Numeric.LinearAlgebra.Util:formatSparse from hmatrix-0.16.1.5

Percentage Accurate: 100.0% → 100.0%

Time: 3.9s

Alternatives: 5

Speedup: 2.0×

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, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\frac{\left|x - y\right|}{\left|y\right|} \]
2. Add Preprocessing

3. Taylor expanded in x around -inf 100.0%

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

5. Simplified 100.0%

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

Alternative 2: 74.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.7 \cdot 10^{+53}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 6.4 \cdot 10^{+83}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3.7e+53) 1.0 (if (<= y 6.4e+83) (fabs (/ x y)) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3.7e+53) {
		tmp = 1.0;
	} else if (y <= 6.4e+83) {
		tmp = fabs((x / y));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-3.7d+53)) then
        tmp = 1.0d0
    else if (y <= 6.4d+83) then
        tmp = abs((x / y))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -3.7e+53) {
		tmp = 1.0;
	} else if (y <= 6.4e+83) {
		tmp = Math.abs((x / y));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -3.7e+53:
		tmp = 1.0
	elif y <= 6.4e+83:
		tmp = math.fabs((x / y))
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3.7e+53)
		tmp = 1.0;
	elseif (y <= 6.4e+83)
		tmp = abs(Float64(x / y));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -3.7e+53)
		tmp = 1.0;
	elseif (y <= 6.4e+83)
		tmp = abs((x / y));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3.7e+53], 1.0, If[LessEqual[y, 6.4e+83], N[Abs[N[(x / y), $MachinePrecision]], $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -3.7e53 or 6.3999999999999998e83 < y

   1. Initial program 100.0%

      \[\frac{\left|x - y\right|}{\left|y\right|} \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 89.4%

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

   8. Applied egg-rr 89.4%

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

   if -3.7e53 < y < 6.3999999999999998e83

   1. Initial program 100.0%

      \[\frac{\left|x - y\right|}{\left|y\right|} \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf 72.2%

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

Alternative 3: 58.9% accurate, 11.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.32 \cdot 10^{+253}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq -1.25 \cdot 10^{+145} \lor \neg \left(x \leq 4.1 \cdot 10^{+105}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.32e+253)
   (* x x)
   (if (or (<= x -1.25e+145) (not (<= x 4.1e+105))) (/ x y) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.32e+253) {
		tmp = x * x;
	} else if ((x <= -1.25e+145) || !(x <= 4.1e+105)) {
		tmp = x / y;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.32d+253)) then
        tmp = x * x
    else if ((x <= (-1.25d+145)) .or. (.not. (x <= 4.1d+105))) then
        tmp = x / y
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.32e+253) {
		tmp = x * x;
	} else if ((x <= -1.25e+145) || !(x <= 4.1e+105)) {
		tmp = x / y;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.32e+253:
		tmp = x * x
	elif (x <= -1.25e+145) or not (x <= 4.1e+105):
		tmp = x / y
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.32e+253)
		tmp = Float64(x * x);
	elseif ((x <= -1.25e+145) || !(x <= 4.1e+105))
		tmp = Float64(x / y);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.32e+253)
		tmp = x * x;
	elseif ((x <= -1.25e+145) || ~((x <= 4.1e+105)))
		tmp = x / y;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.32e+253], N[(x * x), $MachinePrecision], If[Or[LessEqual[x, -1.25e+145], N[Not[LessEqual[x, 4.1e+105]], $MachinePrecision]], N[(x / y), $MachinePrecision], 1.0]]

Derivation

1. Split input into 3 regimes

2. if x < -1.32e253

   1. Initial program 100.0%

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

      1. add-log-exp 74.8%

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

      2. *-un-lft-identity 74.8%

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

      3. log-prod 74.8%

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

      4. metadata-eval 74.8%

         \[\leadsto \color{blue}{0} + \log \left(e^{\frac{\left|x - y\right|}{\left|y\right|}}\right) \]

      5. add-log-exp 100.0%

         \[\leadsto 0 + \color{blue}{\frac{\left|x - y\right|}{\left|y\right|}} \]

      6. add-sqr-sqrt 0.0%

         \[\leadsto 0 + \frac{\left|\color{blue}{\sqrt{x - y} \cdot \sqrt{x - y}}\right|}{\left|y\right|} \]

      7. fabs-sqr 0.0%

         \[\leadsto 0 + \frac{\color{blue}{\sqrt{x - y} \cdot \sqrt{x - y}}}{\left|y\right|} \]

      8. add-sqr-sqrt 0.0%

         \[\leadsto 0 + \frac{\sqrt{x - y} \cdot \sqrt{x - y}}{\left|\color{blue}{\sqrt{y} \cdot \sqrt{y}}\right|} \]

      9. fabs-sqr 0.0%

         \[\leadsto 0 + \frac{\sqrt{x - y} \cdot \sqrt{x - y}}{\color{blue}{\sqrt{y} \cdot \sqrt{y}}} \]

      10. add-sqr-sqrt 0.0%

          \[\leadsto 0 + \frac{\sqrt{x - y} \cdot \sqrt{x - y}}{\color{blue}{y}} \]

      11. add-sqr-sqrt 33.4%

          \[\leadsto 0 + \frac{\color{blue}{x - y}}{y} \]

