Numeric.LinearAlgebra.Util:formatSparse from hmatrix-0.16.1.5

Percentage Accurate: 100.0% → 100.0%

Time: 5.5s

Alternatives: 4

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

\[\begin{array}{l} \\ \left|1 - \frac{x}{y}\right| \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x, y)
	return abs(Float64(1.0 - Float64(x / y)))
end

MATLAB

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

Wolfram

code[x_, y_] := N[Abs[N[(1.0 - N[(x / y), $MachinePrecision]), $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. neg-fabs N/A

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

   2. div-fabs N/A

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

   3. fabs-lowering-fabs.f64 N/A

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

   4. sub-neg N/A

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

   5. +-commutative N/A

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

   6. distribute-neg-in N/A

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

   7. remove-double-neg N/A

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

   8. sub-neg N/A

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

   9. div-sub N/A

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

   10. *-inverses N/A

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

   11. --lowering--.f64 N/A

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

   12. /-lowering-/.f64 100.0

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

4. Applied egg-rr 100.0%

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

Alternative 2: 97.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	double tmp;
	if ((fabs((x - y)) / fabs(y)) <= 2.0) {
		tmp = 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((x - y)) / abs(y)) <= 2.0d0) then
        tmp = 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((x - y)) / Math.abs(y)) <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = Math.abs((x / y));
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

      2. div-fabs N/A

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

      3. fabs-lowering-fabs.f64 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. distribute-neg-in N/A

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

      7. remove-double-neg N/A

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

      8. sub-neg N/A

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

      9. div-sub N/A

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

      10. *-inverses N/A

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

      11. --lowering--.f64 N/A

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

      12. /-lowering-/.f64 100.0

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

   4. Applied egg-rr 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. Simplified 97.2%

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

      1. metadata-eval 97.2

         \[\leadsto \color{blue}{1} \]

   9. Applied egg-rr 97.2%

      \[\leadsto \color{blue}{1} \]

   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. Taylor expanded in x around inf

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

   5. Simplified 97.5%

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

      1. div-fabs N/A

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

      2. fabs-lowering-fabs.f64 N/A

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

      3. /-lowering-/.f64 97.5

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

   7. Applied egg-rr 97.5%

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

Alternative 3: 72.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left|x - y\right|}{\left|y\right|} \leq 50000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (fabs (- x y)) (fabs y)) 50000.0) 1.0 (/ x y)))

C

double code(double x, double y) {
	double tmp;
	if ((fabs((x - y)) / fabs(y)) <= 50000.0) {
		tmp = 1.0;
	} else {
		tmp = 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((x - y)) / abs(y)) <= 50000.0d0) then
        tmp = 1.0d0
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((Math.abs((x - y)) / Math.abs(y)) <= 50000.0) {
		tmp = 1.0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (math.fabs((x - y)) / math.fabs(y)) <= 50000.0:
		tmp = 1.0
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(abs(Float64(x - y)) / abs(y)) <= 50000.0)
		tmp = 1.0;
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((abs((x - y)) / abs(y)) <= 50000.0)
		tmp = 1.0;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Abs[N[(x - y), $MachinePrecision]], $MachinePrecision] / N[Abs[y], $MachinePrecision]), $MachinePrecision], 50000.0], 1.0, N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (fabs.f64 (-.f64 x y)) (fabs.f64 y)) < 5e4

   1. Initial program 100.0%

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

      1. neg-fabs N/A

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

      2. div-fabs N/A

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

      3. fabs-lowering-fabs.f64 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. distribute-neg-in N/A

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

      7. remove-double-neg N/A

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

      8. sub-neg N/A

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

      9. div-sub N/A

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

      10. *-inverses N/A

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

      11. --lowering--.f64 N/A

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

      12. /-lowering-/.f64 100.0

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

   4. Applied egg-rr 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. Simplified 96.6%

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

      1. metadata-eval 96.6

         \[\leadsto \color{blue}{1} \]

   9. Applied egg-rr 96.6%

      \[\leadsto \color{blue}{1} \]

   if 5e4 < (/.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. Taylor expanded in x around inf

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

   5. Simplified 98.0%

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

      1. div-fabs N/A

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

      2. fabs-lowering-fabs.f64 N/A

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

      3. /-lowering-/.f64 98.0

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

   7. Applied egg-rr 98.0%

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

      1. clear-num N/A

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

      2. inv-pow N/A

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

      3. sqr-pow N/A

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

      4. fabs-sqr N/A

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

      5. sqr-pow N/A

         \[\leadsto \color{blue}{{\left(\frac{y}{x}\right)}^{-1}} \]

      6. inv-pow N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{y}{x}}} \]

      7. clear-num N/A

         \[\leadsto \color{blue}{\frac{x}{y}} \]

      8. /-lowering-/.f64 40.5

         \[\leadsto \color{blue}{\frac{x}{y}} \]

   9. Applied egg-rr 40.5%

      \[\leadsto \color{blue}{\frac{x}{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 50.5% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 100.0%

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

   1. neg-fabs N/A

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

   2. div-fabs N/A

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

   3. fabs-lowering-fabs.f64 N/A

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

   4. sub-neg N/A

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

   5. +-commutative N/A

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

   6. distribute-neg-in N/A

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

   7. remove-double-neg N/A

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

   8. sub-neg N/A

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

   9. div-sub N/A

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

   10. *-inverses N/A

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

   11. --lowering--.f64 N/A

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

   12. /-lowering-/.f64 100.0

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

4. Applied egg-rr 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. Simplified 50.8%

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

   1. metadata-eval 50.8

      \[\leadsto \color{blue}{1} \]

9. Applied egg-rr 50.8%

   \[\leadsto \color{blue}{1} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024196 
(FPCore (x y)
  :name "Numeric.LinearAlgebra.Util:formatSparse from hmatrix-0.16.1.5"
  :precision binary64
  (/ (fabs (- x y)) (fabs y)))