Numeric.LinearAlgebra.Util:formatSparse from hmatrix-0.16.1.5

Percentage Accurate: 100.0% → 100.0%

Time: 2.2s

Alternatives: 5

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Final simplification 100.0%

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

Alternative 2: 73.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|\frac{y - x}{y}\right| \leq 2000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (fabs (/ (- y x) y)) 2000000.0) 1.0 (/ (- x) y)))

C

double code(double x, double y) {
	double tmp;
	if (fabs(((y - x) / y)) <= 2000000.0) {
		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) / y)) <= 2000000.0d0) 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) / y)) <= 2000000.0) {
		tmp = 1.0;
	} else {
		tmp = -x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.fabs(((y - x) / y)) <= 2000000.0:
		tmp = 1.0
	else:
		tmp = -x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (abs(Float64(Float64(y - x) / y)) <= 2000000.0)
		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) / y)) <= 2000000.0)
		tmp = 1.0;
	else
		tmp = -x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Abs[N[(N[(y - x), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], 2000000.0], 1.0, N[((-x) / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (fabs.f64 (-.f64 x y)) (fabs.f64 y)) < 2e6

   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. lift-fabs.f64 N/A

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

      4. lift--.f64 N/A

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

      5. fabs-sub N/A

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

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

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

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

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

      8. sqrt-prod N/A

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

      9. rem-square-sqrt N/A

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

      10. sqrt-prod N/A

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

      11. rem-square-sqrt N/A

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

      12. div-sub N/A

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

      13. *-inverses N/A

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

      14. metadata-eval N/A

          \[\leadsto \color{blue}{2 \cdot \frac{1}{2}} - \frac{x}{y} \]

      15. lower--.f64 N/A

          \[\leadsto \color{blue}{2 \cdot \frac{1}{2} - \frac{x}{y}} \]

      16. metadata-eval N/A

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

      17. lower-/.f64 99.1

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 99.0%

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

   if 2e6 < (/.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. lift-fabs.f64 N/A

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

      4. lift--.f64 N/A

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

      5. fabs-sub N/A

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

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

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

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

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

      8. sqrt-prod N/A

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

      9. rem-square-sqrt N/A

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

      10. sqrt-prod N/A

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

      11. rem-square-sqrt N/A

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

      12. div-sub N/A

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

      13. *-inverses N/A

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

      14. metadata-eval N/A

          \[\leadsto \color{blue}{2 \cdot \frac{1}{2}} - \frac{x}{y} \]

      15. lower--.f64 N/A

          \[\leadsto \color{blue}{2 \cdot \frac{1}{2} - \frac{x}{y}} \]

      16. metadata-eval N/A

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

      17. lower-/.f64 49.6

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

   4. Applied rewrites 49.6%

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

   5. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 49.4

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

   7. Applied rewrites 49.4%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|\frac{y - x}{y}\right| \leq 2000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+221}:\\ \;\;\;\;\frac{x}{y} - 1\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -5e+221) (- (/ x y) 1.0) (- 1.0 (/ x y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -5e+221) {
		tmp = (x / y) - 1.0;
	} else {
		tmp = 1.0 - (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 (x <= (-5d+221)) then
        tmp = (x / y) - 1.0d0
    else
        tmp = 1.0d0 - (x / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -5e+221) {
		tmp = (x / y) - 1.0;
	} else {
		tmp = 1.0 - (x / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -5e+221:
		tmp = (x / y) - 1.0
	else:
		tmp = 1.0 - (x / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -5e+221)
		tmp = Float64(Float64(x / y) - 1.0);
	else
		tmp = Float64(1.0 - Float64(x / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -5e+221)
		tmp = (x / y) - 1.0;
	else
		tmp = 1.0 - (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -5e+221], N[(N[(x / y), $MachinePrecision] - 1.0), $MachinePrecision], N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.0000000000000002e221

   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. lift-fabs.f64 N/A

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

      4. div-fabs N/A

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

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

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

      6. sqrt-prod N/A

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

      7. rem-square-sqrt N/A

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

      8. lift--.f64 N/A

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

      9. div-sub N/A

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

      10. *-inverses N/A

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

      11. metadata-eval N/A

          \[\leadsto \frac{x}{y} - \color{blue}{2 \cdot \frac{1}{2}} \]

      12. lower--.f64 N/A

          \[\leadsto \color{blue}{\frac{x}{y} - 2 \cdot \frac{1}{2}} \]

      13. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{x}{y}} - 2 \cdot \frac{1}{2} \]

      14. metadata-eval 68.4

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

   4. Applied rewrites 68.4%

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

   if -5.0000000000000002e221 < 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. lift-fabs.f64 N/A

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

      4. lift--.f64 N/A

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

      5. fabs-sub N/A

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

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

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

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

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

      8. sqrt-prod N/A

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

      9. rem-square-sqrt N/A

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

      10. sqrt-prod N/A

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

      11. rem-square-sqrt N/A

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

      12. div-sub N/A

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

      13. *-inverses N/A

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

      14. metadata-eval N/A

          \[\leadsto \color{blue}{2 \cdot \frac{1}{2}} - \frac{x}{y} \]

      15. lower--.f64 N/A

          \[\leadsto \color{blue}{2 \cdot \frac{1}{2} - \frac{x}{y}} \]

      16. metadata-eval N/A

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

      17. lower-/.f64 74.1

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

   4. Applied rewrites 74.1%

      \[\leadsto \color{blue}{1 - \frac{x}{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 74.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\frac{\left|x - y\right|}{\left|y\right|} \]
2. 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. lift-fabs.f64 N/A

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

   4. lift--.f64 N/A

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

   5. fabs-sub N/A

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

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

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

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

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

   8. sqrt-prod N/A

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

   9. rem-square-sqrt N/A

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

   10. sqrt-prod N/A

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

   11. rem-square-sqrt N/A

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

   12. div-sub N/A

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

   13. *-inverses N/A

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

   14. metadata-eval N/A

       \[\leadsto \color{blue}{2 \cdot \frac{1}{2}} - \frac{x}{y} \]

   15. lower--.f64 N/A

       \[\leadsto \color{blue}{2 \cdot \frac{1}{2} - \frac{x}{y}} \]

   16. metadata-eval N/A

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

   17. lower-/.f64 70.5

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

4. Applied rewrites 70.5%

   \[\leadsto \color{blue}{1 - \frac{x}{y}} \]
5. Add Preprocessing

Alternative 5: 51.2% 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. lift-fabs.f64 N/A

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

   4. lift--.f64 N/A

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

   5. fabs-sub N/A

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

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

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

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

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

   8. sqrt-prod N/A

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

   9. rem-square-sqrt N/A

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

   10. sqrt-prod N/A

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

   11. rem-square-sqrt N/A

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

   12. div-sub N/A

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

   13. *-inverses N/A

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

   14. metadata-eval N/A

       \[\leadsto \color{blue}{2 \cdot \frac{1}{2}} - \frac{x}{y} \]

   15. lower--.f64 N/A

       \[\leadsto \color{blue}{2 \cdot \frac{1}{2} - \frac{x}{y}} \]

   16. metadata-eval N/A

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

   17. lower-/.f64 70.5

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

4. Applied rewrites 70.5%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 44.7%

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

Reproduce

herbie shell --seed 2024326 
(FPCore (x y)
  :name "Numeric.LinearAlgebra.Util:formatSparse from hmatrix-0.16.1.5"
  :precision binary64
  (/ (fabs (- x y)) (fabs y)))