Numeric.LinearAlgebra.Util:formatSparse from hmatrix-0.16.1.5

Percentage Accurate: 100.0% → 100.0%

Time: 4.4s

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: 74.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|\frac{y - x}{y}\right| \leq 500000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (fabs (/ (- y x) y)) 500000.0) 1.0 (/ x y)))

C

double code(double x, double y) {
	double tmp;
	if (fabs(((y - x) / y)) <= 500000.0) {
		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)) <= 500000.0d0) 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)) <= 500000.0) {
		tmp = 1.0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.fabs(((y - x) / y)) <= 500000.0:
		tmp = 1.0
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (abs(Float64(Float64(y - x) / y)) <= 500000.0)
		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) / y)) <= 500000.0)
		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], 500000.0], 1.0, N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (fabs.f64 (-.f64 x y)) (fabs.f64 y)) < 5e5

   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 99.3

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 97.2%

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

   if 5e5 < (/.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-fabs.f64 N/A

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

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

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

      3. sqrt-prod N/A

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

      4. rem-square-sqrt 47.8

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

   4. Applied rewrites 47.8%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-fabs.f64 47.0

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

   7. Applied rewrites 47.0%

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

   9. Applied rewrites 51.2%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|\frac{y - x}{y}\right| \leq 500000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.85 \cdot 10^{+85}:\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{\left|y\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.85e+85) (- 1.0 (/ x y)) (/ (- x y) (fabs y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.85e+85) {
		tmp = 1.0 - (x / y);
	} else {
		tmp = (x - y) / 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 <= 1.85d+85) then
        tmp = 1.0d0 - (x / y)
    else
        tmp = (x - y) / abs(y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 1.85e+85) {
		tmp = 1.0 - (x / y);
	} else {
		tmp = (x - y) / Math.abs(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1.85e+85:
		tmp = 1.0 - (x / y)
	else:
		tmp = (x - y) / math.fabs(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.85e+85)
		tmp = Float64(1.0 - Float64(x / y));
	else
		tmp = Float64(Float64(x - y) / abs(y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 1.85e+85)
		tmp = 1.0 - (x / y);
	else
		tmp = (x - y) / abs(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.85e+85], N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision], N[(N[(x - y), $MachinePrecision] / N[Abs[y], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8500000000000001e85

   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 80.2

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

   4. Applied rewrites 80.2%

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

   if 1.8500000000000001e85 < x

   1. Initial program 100.0%

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

      1. lift-fabs.f64 N/A

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

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

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

      3. sqrt-prod N/A

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

      4. rem-square-sqrt 85.6

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

   4. Applied rewrites 85.6%

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

Alternative 4: 78.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.66 \cdot 10^{+174}:\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left|y\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.66e+174) (- 1.0 (/ x y)) (/ x (fabs y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.66e+174) {
		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 <= 1.66d+174) 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 <= 1.66e+174) {
		tmp = 1.0 - (x / y);
	} else {
		tmp = x / Math.abs(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1.66e+174:
		tmp = 1.0 - (x / y)
	else:
		tmp = x / math.fabs(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.66e+174)
		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 <= 1.66e+174)
		tmp = 1.0 - (x / y);
	else
		tmp = x / abs(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.66e+174], N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision], N[(x / N[Abs[y], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.66e174

   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 78.9

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

   4. Applied rewrites 78.9%

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

   if 1.66e174 < x

   1. Initial program 99.9%

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

      1. lift-fabs.f64 N/A

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

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

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

      3. sqrt-prod N/A

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

      4. rem-square-sqrt 92.1

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

   4. Applied rewrites 92.1%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-fabs.f64 85.5

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

   7. Applied rewrites 85.5%

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

Alternative 5: 52.1% 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 75.6

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

4. Applied rewrites 75.6%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 54.1%

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

Reproduce

herbie shell --seed 2024337 
(FPCore (x y)
  :name "Numeric.LinearAlgebra.Util:formatSparse from hmatrix-0.16.1.5"
  :precision binary64
  (/ (fabs (- x y)) (fabs y)))