Numeric.LinearAlgebra.Util:formatSparse from hmatrix-0.16.1.5

Percentage Accurate: 100.0% → 100.0%

Time: 6.9s

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. 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-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. div-sub N/A

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

   4. *-inverses N/A

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

   5. lower--.f64 N/A

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

   6. lower-/.f64 100.0

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

6. Applied rewrites 100.0%

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

7. Final simplification 100.0%

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

Alternative 2: 99.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left|y - x\right|}{\left|y\right|} \leq 40000000:\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-x}{y}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (fabs (- y x)) (fabs y)) 40000000.0)
   (- 1.0 (/ x y))
   (fabs (/ (- x) y))))

C

double code(double x, double y) {
	double tmp;
	if ((fabs((y - x)) / fabs(y)) <= 40000000.0) {
		tmp = 1.0 - (x / y);
	} else {
		tmp = fabs((-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((y - x)) / abs(y)) <= 40000000.0d0) then
        tmp = 1.0d0 - (x / y)
    else
        tmp = abs((-x / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((Math.abs((y - x)) / Math.abs(y)) <= 40000000.0) {
		tmp = 1.0 - (x / y);
	} else {
		tmp = Math.abs((-x / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (math.fabs((y - x)) / math.fabs(y)) <= 40000000.0:
		tmp = 1.0 - (x / y)
	else:
		tmp = math.fabs((-x / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(abs(Float64(y - x)) / abs(y)) <= 40000000.0)
		tmp = Float64(1.0 - Float64(x / y));
	else
		tmp = abs(Float64(Float64(-x) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((abs((y - x)) / abs(y)) <= 40000000.0)
		tmp = 1.0 - (x / y);
	else
		tmp = abs((-x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Abs[N[(y - x), $MachinePrecision]], $MachinePrecision] / N[Abs[y], $MachinePrecision]), $MachinePrecision], 40000000.0], N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision], N[Abs[N[((-x) / y), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (fabs.f64 (-.f64 x y)) (fabs.f64 y)) < 4e7

   1. Initial program 99.9%

      \[\frac{\left|x - y\right|}{\left|y\right|} \]
   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 99.9

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

   4. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\left|\frac{y - x}{y}\right|} \]
   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. lift-/.f64 N/A

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

      5. fabs-div N/A

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

      6. metadata-eval N/A

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

      7. lift-fabs.f64 N/A

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

      8. inv-pow N/A

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

      9. metadata-eval N/A

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

      10. pow-pow N/A

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

      11. pow2 N/A

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

      12. lift-fabs.f64 N/A

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

      13. lift-fabs.f64 N/A

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

      14. sqr-abs N/A

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

      15. pow-prod-down N/A

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

      16. pow-prod-up N/A

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

      17. metadata-eval N/A

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

      18. inv-pow N/A

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

      19. lift-/.f64 N/A

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

      20. clear-num N/A

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

      21. lift--.f64 N/A

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

      22. div-sub N/A

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

      23. *-inverses N/A

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

      24. lower--.f64 N/A

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

      25. lower-/.f64 97.9

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

   6. Applied rewrites 97.9%

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

   if 4e7 < (/.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. 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. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 99.5

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

   7. Applied rewrites 99.5%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left|y - x\right|}{\left|y\right|} \leq 40000000:\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left|y - x\right|}{\left|y\right|} \leq 2000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (fabs (- y x)) (fabs y)) 2000.0) 1.0 (/ (- x) y)))

C

double code(double x, double y) {
	double tmp;
	if ((fabs((y - x)) / fabs(y)) <= 2000.0) {
		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((y - x)) / abs(y)) <= 2000.0d0) 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((y - x)) / Math.abs(y)) <= 2000.0) {
		tmp = 1.0;
	} else {
		tmp = -x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (math.fabs((y - x)) / math.fabs(y)) <= 2000.0:
		tmp = 1.0
	else:
		tmp = -x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(abs(Float64(y - x)) / abs(y)) <= 2000.0)
		tmp = 1.0;
	else
		tmp = Float64(Float64(-x) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((abs((y - x)) / abs(y)) <= 2000.0)
		tmp = 1.0;
	else
		tmp = -x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Abs[N[(y - x), $MachinePrecision]], $MachinePrecision] / N[Abs[y], $MachinePrecision]), $MachinePrecision], 2000.0], 1.0, N[((-x) / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (fabs.f64 (-.f64 x y)) (fabs.f64 y)) < 2e3

