Numeric.LinearAlgebra.Util:formatSparse from hmatrix-0.16.1.5

Percentage Accurate: 100.0% → 100.0%

Time: 3.3s

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

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

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. Taylor expanded in x around -inf 100.0%

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

   1. fabs-neg 100.0%

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

   2. mul-1-neg 100.0%

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

   3. sub-neg 100.0%

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

   4. fabs-div 100.0%

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

   5. div-sub 100.0%

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

   6. *-inverses 100.0%

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

5. Simplified 100.0%

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

6. Final simplification 100.0%

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

Alternative 2: 75.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2800000000000:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -6.2 \cdot 10^{-53} \lor \neg \left(y \leq -2 \cdot 10^{-70}\right) \land y \leq 9.5 \cdot 10^{-45}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2800000000000.0)
   1.0
   (if (or (<= y -6.2e-53) (and (not (<= y -2e-70)) (<= y 9.5e-45)))
     (fabs (/ x y))
     1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2800000000000.0) {
		tmp = 1.0;
	} else if ((y <= -6.2e-53) || (!(y <= -2e-70) && (y <= 9.5e-45))) {
		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 <= (-2800000000000.0d0)) then
        tmp = 1.0d0
    else if ((y <= (-6.2d-53)) .or. (.not. (y <= (-2d-70))) .and. (y <= 9.5d-45)) 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 <= -2800000000000.0) {
		tmp = 1.0;
	} else if ((y <= -6.2e-53) || (!(y <= -2e-70) && (y <= 9.5e-45))) {
		tmp = Math.abs((x / y));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2800000000000.0:
		tmp = 1.0
	elif (y <= -6.2e-53) or (not (y <= -2e-70) and (y <= 9.5e-45)):
		tmp = math.fabs((x / y))
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2800000000000.0)
		tmp = 1.0;
	elseif ((y <= -6.2e-53) || (!(y <= -2e-70) && (y <= 9.5e-45)))
		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 <= -2800000000000.0)
		tmp = 1.0;
	elseif ((y <= -6.2e-53) || (~((y <= -2e-70)) && (y <= 9.5e-45)))
		tmp = abs((x / y));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2800000000000.0], 1.0, If[Or[LessEqual[y, -6.2e-53], And[N[Not[LessEqual[y, -2e-70]], $MachinePrecision], LessEqual[y, 9.5e-45]]], N[Abs[N[(x / y), $MachinePrecision]], $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -2.8e12 or -6.20000000000000031e-53 < y < -1.99999999999999999e-70 or 9.5000000000000002e-45 < 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%

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

      1. fabs-neg 100.0%

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

      2. mul-1-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. fabs-div 100.0%

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

      5. div-sub 100.0%

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

      6. *-inverses 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 81.4%

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

   if -2.8e12 < y < -6.20000000000000031e-53 or -1.99999999999999999e-70 < y < 9.5000000000000002e-45

   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%

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

      1. fabs-neg 100.0%

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

      2. mul-1-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. fabs-div 100.0%

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

      5. div-sub 100.0%

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

      6. *-inverses 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf 85.8%

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

      1. associate-*r/ 85.8%

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

      2. neg-mul-1 85.8%

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

   8. Simplified 85.8%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2800000000000:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -6.2 \cdot 10^{-53} \lor \neg \left(y \leq -2 \cdot 10^{-70}\right) \land y \leq 9.5 \cdot 10^{-45}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 59.1% accurate, 15.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{-145}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{-61}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -4.8e-145) 1.0 (if (<= y 2.5e-61) (/ x y) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -4.8e-145) {
		tmp = 1.0;
	} else if (y <= 2.5e-61) {
		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 (y <= (-4.8d-145)) then
        tmp = 1.0d0
    else if (y <= 2.5d-61) 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 (y <= -4.8e-145) {
		tmp = 1.0;
	} else if (y <= 2.5e-61) {
		tmp = x / y;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -4.8e-145:
		tmp = 1.0
	elif y <= 2.5e-61:
		tmp = x / y
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -4.8e-145)
		tmp = 1.0;
	elseif (y <= 2.5e-61)
		tmp = Float64(x / y);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -4.8e-145)
		tmp = 1.0;
	elseif (y <= 2.5e-61)
		tmp = x / y;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -4.8e-145], 1.0, If[LessEqual[y, 2.5e-61], N[(x / y), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -4.8000000000000003e-145 or 2.4999999999999999e-61 < 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%

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

      1. fabs-neg 100.0%

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

      2. mul-1-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. fabs-div 100.0%

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

      5. div-sub 100.0%

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

      6. *-inverses 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 72.8%

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

   if -4.8000000000000003e-145 < y < 2.4999999999999999e-61

   1. Initial program 100.0%

      \[\frac{\left|x - y\right|}{\left|y\right|} \]
   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 41.0%

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

      3. fabs-sqr 41.0%

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

      4. add-sqr-sqrt 41.4%

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

      5. *-commutative 41.4%

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

      6. add-sqr-sqrt 21.2%

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

      7. fabs-sqr 21.2%

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

      8. add-sqr-sqrt 44.9%

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

   4. Applied egg-rr 44.9%

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

   5. Taylor expanded in y around 0 45.2%

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

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{-145}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{-61}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 26.7% 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. div-inv 99.7%

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

   2. add-sqr-sqrt 46.1%

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

   3. fabs-sqr 46.1%

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

   4. add-sqr-sqrt 46.9%

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

   5. *-commutative 46.9%

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

   6. add-sqr-sqrt 9.8%

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

   7. fabs-sqr 9.8%

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

   8. add-sqr-sqrt 24.3%

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

4. Applied egg-rr 24.3%

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

5. Taylor expanded in y around 0 25.2%

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

6. Final simplification 25.2%

   \[\leadsto \frac{x}{y} \]
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. div-inv 99.7%

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

   2. add-sqr-sqrt 46.1%

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

   3. fabs-sqr 46.1%

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

   4. add-sqr-sqrt 46.9%

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

   5. *-commutative 46.9%

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

   6. add-sqr-sqrt 9.8%

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

   7. fabs-sqr 9.8%

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

   8. add-sqr-sqrt 24.3%

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

4. Applied egg-rr 24.3%

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

5. Taylor expanded in y around inf 1.3%

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

6. Final simplification 1.3%

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

Reproduce

herbie shell --seed 2024078 
(FPCore (x y)
  :name "Numeric.LinearAlgebra.Util:formatSparse from hmatrix-0.16.1.5"
  :precision binary64
  (/ (fabs (- x y)) (fabs y)))