Numeric.LinearAlgebra.Util:formatSparse from hmatrix-0.16.1.5

Percentage Accurate: 100.0% → 100.0%

Time: 5.5s

Alternatives: 5

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[Abs[N[(N[(x - y), $MachinePrecision] / y), $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. lift-/.f64 N/A

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

   2. lift-fabs.f64 N/A

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

   3. neg-fabs N/A

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

   4. lift-fabs.f64 N/A

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

   5. div-fabs N/A

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

   6. lower-fabs.f64 N/A

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

   7. neg-sub0 N/A

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

   8. lift--.f64 N/A

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

   9. associate--r- N/A

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

   10. neg-sub0 N/A

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

   11. +-commutative N/A

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

   12. sub-neg N/A

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

   13. lower-/.f64 N/A

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

   14. lower--.f64 100.0

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

4. Applied rewrites 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 74.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\frac{x - y}{y}\right|\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{+117} \lor \neg \left(t\_0 \leq 5 \cdot 10^{+273}\right):\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fabs (/ (- x y) y))))
   (if (or (<= t_0 2e+117) (not (<= t_0 5e+273))) (- 1.0 (/ x y)) (/ x y))))

C

double code(double x, double y) {
	double t_0 = fabs(((x - y) / y));
	double tmp;
	if ((t_0 <= 2e+117) || !(t_0 <= 5e+273)) {
		tmp = 1.0 - (x / y);
	} 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) :: t_0
    real(8) :: tmp
    t_0 = abs(((x - y) / y))
    if ((t_0 <= 2d+117) .or. (.not. (t_0 <= 5d+273))) then
        tmp = 1.0d0 - (x / y)
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.abs(((x - y) / y));
	double tmp;
	if ((t_0 <= 2e+117) || !(t_0 <= 5e+273)) {
		tmp = 1.0 - (x / y);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.fabs(((x - y) / y))
	tmp = 0
	if (t_0 <= 2e+117) or not (t_0 <= 5e+273):
		tmp = 1.0 - (x / y)
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	t_0 = abs(Float64(Float64(x - y) / y))
	tmp = 0.0
	if ((t_0 <= 2e+117) || !(t_0 <= 5e+273))
		tmp = Float64(1.0 - Float64(x / y));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = abs(((x - y) / y));
	tmp = 0.0;
	if ((t_0 <= 2e+117) || ~((t_0 <= 5e+273)))
		tmp = 1.0 - (x / y);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Abs[N[(N[(x - y), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[t$95$0, 2e+117], N[Not[LessEqual[t$95$0, 5e+273]], $MachinePrecision]], N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (fabs.f64 (-.f64 x y)) (fabs.f64 y)) < 2.0000000000000001e117 or 4.99999999999999961e273 < (/.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. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-fabs.f64 N/A

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

      3. neg-fabs N/A

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

      4. lift-fabs.f64 N/A

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

      5. div-fabs N/A

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

      6. lower-fabs.f64 N/A

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

      7. neg-sub0 N/A

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

      8. lift--.f64 N/A

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

      9. associate--r- N/A

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

      10. neg-sub0 N/A

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

      11. +-commutative N/A

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

      12. sub-neg N/A

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

      13. lower-/.f64 N/A

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

      14. lower--.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-fabs.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. inv-pow N/A

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

      5. sqr-pow N/A

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

      6. fabs-sqr N/A

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

      7. sqr-pow N/A

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

      8. inv-pow N/A

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

      9. clear-num N/A

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

      10. lift--.f64 N/A

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

      11. div-sub N/A

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

      12. *-inverses N/A

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

      13. metadata-eval N/A

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

      14. lower--.f64 N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 86.8

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

   6. Applied rewrites 86.8%

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

   if 2.0000000000000001e117 < (/.f64 (fabs.f64 (-.f64 x y)) (fabs.f64 y)) < 4.99999999999999961e273

   1. Initial program 100.0%

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

      1. lift-/.f64 N/A

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

      2. lift-fabs.f64 N/A

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

      3. neg-fabs N/A

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

      4. lift-fabs.f64 N/A

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

      5. div-fabs N/A

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

      6. lower-fabs.f64 N/A

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

      7. neg-sub0 N/A

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

      8. lift--.f64 N/A

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

      9. associate--r- N/A

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

      10. neg-sub0 N/A

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

      11. +-commutative N/A

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

      12. sub-neg N/A

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

      13. lower-/.f64 N/A

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

      14. lower--.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-fabs.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. inv-pow N/A

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

      5. sqr-pow N/A

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

      6. fabs-sqr N/A

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

      7. sqr-pow N/A

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

      8. inv-pow N/A

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

      9. frac-2neg N/A

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

      10. clear-num N/A

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

      11. div-inv N/A

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

      12. inv-pow N/A

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

      13. sqr-pow N/A

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

      14. pow-prod-down N/A

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

      15. sqr-neg N/A

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

      16. pow-prod-down N/A

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

      17. sqr-pow N/A

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

      18. inv-pow N/A

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

      19. div-inv N/A

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

      20. lift--.f64 N/A

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

      21. flip-- N/A

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

      22. distribute-neg-frac2 N/A

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

   6. Applied rewrites 58.5%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 1.0%

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

   10. Taylor expanded in x around inf

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

       1. lower-/.f64 58.5

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

   12. Applied rewrites 58.5%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|\frac{x - y}{y}\right| \leq 2 \cdot 10^{+117} \lor \neg \left(\left|\frac{x - y}{y}\right| \leq 5 \cdot 10^{+273}\right):\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|\frac{x - y}{y}\right| \leq 10^{+26}:\\ \;\;\;\;\left|1\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (fabs (/ (- x y) y)) 1e+26) (fabs 1.0) (/ x y)))

