} \leq 2:\\ \;\;\;\;\left|1\right|\\ \mathbf{else}:\\ \;\;\;\;\left|-\frac{x}{y}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (fabs (- y x)) (fabs y)) 2.0) (fabs 1.0) (fabs (- (/ x y)))))

C

double code(double x, double y) {
	double tmp;
	if ((fabs((y - x)) / fabs(y)) <= 2.0) {
		tmp = fabs(1.0);
	} else {
		tmp = fabs(-(x / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((abs((y - x)) / abs(y)) <= 2.0d0) then
        tmp = abs(1.0d0)
    else
        tmp = abs(-(x / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((Math.abs((y - x)) / Math.abs(y)) <= 2.0) {
		tmp = Math.abs(1.0);
	} else {
		tmp = Math.abs(-(x / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (math.fabs((y - x)) / math.fabs(y)) <= 2.0:
		tmp = math.fabs(1.0)
	else:
		tmp = math.fabs(-(x / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(abs(Float64(y - x)) / abs(y)) <= 2.0)
		tmp = abs(1.0);
	else
		tmp = abs(Float64(-Float64(x / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((abs((y - x)) / abs(y)) <= 2.0)
		tmp = abs(1.0);
	else
		tmp = abs(-(x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Abs[N[(y - x), $MachinePrecision]], $MachinePrecision] / N[Abs[y], $MachinePrecision]), $MachinePrecision], 2.0], N[Abs[1.0], $MachinePrecision], N[Abs[(-N[(x / y), $MachinePrecision])], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (fabs.f64 (-.f64 x y)) (fabs.f64 y)) < 2

   1. Initial program 100.0%

      \[\frac{\left|x - y\right|}{\left|y\right|} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left|x - y\right|}{\left|y\right|}} \]

      2. lift-fabs.f64 N/A

         \[\leadsto \frac{\color{blue}{\left|x - y\right|}}{\left|y\right|} \]

      3. neg-fabs N/A

         \[\leadsto \frac{\color{blue}{\left|\mathsf{neg}\left(\left(x - y\right)\right)\right|}}{\left|y\right|} \]

      4. lift-fabs.f64 N/A

         \[\leadsto \frac{\left|\mathsf{neg}\left(\left(x - y\right)\right)\right|}{\color{blue}{\left|y\right|}} \]

      5. div-fabs N/A

         \[\leadsto \color{blue}{\left|\frac{\mathsf{neg}\left(\left(x - y\right)\right)}{y}\right|} \]

      6. lower-fabs.f64 N/A

         \[\leadsto \color{blue}{\left|\frac{\mathsf{neg}\left(\left(x - y\right)\right)}{y}\right|} \]

      7. lift--.f64 N/A

         \[\leadsto \left|\frac{\mathsf{neg}\left(\color{blue}{\left(x - y\right)}\right)}{y}\right| \]

      8. sub-neg N/A

         \[\leadsto \left|\frac{\mathsf{neg}\left(\color{blue}{\left(x + \left(\mathsf{neg}\left(y\right)\right)\right)}\right)}{y}\right| \]

      9. +-commutative N/A

         \[\leadsto \left|\frac{\mathsf{neg}\left(\color{blue}{\left(\left(\mathsf{neg}\left(y\right)\right) + x\right)}\right)}{y}\right| \]

      10. distribute-neg-in N/A

          \[\leadsto \left|\frac{\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right) + \left(\mathsf{neg}\left(x\right)\right)}}{y}\right| \]

      11. remove-double-neg N/A

          \[\leadsto \left|\frac{\color{blue}{y} + \left(\mathsf{neg}\left(x\right)\right)}{y}\right| \]

      12. sub-neg N/A

          \[\leadsto \left|\frac{\color{blue}{y - x}}{y}\right| \]

      13. div-sub N/A

          \[\leadsto \left|\color{blue}{\frac{y}{y} - \frac{x}{y}}\right| \]

