Numeric.LinearAlgebra.Util:formatSparse from hmatrix-0.16.1.5

Percentage Accurate: 100.0% → 100.0%

Time: 6.3s

Alternatives: 6

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. lower-/.f64 N/A

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

   8. lift--.f64 N/A

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

   9. sub-neg N/A

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

   10. +-commutative N/A

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

   11. distribute-neg-in N/A

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

   12. remove-double-neg N/A

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

   13. sub-neg N/A

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

   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: 73.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|\frac{x - y}{y}\right| \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (fabs (/ (- x y) y)) 2.0) 1.0 (/ x y)))

C

double code(double x, double y) {
	double tmp;
	if (fabs(((x - y) / y)) <= 2.0) {
		tmp = 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)) <= 2.0d0) then
        tmp = 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)) <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.fabs(((x - y) / y)) <= 2.0:
		tmp = 1.0
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (abs(Float64(Float64(x - y) / y)) <= 2.0)
		tmp = 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)) <= 2.0)
		tmp = 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], 2.0], 1.0, N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   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. fabs-fabs-rev N/A

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

      4. lift-fabs.f64 N/A

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

      5. lift-fabs.f64 N/A

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

      6. fabs-fabs-rev N/A

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

      7. lift-fabs.f64 N/A

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

      8. div-fabs N/A

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

      9. frac-2neg N/A

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

      10. div-inv N/A

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

      11. fabs-mul N/A

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

      12. neg-fabs N/A

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

      13. lift-fabs.f64 N/A

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

      14. fabs-fabs-rev N/A

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

      15. rem-sqrt-square-rev N/A

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

      16. sqrt-prod N/A

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

      17. rem-square-sqrt N/A

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

      18. lower-*.f64 N/A

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

      19. lower-fabs.f64 N/A

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

      20. neg-mul-1 N/A

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

      21. lift-fabs.f64 N/A

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

      22. rem-sqrt-square-rev N/A

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

      23. sqrt-prod N/A

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

      24. rem-square-sqrt N/A

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

   4. Applied rewrites 42.8%

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

   5. Taylor expanded in x around inf

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

      1. fabs-div N/A

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

      2. metadata-eval N/A

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

      3. associate-*r/ N/A

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

      4. *-rgt-identity N/A

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

      5. lower-/.f64 N/A

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

      6. lower-fabs.f64 3.5

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

   7. Applied rewrites 3.5%

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

   9. Applied rewrites 3.6%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 98.4%

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

   if 2 < (/.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. fabs-fabs-rev N/A

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

      4. lift-fabs.f64 N/A

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

      5. lift-fabs.f64 N/A

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

      6. fabs-fabs-rev N/A

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

      7. lift-fabs.f64 N/A

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

      8. div-fabs N/A

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

      9. frac-2neg N/A

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

      10. div-inv N/A

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

      11. fabs-mul N/A

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

      12. neg-fabs N/A

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

      13. lift-fabs.f64 N/A

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

      14. fabs-fabs-rev N/A

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

      15. rem-sqrt-square-rev N/A

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

      16. sqrt-prod N/A

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

      17. rem-square-sqrt N/A

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

      18. lower-*.f64 N/A

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

      19. lower-fabs.f64 N/A

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

      20. neg-mul-1 N/A

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

      21. lift-fabs.f64 N/A

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

      22. rem-sqrt-square-rev N/A

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

      23. sqrt-prod N/A

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

      24. rem-square-sqrt N/A

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

   4. Applied rewrites 51.7%

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

   5. Taylor expanded in x around inf

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

      1. fabs-div N/A

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

      2. metadata-eval N/A

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

      3. associate-*r/ N/A

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

      4. *-rgt-identity N/A

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

      5. lower-/.f64 N/A

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

      6. lower-fabs.f64 50.3

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

   7. Applied rewrites 50.3%

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

   9. Applied rewrites 51.0%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|\frac{x - y}{y}\right| \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{+199}:\\ \;\;\;\;\frac{x}{y} - 1\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{+48}:\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left|y\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -7.5e+199)
   (- (/ x y) 1.0)
   (if (<= x 5.8e+48) (- 1.0 (/ x y)) (/ x (fabs y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -7.5e+199) {
		tmp = (x / y) - 1.0;
	} else if (x <= 5.8e+48) {
		tmp = 1.0 - (x / y);
	} else {
		tmp = x / fabs(y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-7.5d+199)) then
        tmp = (x / y) - 1.0d0
    else if (x <= 5.8d+48) then
        tmp = 1.0d0 - (x / y)
    else
        tmp = x / abs(y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -7.5e+199) {
		tmp = (x / y) - 1.0;
	} else if (x <= 5.8e+48) {
		tmp = 1.0 - (x / y);
	} else {
		tmp = x / Math.abs(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -7.5e+199:
		tmp = (x / y) - 1.0
	elif x <= 5.8e+48:
		tmp = 1.0 - (x / y)
	else:
		tmp = x / math.fabs(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -7.5e+199)
		tmp = Float64(Float64(x / y) - 1.0);
	elseif (x <= 5.8e+48)
		tmp = Float64(1.0 - Float64(x / y));
	else
		tmp = Float64(x / abs(y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -7.5e+199)
		tmp = (x / y) - 1.0;
	elseif (x <= 5.8e+48)
		tmp = 1.0 - (x / y);
	else
		tmp = x / abs(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -7.5e+199], N[(N[(x / y), $MachinePrecision] - 1.0), $MachinePrecision], If[LessEqual[x, 5.8e+48], N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision], N[(x / N[Abs[y], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.49999999999999977e199