   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. 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. lift-/.f64 N/A

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

      5. fabs-div N/A

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

      6. metadata-eval N/A

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

      7. lift-fabs.f64 N/A

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

      8. inv-pow N/A

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

      9. metadata-eval N/A

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

      10. pow-pow N/A

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

      11. pow2 N/A

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

      12. lift-fabs.f64 N/A

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

      13. lift-fabs.f64 N/A

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

      14. sqr-abs N/A

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

      15. pow-prod-down N/A

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

      16. pow-prod-up N/A

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

      17. metadata-eval N/A

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

      18. inv-pow N/A

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

      19. lift-/.f64 N/A

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

      20. clear-num N/A

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

      21. lift-/.f64 99.3

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

   6. Applied rewrites 99.3%

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

   7. Taylor expanded in y around inf

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

   9. Applied rewrites 96.3%

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

   if 2e3 < (/.f64 (fabs.f64 (-.f64 x y)) (fabs.f64 y))

   1. Initial program 99.9%

      \[\frac{\left|x - y\right|}{\left|y\right|} \]
   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 99.9

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

   4. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\left|\frac{y - x}{y}\right|} \]
   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. lift-/.f64 N/A

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

      5. fabs-div N/A

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

      6. metadata-eval N/A

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

      7. lift-fabs.f64 N/A

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

      8. inv-pow N/A

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

      9. metadata-eval N/A

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

      10. pow-pow N/A

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

      11. pow2 N/A

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

      12. lift-fabs.f64 N/A

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

      13. lift-fabs.f64 N/A

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

      14. sqr-abs N/A

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

      15. pow-prod-down N/A

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

      16. pow-prod-up N/A

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

      17. metadata-eval N/A

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

      18. inv-pow N/A

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

      19. lift-/.f64 N/A

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

      20. clear-num N/A

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

      21. lift-/.f64 54.3

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

   6. Applied rewrites 54.3%

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

   7. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 52.4

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

   9. Applied rewrites 52.4%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left|y - x\right|}{\left|y\right|} \leq 2000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(abs(Float64(y - x)) / abs(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((y - x)) / abs(y)) <= 2.0)
		tmp = 1.0;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

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

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

      5. fabs-div N/A

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

      6. metadata-eval N/A

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

      7. lift-fabs.f64 N/A

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

      8. inv-pow N/A

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

      9. metadata-eval N/A

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

      10. pow-pow N/A

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

      11. pow2 N/A

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

      12. lift-fabs.f64 N/A

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

      13. lift-fabs.f64 N/A

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

      14. sqr-abs N/A

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

      15. pow-prod-down N/A

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

      16. pow-prod-up N/A

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

      17. metadata-eval N/A

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

      18. inv-pow N/A

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

      19. lift-/.f64 N/A

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

      20. clear-num N/A

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

      21. lift-/.f64 100.0

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in y around inf

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

   9. Applied rewrites 96.9%

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

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

   1. Initial program 99.9%

      \[\frac{\left|x - y\right|}{\left|y\right|} \]
   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 99.9

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

   4. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\left|\frac{y - x}{y}\right|} \]
   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. lift-/.f64 N/A

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

      5. fabs-div N/A

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

      6. metadata-eval N/A

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

      7. lift-fabs.f64 N/A

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

      8. inv-pow N/A

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

      9. metadata-eval N/A

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

      10. pow-pow N/A

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

      11. pow2 N/A

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

      12. lift-fabs.f64 N/A

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

      13. lift-fabs.f64 N/A

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

      14. sqr-abs N/A

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

      15. pow-prod-down N/A

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

      16. pow-prod-up N/A

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

      17. metadata-eval N/A

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

      18. inv-pow N/A

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

      19. lift-/.f64 N/A

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

      20. clear-num N/A

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

      21. lift-/.f64 53.9

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

   6. Applied rewrites 53.9%

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

   7. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 52.0

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

   9. Applied rewrites 52.0%

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

   11. Applied rewrites 45.0%

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

4. Final simplification 72.0%

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

Alternative 5: 74.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{x}{y}\\ \mathbf{if}\;x \leq -9 \cdot 10^{+247}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -1.7 \cdot 10^{+202}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ x y))))
   (if (<= x -9e+247) t_0 (if (<= x -1.7e+202) (/ x y) t_0))))

