Numeric.LinearAlgebra.Util:formatSparse from hmatrix-0.16.1.5

Percentage Accurate: 100.0% → 100.0%

Time: 3.2s

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. Step-by-step derivation

   1. div-fabs 100.0%

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

   2. div-sub 100.0%

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

   3. pow1 100.0%

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

   4. pow1 100.0%

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

   5. pow-div 100.0%

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

   6. metadata-eval 100.0%

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

   7. metadata-eval 100.0%

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

4. Applied egg-rr 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 76.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{-7} \lor \neg \left(x \leq 66000000000\right):\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -8e-7) (not (<= x 66000000000.0))) (fabs (/ x y)) 1.0))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -8e-7) || !(x <= 66000000000.0)) {
		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 ((x <= (-8d-7)) .or. (.not. (x <= 66000000000.0d0))) 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 ((x <= -8e-7) || !(x <= 66000000000.0)) {
		tmp = Math.abs((x / y));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -8e-7) or not (x <= 66000000000.0):
		tmp = math.fabs((x / y))
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -8e-7) || !(x <= 66000000000.0))
		tmp = abs(Float64(x / y));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -8e-7) || ~((x <= 66000000000.0)))
		tmp = abs((x / y));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -8e-7], N[Not[LessEqual[x, 66000000000.0]], $MachinePrecision]], N[Abs[N[(x / y), $MachinePrecision]], $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < -7.9999999999999996e-7 or 6.6e10 < x

   1. Initial program 100.0%

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

      1. div-fabs 100.0%

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

      2. div-sub 100.0%

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

      3. pow1 100.0%

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

      4. pow1 100.0%

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

      5. pow-div 100.0%

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

      6. metadata-eval 100.0%

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

      7. metadata-eval 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf 83.6%

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

   if -7.9999999999999996e-7 < x < 6.6e10

   1. Initial program 100.0%

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

      1. div-fabs 100.0%

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

      2. div-sub 100.0%

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

      3. pow1 100.0%

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

      4. pow1 100.0%

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

      5. pow-div 100.0%

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

      6. metadata-eval 100.0%

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

      7. metadata-eval 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0 81.3%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{-7} \lor \neg \left(x \leq 66000000000\right):\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 58.2% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{+103} \lor \neg \left(x \leq 3 \cdot 10^{+14}\right) \land \left(x \leq 1.5 \cdot 10^{+44} \lor \neg \left(x \leq 1.8 \cdot 10^{+171}\right)\right):\\ \;\;\;\;\frac{x}{y} + -1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -3.8e+103)
         (and (not (<= x 3e+14)) (or (<= x 1.5e+44) (not (<= x 1.8e+171)))))
   (+ (/ x y) -1.0)
   1.0))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -3.8e+103) || (!(x <= 3e+14) && ((x <= 1.5e+44) || !(x <= 1.8e+171)))) {
		tmp = (x / y) + -1.0;
	} 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 <= (-3.8d+103)) .or. (.not. (x <= 3d+14)) .and. (x <= 1.5d+44) .or. (.not. (x <= 1.8d+171))) then
        tmp = (x / y) + (-1.0d0)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -3.8e+103) || (!(x <= 3e+14) && ((x <= 1.5e+44) || !(x <= 1.8e+171)))) {
		tmp = (x / y) + -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -3.8e+103) or (not (x <= 3e+14) and ((x <= 1.5e+44) or not (x <= 1.8e+171))):
		tmp = (x / y) + -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -3.8e+103) || (!(x <= 3e+14) && ((x <= 1.5e+44) || !(x <= 1.8e+171))))
		tmp = Float64(Float64(x / y) + -1.0);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -3.8e+103) || (~((x <= 3e+14)) && ((x <= 1.5e+44) || ~((x <= 1.8e+171)))))
		tmp = (x / y) + -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -3.8e+103], And[N[Not[LessEqual[x, 3e+14]], $MachinePrecision], Or[LessEqual[x, 1.5e+44], N[Not[LessEqual[x, 1.8e+171]], $MachinePrecision]]]], N[(N[(x / y), $MachinePrecision] + -1.0), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < -3.7999999999999997e103 or 3e14 < x < 1.49999999999999993e44 or 1.80000000000000009e171 < x

   1. Initial program 100.0%

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

      1. add-sqr-sqrt 47.0%

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

      2. fabs-sqr 47.0%

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

      3. add-sqr-sqrt 29.7%

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

      4. fabs-sqr 29.7%

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

      5. add-sqr-sqrt 30.0%

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

      6. add-sqr-sqrt 51.7%

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

      7. div-sub 51.7%

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

      8. pow1 51.7%

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

      9. pow1 51.7%

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

      10. pow-div 51.7%

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

      11. metadata-eval 51.7%

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

      12. metadata-eval 51.7%

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

   4. Applied egg-rr 51.7%

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

   if -3.7999999999999997e103 < x < 3e14 or 1.49999999999999993e44 < x < 1.80000000000000009e171

   1. Initial program 100.0%

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

      1. div-fabs 100.0%

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

      2. div-sub 100.0%

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

      3. pow1 100.0%

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

      4. pow1 100.0%

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

      5. pow-div 100.0%

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

      6. metadata-eval 100.0%

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

      7. metadata-eval 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0 72.5%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{+103} \lor \neg \left(x \leq 3 \cdot 10^{+14}\right) \land \left(x \leq 1.5 \cdot 10^{+44} \lor \neg \left(x \leq 1.8 \cdot 10^{+171}\right)\right):\\ \;\;\;\;\frac{x}{y} + -1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 25.7% accurate, 68.3× speedup

Math

\[\begin{array}{l} \\ \frac{x}{y} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return x / y

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. add-sqr-sqrt 45.1%

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

   2. fabs-sqr 45.1%

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

   3. add-sqr-sqrt 11.6%

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

   4. fabs-sqr 11.6%

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

   5. add-sqr-sqrt 12.2%

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

   6. add-sqr-sqrt 24.0%

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

   7. div-sub 24.0%

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

   8. pow1 24.0%

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

   9. pow1 24.0%

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

   10. pow-div 24.0%

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

   11. metadata-eval 24.0%

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

   12. metadata-eval 24.0%

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

4. Applied egg-rr 24.0%

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

5. Taylor expanded in x around inf 24.4%

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

6. Final simplification 24.4%

   \[\leadsto \frac{x}{y} \]
7. Add Preprocessing

Alternative 5: 1.3% 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. Step-by-step derivation

   1. add-sqr-sqrt 45.1%

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

   2. fabs-sqr 45.1%

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

   3. add-sqr-sqrt 11.6%

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

   4. fabs-sqr 11.6%

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

   5. add-sqr-sqrt 12.2%

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

   6. add-sqr-sqrt 24.0%

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

   7. div-sub 24.0%

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

   8. pow1 24.0%

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

   9. pow1 24.0%

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

   10. pow-div 24.0%

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

   11. metadata-eval 24.0%

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

   12. metadata-eval 24.0%

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

4. Applied egg-rr 24.0%

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

5. Taylor expanded in x around 0 1.3%

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

6. Final simplification 1.3%

   \[\leadsto -1 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024017 
(FPCore (x y)
  :name "Numeric.LinearAlgebra.Util:formatSparse from hmatrix-0.16.1.5"
  :precision binary64
  (/ (fabs (- x y)) (fabs y)))