   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. rem-sqrt-square-rev N/A

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

      4. sqrt-prod N/A

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

      5. rem-square-sqrt N/A

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

      6. lift--.f64 N/A

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

      7. lift-fabs.f64 N/A

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

      8. rem-sqrt-square-rev N/A

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

      9. sqrt-prod N/A

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

      10. rem-square-sqrt N/A

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

      11. div-sub N/A

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

      12. *-inverses N/A

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

      13. lower--.f64 N/A

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

      14. lower-/.f64 69.7

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

   4. Applied rewrites 69.7%

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

   if -7.49999999999999977e199 < x < 5.7999999999999998e48

   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. lower-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. distribute-neg-in N/A

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

      12. remove-double-neg N/A

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

      13. sub-neg N/A

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

      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. rem-sqrt-square-rev N/A

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

      3. sqrt-prod N/A

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

      4. rem-square-sqrt 84.6

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

      5. lift-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. div-sub N/A

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

      8. *-inverses N/A

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

      9. lift-/.f64 N/A

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

      10. lower--.f64 84.6

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

   6. Applied rewrites 84.6%

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

   if 5.7999999999999998e48 < x

   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. fabs-fabs-rev N/A

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

      4. lift-fabs.f64 N/A

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

      5. lift-fabs.f64 N/A

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

      6. fabs-fabs-rev N/A

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

      7. lift-fabs.f64 N/A

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

      8. div-fabs N/A

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

      9. frac-2neg N/A

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

      10. div-inv N/A

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

      11. fabs-mul N/A

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

      12. neg-fabs N/A

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

      13. lift-fabs.f64 N/A

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

      14. fabs-fabs-rev N/A

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

      15. rem-sqrt-square-rev N/A

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

      16. sqrt-prod N/A

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

      17. rem-square-sqrt N/A

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

      18. lower-*.f64 N/A

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

      19. lower-fabs.f64 N/A

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

      20. neg-mul-1 N/A

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

      21. lift-fabs.f64 N/A

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

      22. rem-sqrt-square-rev N/A

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

      23. sqrt-prod N/A

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

      24. rem-square-sqrt N/A

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

   4. Applied rewrites 91.3%

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

   5. Taylor expanded in x around inf

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

      1. fabs-div N/A

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

      2. metadata-eval N/A

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

      3. associate-*r/ N/A

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

      4. *-rgt-identity N/A

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

      5. lower-/.f64 N/A

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

      6. lower-fabs.f64 87.4

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

   7. Applied rewrites 87.4%

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

Alternative 4: 78.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{+199}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{+48}:\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left|y\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -7.5e+199)
   (/ x y)
   (if (<= x 5.8e+48) (- 1.0 (/ x y)) (/ x (fabs y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -7.5e+199) {
		tmp = x / y;
	} else if (x <= 5.8e+48) {
		tmp = 1.0 - (x / y);
	} else {
		tmp = x / fabs(y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-7.5d+199)) then
        tmp = x / y
    else if (x <= 5.8d+48) then
        tmp = 1.0d0 - (x / y)
    else
        tmp = x / abs(y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -7.5e+199) {
		tmp = x / y;
	} else if (x <= 5.8e+48) {
		tmp = 1.0 - (x / y);
	} else {
		tmp = x / Math.abs(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -7.5e+199:
		tmp = x / y
	elif x <= 5.8e+48:
		tmp = 1.0 - (x / y)
	else:
		tmp = x / math.fabs(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -7.5e+199)
		tmp = Float64(x / y);
	elseif (x <= 5.8e+48)
		tmp = Float64(1.0 - Float64(x / y));
	else
		tmp = Float64(x / abs(y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -7.5e+199)
		tmp = x / y;
	elseif (x <= 5.8e+48)
		tmp = 1.0 - (x / y);
	else
		tmp = x / abs(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -7.5e+199], N[(x / y), $MachinePrecision], If[LessEqual[x, 5.8e+48], N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision], N[(x / N[Abs[y], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.49999999999999977e199