C

double code(double x, double y) {
	double t_0 = 1.0 - (x / y);
	double tmp;
	if (x <= -9e+247) {
		tmp = t_0;
	} else if (x <= -1.7e+202) {
		tmp = x / y;
	} else {
		tmp = t_0;
	}
	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 = 1.0d0 - (x / y)
    if (x <= (-9d+247)) then
        tmp = t_0
    else if (x <= (-1.7d+202)) then
        tmp = x / y
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 - (x / y);
	double tmp;
	if (x <= -9e+247) {
		tmp = t_0;
	} else if (x <= -1.7e+202) {
		tmp = x / y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 - (x / y)
	tmp = 0
	if x <= -9e+247:
		tmp = t_0
	elif x <= -1.7e+202:
		tmp = x / y
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 - Float64(x / y))
	tmp = 0.0
	if (x <= -9e+247)
		tmp = t_0;
	elseif (x <= -1.7e+202)
		tmp = Float64(x / y);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 - (x / y);
	tmp = 0.0;
	if (x <= -9e+247)
		tmp = t_0;
	elseif (x <= -1.7e+202)
		tmp = x / y;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -9e+247], t$95$0, If[LessEqual[x, -1.7e+202], N[(x / y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -9.00000000000000004e247 or -1.7e202 < 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. 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. 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. lift-/.f64 N/A

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

      5. fabs-div N/A

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

      6. metadata-eval N/A

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

      7. lift-fabs.f64 N/A

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

      8. inv-pow N/A

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

      9. metadata-eval N/A

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

      10. pow-pow N/A

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

      11. pow2 N/A

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

      12. lift-fabs.f64 N/A

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

      13. lift-fabs.f64 N/A

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

      14. sqr-abs N/A

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

      15. pow-prod-down N/A

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

      16. pow-prod-up N/A

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

      17. metadata-eval N/A

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

      18. inv-pow N/A

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

      19. lift-/.f64 N/A

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

      20. clear-num N/A

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

      21. lift--.f64 N/A

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

      22. div-sub N/A

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

      23. *-inverses N/A

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

      24. lower--.f64 N/A

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

      25. lower-/.f64 81.6

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

   6. Applied rewrites 81.6%

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

   if -9.00000000000000004e247 < x < -1.7e202

   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. 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. lift-/.f64 N/A

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

      5. fabs-div N/A

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

      6. metadata-eval N/A

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

      7. lift-fabs.f64 N/A

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

      8. inv-pow N/A

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

      9. metadata-eval N/A

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

      10. pow-pow N/A

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

      11. pow2 N/A

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

      12. lift-fabs.f64 N/A

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

      13. lift-fabs.f64 N/A

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

      14. sqr-abs N/A

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

      15. pow-prod-down N/A

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

      16. pow-prod-up N/A

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

      17. metadata-eval N/A

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

      18. inv-pow N/A

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

      19. lift-/.f64 N/A

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

      20. clear-num N/A

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

      21. lift-/.f64 7.9

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

   6. Applied rewrites 7.9%

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

   7. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 7.9

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

   9. Applied rewrites 7.9%

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

   11. Applied rewrites 92.4%

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

Alternative 6: 51.4% 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. 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. 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. lift-/.f64 N/A

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

   5. fabs-div N/A

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

   6. metadata-eval N/A

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

   7. lift-fabs.f64 N/A

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

   8. inv-pow N/A

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

   9. metadata-eval N/A

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

   10. pow-pow N/A

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

   11. pow2 N/A

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

   12. lift-fabs.f64 N/A

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

   13. lift-fabs.f64 N/A

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

   14. sqr-abs N/A

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

   15. pow-prod-down N/A

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

   16. pow-prod-up N/A

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

   17. metadata-eval N/A

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

   18. inv-pow N/A

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

   19. lift-/.f64 N/A

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

   20. clear-num N/A

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

   21. lift-/.f64 77.8

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

6. Applied rewrites 77.8%

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

7. Taylor expanded in y around inf

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

9. Applied rewrites 52.9%

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

Reproduce

herbie shell --seed 2024268 
(FPCore (x y)
  :name "Numeric.LinearAlgebra.Util:formatSparse from hmatrix-0.16.1.5"
  :precision binary64
  (/ (fabs (- x y)) (fabs y)))