      14. *-inverses N/A

          \[\leadsto \left|\color{blue}{1} - \frac{x}{y}\right| \]

      15. lower--.f64 N/A

          \[\leadsto \left|\color{blue}{1 - \frac{x}{y}}\right| \]

      16. lower-/.f64 100.0

          \[\leadsto \left|1 - \color{blue}{\frac{x}{y}}\right| \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\left|1 - \frac{x}{y}\right|} \]

   5. Taylor expanded in x around 0

      \[\leadsto \left|\color{blue}{1}\right| \]
   6. Step-by-step derivation

   7. Applied rewrites 97.8%

      \[\leadsto \left|\color{blue}{1}\right| \]

   if 2 < (/.f64 (fabs.f64 (-.f64 x y)) (fabs.f64 y))

   1. Initial program 100.0%

      \[\frac{\left|x - y\right|}{\left|y\right|} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left|x - y\right|}{\left|y\right|}} \]

      2. lift-fabs.f64 N/A

         \[\leadsto \frac{\color{blue}{\left|x - y\right|}}{\left|y\right|} \]

      3. neg-fabs N/A

         \[\leadsto \frac{\color{blue}{\left|\mathsf{neg}\left(\left(x - y\right)\right)\right|}}{\left|y\right|} \]

      4. lift-fabs.f64 N/A

         \[\leadsto \frac{\left|\mathsf{neg}\left(\left(x - y\right)\right)\right|}{\color{blue}{\left|y\right|}} \]

      5. div-fabs N/A

         \[\leadsto \color{blue}{\left|\frac{\mathsf{neg}\left(\left(x - y\right)\right)}{y}\right|} \]

      6. lower-fabs.f64 N/A

         \[\leadsto \color{blue}{\left|\frac{\mathsf{neg}\left(\left(x - y\right)\right)}{y}\right|} \]

      7. lift--.f64 N/A

         \[\leadsto \left|\frac{\mathsf{neg}\left(\color{blue}{\left(x - y\right)}\right)}{y}\right| \]

      8. sub-neg N/A

         \[\leadsto \left|\frac{\mathsf{neg}\left(\color{blue}{\left(x + \left(\mathsf{neg}\left(y\right)\right)\right)}\right)}{y}\right| \]

      9. +-commutative N/A

         \[\leadsto \left|\frac{\mathsf{neg}\left(\color{blue}{\left(\left(\mathsf{neg}\left(y\right)\right) + x\right)}\right)}{y}\right| \]

      10. distribute-neg-in N/A

          \[\leadsto \left|\frac{\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right) + \left(\mathsf{neg}\left(x\right)\right)}}{y}\right| \]

      11. remove-double-neg N/A

          \[\leadsto \left|\frac{\color{blue}{y} + \left(\mathsf{neg}\left(x\right)\right)}{y}\right| \]

      12. sub-neg N/A

          \[\leadsto \left|\frac{\color{blue}{y - x}}{y}\right| \]

      13. div-sub N/A

          \[\leadsto \left|\color{blue}{\frac{y}{y} - \frac{x}{y}}\right| \]

      14. *-inverses N/A

          \[\leadsto \left|\color{blue}{1} - \frac{x}{y}\right| \]

      15. lower--.f64 N/A

          \[\leadsto \left|\color{blue}{1 - \frac{x}{y}}\right| \]

      16. lower-/.f64 100.0

          \[\leadsto \left|1 - \color{blue}{\frac{x}{y}}\right| \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\left|1 - \frac{x}{y}\right|} \]

   5. Taylor expanded in x around inf

      \[\leadsto \left|\color{blue}{-1 \cdot \frac{x}{y}}\right| \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 98.1

         \[\leadsto \left|\frac{x}{\color{blue}{-y}}\right| \]