   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. fabs-fabs-rev N/A

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

      4. lift-fabs.f64 N/A

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

      5. lift-fabs.f64 N/A

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

      6. fabs-fabs-rev N/A

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

      7. lift-fabs.f64 N/A

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

      8. div-fabs N/A

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

      9. frac-2neg N/A

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

      10. div-inv N/A

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

      11. fabs-mul N/A

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

      12. neg-fabs N/A

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

      13. lift-fabs.f64 N/A

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

      14. fabs-fabs-rev N/A

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

      15. rem-sqrt-square-rev N/A

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

      16. sqrt-prod N/A

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

      17. rem-square-sqrt N/A

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

      18. lower-*.f64 N/A

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

      19. lower-fabs.f64 N/A

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

      20. neg-mul-1 N/A

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

      21. lift-fabs.f64 N/A

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

      22. rem-sqrt-square-rev N/A

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

      23. sqrt-prod N/A

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

      24. rem-square-sqrt N/A

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

   4. Applied rewrites 4.9%

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

   5. Taylor expanded in x around inf

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

      1. fabs-div N/A

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

      2. metadata-eval N/A

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

      3. associate-*r/ N/A

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

      4. *-rgt-identity N/A

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

      5. lower-/.f64 N/A

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

      6. lower-fabs.f64 0.6

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

   7. Applied rewrites 0.6%

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

   9. Applied rewrites 67.0%

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

   if -7.49999999999999977e199 < x < 5.7999999999999998e48

   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. lower-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. distribute-neg-in N/A

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

      12. remove-double-neg N/A

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

      13. sub-neg N/A

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

      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. rem-sqrt-square-rev N/A

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

      3. sqrt-prod N/A

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

      4. rem-square-sqrt 84.6

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

      5. lift-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. div-sub N/A

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

      8. *-inverses N/A

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

      9. lift-/.f64 N/A

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

      10. lower--.f64 84.6

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

   6. Applied rewrites 84.6%

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

   if 5.7999999999999998e48 < x

   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. fabs-fabs-rev N/A

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

      4. lift-fabs.f64 N/A

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

      5. lift-fabs.f64 N/A

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

      6. fabs-fabs-rev N/A

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

      7. lift-fabs.f64 N/A

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

      8. div-fabs N/A

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

      9. frac-2neg N/A

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

      10. div-inv N/A

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

      11. fabs-mul N/A

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

      12. neg-fabs N/A

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

      13. lift-fabs.f64 N/A

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

      14. fabs-fabs-rev N/A

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

      15. rem-sqrt-square-rev N/A

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

      16. sqrt-prod N/A

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

      17. rem-square-sqrt N/A

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

      18. lower-*.f64 N/A

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

      19. lower-fabs.f64 N/A

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

      20. neg-mul-1 N/A

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

      21. lift-fabs.f64 N/A

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

      22. rem-sqrt-square-rev N/A

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

      23. sqrt-prod N/A

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

      24. rem-square-sqrt N/A

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

   4. Applied rewrites 91.3%

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

   5. Taylor expanded in x around inf

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

      1. fabs-div N/A

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

      2. metadata-eval N/A

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

      3. associate-*r/ N/A

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

      4. *-rgt-identity N/A

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

      5. lower-/.f64 N/A

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

      6. lower-fabs.f64 87.4

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

   7. Applied rewrites 87.4%

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

Alternative 5: 51.5% 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. fabs-fabs-rev N/A

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

   4. lift-fabs.f64 N/A

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

   5. lift-fabs.f64 N/A

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

   6. fabs-fabs-rev N/A

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

   7. lift-fabs.f64 N/A

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

   8. div-fabs N/A

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

   9. frac-2neg N/A

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

   10. div-inv N/A

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

   11. fabs-mul N/A

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

   12. neg-fabs N/A

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

   13. lift-fabs.f64 N/A

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

   14. fabs-fabs-rev N/A

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

   15. rem-sqrt-square-rev N/A

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

   16. sqrt-prod N/A

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

   17. rem-square-sqrt N/A

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

   18. lower-*.f64 N/A

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

   19. lower-fabs.f64 N/A

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

   20. neg-mul-1 N/A

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

   21. lift-fabs.f64 N/A

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

   22. rem-sqrt-square-rev N/A

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

   23. sqrt-prod N/A

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

   24. rem-square-sqrt N/A

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

4. Applied rewrites 47.3%

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

5. Taylor expanded in x around inf

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

   1. fabs-div N/A

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

   2. metadata-eval N/A

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

   3. associate-*r/ N/A

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

   4. *-rgt-identity N/A

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

   5. lower-/.f64 N/A

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

   6. lower-fabs.f64 27.2

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

7. Applied rewrites 27.2%

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

9. Applied rewrites 27.6%

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

10. Taylor expanded in x around 0

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

12. Applied rewrites 51.1%

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

Alternative 6: 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. rem-sqrt-square-rev N/A

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

   4. sqrt-prod N/A

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

   5. rem-square-sqrt N/A

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

   6. lift--.f64 N/A

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

   7. lift-fabs.f64 N/A

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

   8. rem-sqrt-square-rev N/A

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

   9. sqrt-prod N/A

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

   10. rem-square-sqrt N/A

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

   11. div-sub N/A

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

   12. *-inverses N/A

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

   13. lower--.f64 N/A

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

   14. lower-/.f64 27.5

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

4. Applied rewrites 27.5%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 1.3%

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

Reproduce

herbie shell --seed 2024298 
(FPCore (x y)
  :name "Numeric.LinearAlgebra.Util:formatSparse from hmatrix-0.16.1.5"
  :precision binary64
  (/ (fabs (- x y)) (fabs y)))