   4. Applied egg-rr 33.4%

      \[\leadsto \color{blue}{0 + \frac{x - y}{y}} \]

   5. Taylor expanded in x around inf 33.4%

      \[\leadsto 0 + \frac{\color{blue}{x}}{y} \]

   7. Applied egg-rr 74.8%

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

   if -1.32e253 < x < -1.24999999999999992e145 or 4.1000000000000002e105 < x

   1. Initial program 99.9%

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

      1. div-inv 99.8%

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

      2. add-sqr-sqrt 69.0%

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

      3. fabs-sqr 69.0%

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

      4. add-sqr-sqrt 69.5%

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

      5. *-commutative 69.5%

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

      6. add-sqr-sqrt 42.7%

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

      7. fabs-sqr 42.7%

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

      8. add-sqr-sqrt 61.5%

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

   4. Applied egg-rr 61.5%

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

   5. Taylor expanded in x around inf 62.3%

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

   6. Taylor expanded in y around 0 62.3%

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

   if -1.24999999999999992e145 < x < 4.1000000000000002e105

   1. Initial program 100.0%

      \[\frac{\left|x - y\right|}{\left|y\right|} \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 66.8%

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

   8. Applied egg-rr 66.8%

      \[\leadsto \color{blue}{1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.32 \cdot 10^{+253}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq -1.25 \cdot 10^{+145} \lor \neg \left(x \leq 4.1 \cdot 10^{+105}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 57.9% accurate, 15.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{+172} \lor \neg \left(x \leq 6.5 \cdot 10^{+174}\right):\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -2.9e+172) (not (<= x 6.5e+174))) (* x x) 1.0))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -2.9e+172) || !(x <= 6.5e+174)) {
		tmp = x * x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-2.9d+172)) .or. (.not. (x <= 6.5d+174))) then
        tmp = x * x
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -2.9e+172) || !(x <= 6.5e+174)) {
		tmp = x * x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -2.9e+172) or not (x <= 6.5e+174):
		tmp = x * x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -2.9e+172) || !(x <= 6.5e+174))
		tmp = Float64(x * x);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -2.9e+172) || ~((x <= 6.5e+174)))
		tmp = x * x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -2.9e+172], N[Not[LessEqual[x, 6.5e+174]], $MachinePrecision]], N[(x * x), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < -2.8999999999999999e172 or 6.5000000000000001e174 < x

   1. Initial program 99.9%

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

      1. add-log-exp 49.5%

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

      2. *-un-lft-identity 49.5%

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

      3. log-prod 49.5%

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

      4. metadata-eval 49.5%

         \[\leadsto \color{blue}{0} + \log \left(e^{\frac{\left|x - y\right|}{\left|y\right|}}\right) \]

      5. add-log-exp 99.9%

         \[\leadsto 0 + \color{blue}{\frac{\left|x - y\right|}{\left|y\right|}} \]

      6. add-sqr-sqrt 46.4%

         \[\leadsto 0 + \frac{\left|\color{blue}{\sqrt{x - y} \cdot \sqrt{x - y}}\right|}{\left|y\right|} \]

      7. fabs-sqr 46.4%

         \[\leadsto 0 + \frac{\color{blue}{\sqrt{x - y} \cdot \sqrt{x - y}}}{\left|y\right|} \]

      8. add-sqr-sqrt 30.8%

         \[\leadsto 0 + \frac{\sqrt{x - y} \cdot \sqrt{x - y}}{\left|\color{blue}{\sqrt{y} \cdot \sqrt{y}}\right|} \]

      9. fabs-sqr 30.8%

         \[\leadsto 0 + \frac{\sqrt{x - y} \cdot \sqrt{x - y}}{\color{blue}{\sqrt{y} \cdot \sqrt{y}}} \]

      10. add-sqr-sqrt 31.0%

          \[\leadsto 0 + \frac{\sqrt{x - y} \cdot \sqrt{x - y}}{\color{blue}{y}} \]

      11. add-sqr-sqrt 58.0%

          \[\leadsto 0 + \frac{\color{blue}{x - y}}{y} \]

   4. Applied egg-rr 58.0%

      \[\leadsto \color{blue}{0 + \frac{x - y}{y}} \]

   5. Taylor expanded in x around inf 58.2%

      \[\leadsto 0 + \frac{\color{blue}{x}}{y} \]

   7. Applied egg-rr 45.2%

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

   if -2.8999999999999999e172 < x < 6.5000000000000001e174

   1. Initial program 100.0%

      \[\frac{\left|x - y\right|}{\left|y\right|} \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 63.9%

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

   8. Applied egg-rr 63.9%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{+172} \lor \neg \left(x \leq 6.5 \cdot 10^{+174}\right):\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 51.3% accurate, 205.0× 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. Taylor expanded in x around -inf 100.0%

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

5. Simplified 100.0%

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

6. Taylor expanded in x around 0 54.0%

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

8. Applied egg-rr 54.0%

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

Reproduce

herbie shell --seed 2024170 
(FPCore (x y)
  :name "Numeric.LinearAlgebra.Util:formatSparse from hmatrix-0.16.1.5"
  :precision binary64
  (/ (fabs (- x y)) (fabs y)))