Numeric.LinearAlgebra.Util:formatSparse from hmatrix-0.16.1.5

Percentage Accurate: 100.0% → 100.0%

Time: 2.2s

Alternatives: 4

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. Step-by-step derivation

   1. fabs-sub 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in y around -inf 100.0%

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

   1. mul-1-neg 100.0%

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

   2. sub-neg 100.0%

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

   3. fabs-neg 100.0%

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

   4. fabs-div 100.0%

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

   5. div-sub 100.0%

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

   6. *-inverses 100.0%

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

6. Simplified 100.0%

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

7. Final simplification 100.0%

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

Alternative 2: 57.9% accurate, 16.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.0028:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 10^{-111} \lor \neg \left(y \leq 3.8 \cdot 10^{-81}\right) \land y \leq 10^{-32}:\\ \;\;\;\;\frac{-x}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -0.0028)
   1.0
   (if (or (<= y 1e-111) (and (not (<= y 3.8e-81)) (<= y 1e-32)))
     (/ (- x) y)
     1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -0.0028) {
		tmp = 1.0;
	} else if ((y <= 1e-111) || (!(y <= 3.8e-81) && (y <= 1e-32))) {
		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 <= (-0.0028d0)) then
        tmp = 1.0d0
    else if ((y <= 1d-111) .or. (.not. (y <= 3.8d-81)) .and. (y <= 1d-32)) 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 <= -0.0028) {
		tmp = 1.0;
	} else if ((y <= 1e-111) || (!(y <= 3.8e-81) && (y <= 1e-32))) {
		tmp = -x / y;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -0.0028:
		tmp = 1.0
	elif (y <= 1e-111) or (not (y <= 3.8e-81) and (y <= 1e-32)):
		tmp = -x / y
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -0.0028)
		tmp = 1.0;
	elseif ((y <= 1e-111) || (!(y <= 3.8e-81) && (y <= 1e-32)))
		tmp = Float64(Float64(-x) / y);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -0.0028)
		tmp = 1.0;
	elseif ((y <= 1e-111) || (~((y <= 3.8e-81)) && (y <= 1e-32)))
		tmp = -x / y;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -0.0028], 1.0, If[Or[LessEqual[y, 1e-111], And[N[Not[LessEqual[y, 3.8e-81]], $MachinePrecision], LessEqual[y, 1e-32]]], N[((-x) / y), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -0.00279999999999999997 or 1.00000000000000009e-111 < y < 3.7999999999999999e-81 or 1.00000000000000006e-32 < y

   1. Initial program 99.9%

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

      1. fabs-sub 99.9%

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

   3. Simplified 99.9%

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

      1. div-inv 99.7%

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

      2. *-commutative 99.7%

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

      3. add-sqr-sqrt 54.1%

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

      4. fabs-sqr 54.1%

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

      5. add-sqr-sqrt 55.0%

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

      6. add-sqr-sqrt 42.4%

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

      7. fabs-sqr 42.4%

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

      8. add-sqr-sqrt 80.4%

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

   5. Applied egg-rr 80.4%

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

   6. Taylor expanded in y around inf 69.8%

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

   if -0.00279999999999999997 < y < 1.00000000000000009e-111 or 3.7999999999999999e-81 < y < 1.00000000000000006e-32

   1. Initial program 100.0%

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

      1. fabs-sub 100.0%

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

   3. Simplified 100.0%

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

      1. div-inv 99.7%

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

      2. *-commutative 99.7%

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

      3. add-sqr-sqrt 51.0%

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

      4. fabs-sqr 51.0%

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

      5. add-sqr-sqrt 51.5%

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

      6. add-sqr-sqrt 32.6%

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

      7. fabs-sqr 32.6%

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

      8. add-sqr-sqrt 59.3%

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

   5. Applied egg-rr 59.3%

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

   6. Taylor expanded in y around 0 46.0%

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

      1. associate-*r/ 46.0%

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

      2. neg-mul-1 46.0%

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

   8. Simplified 46.0%

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

4. Final simplification 59.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.0028:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 10^{-111} \lor \neg \left(y \leq 3.8 \cdot 10^{-81}\right) \land y \leq 10^{-32}:\\ \;\;\;\;\frac{-x}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 3: 76.1% accurate, 41.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. fabs-sub 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in y around -inf 100.0%

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

   1. mul-1-neg 100.0%

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

   2. sub-neg 100.0%

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

   3. fabs-neg 100.0%

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

   4. fabs-sub 100.0%

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

   5. fabs-div 100.0%

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

   6. rem-square-sqrt 70.9%

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

   7. fabs-sqr 70.9%

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

   8. rem-square-sqrt 71.2%

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

   9. div-sub 71.3%

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

   10. *-inverses 71.3%

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

6. Simplified 71.3%

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

7. Final simplification 71.3%

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

Alternative 4: 52.2% 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. Step-by-step derivation

   1. fabs-sub 100.0%

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

3. Simplified 100.0%

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

   1. div-inv 99.7%

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

   2. *-commutative 99.7%

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

   3. add-sqr-sqrt 52.7%

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

   4. fabs-sqr 52.7%

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

   5. add-sqr-sqrt 53.5%

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

   6. add-sqr-sqrt 38.1%

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

   7. fabs-sqr 38.1%

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

   8. add-sqr-sqrt 71.0%

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

5. Applied egg-rr 71.0%

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

6. Taylor expanded in y around inf 46.4%

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

7. Final simplification 46.4%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023301 
(FPCore (x y)
  :name "Numeric.LinearAlgebra.Util:formatSparse from hmatrix-0.16.1.5"
  :precision binary64
  (/ (fabs (- x y)) (fabs y)))