   7. Applied rewrites 98.1%

      \[\leadsto \left|\color{blue}{\frac{x}{-y}}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left|y - x\right|}{\left|y\right|} \leq 2:\\ \;\;\;\;\left|1\right|\\ \mathbf{else}:\\ \;\;\;\;\left|-\frac{x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 52.0% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \left|1\right| \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (fabs 1.0))

C

double code(double x, double y) {
	return fabs(1.0);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = abs(1.0d0)
end function

Java

public static double code(double x, double y) {
	return Math.abs(1.0);
}

Python

def code(x, y):
	return math.fabs(1.0)

Julia

function code(x, y)
	return abs(1.0)
end

MATLAB

function tmp = code(x, y)
	tmp = abs(1.0);
end

Wolfram

code[x_, y_] := N[Abs[1.0], $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{\left|x - y\right|}{\left|y\right|} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\left|x - y\right|}{\left|y\right|}} \]

   2. lift-fabs.f64 N/A

      \[\leadsto \frac{\color{blue}{\left|x - y\right|}}{\left|y\right|} \]

   3. neg-fabs N/A

      \[\leadsto \frac{\color{blue}{\left|\mathsf{neg}\left(\left(x - y\right)\right)\right|}}{\left|y\right|} \]

   4. lift-fabs.f64 N/A

      \[\leadsto \frac{\left|\mathsf{neg}\left(\left(x - y\right)\right)\right|}{\color{blue}{\left|y\right|}} \]

   5. div-fabs N/A

      \[\leadsto \color{blue}{\left|\frac{\mathsf{neg}\left(\left(x - y\right)\right)}{y}\right|} \]

   6. lower-fabs.f64 N/A

      \[\leadsto \color{blue}{\left|\frac{\mathsf{neg}\left(\left(x - y\right)\right)}{y}\right|} \]

   7. lift--.f64 N/A

      \[\leadsto \left|\frac{\mathsf{neg}\left(\color{blue}{\left(x - y\right)}\right)}{y}\right| \]

   8. sub-neg N/A

      \[\leadsto \left|\frac{\mathsf{neg}\left(\color{blue}{\left(x + \left(\mathsf{neg}\left(y\right)\right)\right)}\right)}{y}\right| \]

   9. +-commutative N/A

      \[\leadsto \left|\frac{\mathsf{neg}\left(\color{blue}{\left(\left(\mathsf{neg}\left(y\right)\right) + x\right)}\right)}{y}\right| \]

   10. distribute-neg-in N/A

       \[\leadsto \left|\frac{\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right) + \left(\mathsf{neg}\left(x\right)\right)}}{y}\right| \]

   11. remove-double-neg N/A

       \[\leadsto \left|\frac{\color{blue}{y} + \left(\mathsf{neg}\left(x\right)\right)}{y}\right| \]

   12. sub-neg N/A

       \[\leadsto \left|\frac{\color{blue}{y - x}}{y}\right| \]

   13. div-sub N/A

       \[\leadsto \left|\color{blue}{\frac{y}{y} - \frac{x}{y}}\right| \]

   14. *-inverses N/A

       \[\leadsto \left|\color{blue}{1} - \frac{x}{y}\right| \]

   15. lower--.f64 N/A

       \[\leadsto \left|\color{blue}{1 - \frac{x}{y}}\right| \]

   16. lower-/.f64 100.0

       \[\leadsto \left|1 - \color{blue}{\frac{x}{y}}\right| \]

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\left|1 - \frac{x}{y}\right|} \]

5. Taylor expanded in x around 0

   \[\leadsto \left|\color{blue}{1}\right| \]
6. Step-by-step derivation

7. Applied rewrites 55.1%

   \[\leadsto \left|\color{blue}{1}\right| \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024233 
(FPCore (x y)
  :name "Numeric.LinearAlgebra.Util:formatSparse from hmatrix-0.16.1.5"
  :precision binary64
  (/ (fabs (- x y)) (fabs y)))