C

double code(double x, double y) {
	double tmp;
	if (fabs(((x - y) / y)) <= 1e+26) {
		tmp = fabs(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) / y)) <= 1d+26) then
        tmp = abs(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) / y)) <= 1e+26) {
		tmp = Math.abs(1.0);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.fabs(((x - y) / y)) <= 1e+26:
		tmp = math.fabs(1.0)
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (abs(Float64(Float64(x - y) / y)) <= 1e+26)
		tmp = abs(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) / y)) <= 1e+26)
		tmp = abs(1.0);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Abs[N[(N[(x - y), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], 1e+26], N[Abs[1.0], $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (fabs.f64 (-.f64 x y)) (fabs.f64 y)) < 1.00000000000000005e26

   1. Initial program 100.0%

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

      1. lift-/.f64 N/A

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

      2. lift-fabs.f64 N/A

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

      3. neg-fabs N/A

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

      4. lift-fabs.f64 N/A

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

      5. div-fabs N/A

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

      6. lower-fabs.f64 N/A

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

      7. neg-sub0 N/A

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

      8. lift--.f64 N/A

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

      9. associate--r- N/A

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

      10. neg-sub0 N/A

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

      11. +-commutative N/A

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

      12. sub-neg N/A

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

      13. lower-/.f64 N/A

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

      14. lower--.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 94.5%

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

   if 1.00000000000000005e26 < (/.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. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-fabs.f64 N/A

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

      3. neg-fabs N/A

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

      4. lift-fabs.f64 N/A

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

      5. div-fabs N/A

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

      6. lower-fabs.f64 N/A

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

      7. neg-sub0 N/A

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

      8. lift--.f64 N/A

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

      9. associate--r- N/A

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

      10. neg-sub0 N/A

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

      11. +-commutative N/A

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

      12. sub-neg N/A

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

      13. lower-/.f64 N/A

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

      14. lower--.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-fabs.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. inv-pow N/A

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

      5. sqr-pow N/A

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

      6. fabs-sqr N/A

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

      7. sqr-pow N/A

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

      8. inv-pow N/A

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

      9. frac-2neg N/A

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

      10. clear-num N/A

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

      11. div-inv N/A

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

      12. inv-pow N/A

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

      13. sqr-pow N/A

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

      14. pow-prod-down N/A

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

      15. sqr-neg N/A

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

      16. pow-prod-down N/A

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

      17. sqr-pow N/A

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

      18. inv-pow N/A

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

      19. div-inv N/A

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

      20. lift--.f64 N/A

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

      21. flip-- N/A

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

      22. distribute-neg-frac2 N/A

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

   6. Applied rewrites 48.3%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 1.0%

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

   10. Taylor expanded in x around inf

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

       1. lower-/.f64 48.3

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

   12. Applied rewrites 48.3%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|\frac{x - y}{y}\right| \leq 10^{+26}:\\ \;\;\;\;\left|1\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 51.1% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \left|1\right| \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x, y)
	return abs(1.0)
end

MATLAB

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

Wolfram

code[x_, y_] := N[Abs[1.0], $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. lift-/.f64 N/A

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

   2. lift-fabs.f64 N/A

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

   3. neg-fabs N/A

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

   4. lift-fabs.f64 N/A

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

   5. div-fabs N/A

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

   6. lower-fabs.f64 N/A

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

   7. neg-sub0 N/A

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

   8. lift--.f64 N/A

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

   9. associate--r- N/A

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

   10. neg-sub0 N/A

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

   11. +-commutative N/A

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

   12. sub-neg N/A

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

   13. lower-/.f64 N/A

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

   14. lower--.f64 100.0

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 50.7%

   \[\leadsto \left|\color{blue}{1}\right| \]
8. Add Preprocessing

Alternative 5: 1.3% 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. lift-/.f64 N/A

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

   2. lift-fabs.f64 N/A

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

   3. neg-fabs N/A

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

   4. lift-fabs.f64 N/A

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

   5. div-fabs N/A

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

   6. lower-fabs.f64 N/A

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

   7. neg-sub0 N/A

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

   8. lift--.f64 N/A

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

   9. associate--r- N/A

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

   10. neg-sub0 N/A

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

   11. +-commutative N/A

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

   12. sub-neg N/A

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

   13. lower-/.f64 N/A

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

   14. lower--.f64 100.0

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

4. Applied rewrites 100.0%

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

   1. lift-fabs.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. inv-pow N/A

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

   5. sqr-pow N/A

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

   6. fabs-sqr N/A

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

   7. sqr-pow N/A

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

   8. inv-pow N/A

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

   9. frac-2neg N/A

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

   10. clear-num N/A

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

   11. div-inv N/A

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

   12. inv-pow N/A

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

   13. sqr-pow N/A

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

   14. pow-prod-down N/A

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

   15. sqr-neg N/A

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

   16. pow-prod-down N/A

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

   17. sqr-pow N/A

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

   18. inv-pow N/A

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

   19. div-inv N/A

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

   20. lift--.f64 N/A

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

   21. flip-- N/A

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

   22. distribute-neg-frac2 N/A

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

6. Applied rewrites 24.8%

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

7. Taylor expanded in x around 0

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

9. Applied rewrites 1.3%

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

Reproduce

herbie shell --seed 2024313 
(FPCore (x y)
  :name "Numeric.LinearAlgebra.Util:formatSparse from hmatrix-0.16.1.5"
  :precision binary64
  (/ (fabs (- x y)) (fabs y)))