Numeric.LinearAlgebra.Util:formatSparse from hmatrix-0.16.1.5

Percentage Accurate: 100.0% → 100.0%

Time: 3.6s

Alternatives: 7

Speedup: 2.0×

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, 2.0× 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. Step-by-step derivation

   1. div-fabs 100.0%

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

   2. div-sub 100.0%

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

   3. pow1 100.0%

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

   4. pow1 100.0%

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

   5. pow-div 100.0%

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

   6. metadata-eval 100.0%

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

   7. metadata-eval 100.0%

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

3. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 74.7% accurate, 13.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{+114} \lor \neg \left(x \leq -780000000\right) \land \left(x \leq -6.5 \cdot 10^{-236} \lor \neg \left(x \leq 8.6 \cdot 10^{-103}\right) \land x \leq 3.4 \cdot 10^{+85}\right):\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -2.1e+114)
         (and (not (<= x -780000000.0))
              (or (<= x -6.5e-236)
                  (and (not (<= x 8.6e-103)) (<= x 3.4e+85)))))
   (- 1.0 (/ x y))
   (/ (+ x y) y)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -2.1e+114) || (!(x <= -780000000.0) && ((x <= -6.5e-236) || (!(x <= 8.6e-103) && (x <= 3.4e+85))))) {
		tmp = 1.0 - (x / y);
	} else {
		tmp = (x + y) / 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 <= (-2.1d+114)) .or. (.not. (x <= (-780000000.0d0))) .and. (x <= (-6.5d-236)) .or. (.not. (x <= 8.6d-103)) .and. (x <= 3.4d+85)) then
        tmp = 1.0d0 - (x / y)
    else
        tmp = (x + y) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -2.1e+114) || (!(x <= -780000000.0) && ((x <= -6.5e-236) || (!(x <= 8.6e-103) && (x <= 3.4e+85))))) {
		tmp = 1.0 - (x / y);
	} else {
		tmp = (x + y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -2.1e+114) or (not (x <= -780000000.0) and ((x <= -6.5e-236) or (not (x <= 8.6e-103) and (x <= 3.4e+85)))):
		tmp = 1.0 - (x / y)
	else:
		tmp = (x + y) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -2.1e+114) || (!(x <= -780000000.0) && ((x <= -6.5e-236) || (!(x <= 8.6e-103) && (x <= 3.4e+85)))))
		tmp = Float64(1.0 - Float64(x / y));
	else
		tmp = Float64(Float64(x + y) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -2.1e+114) || (~((x <= -780000000.0)) && ((x <= -6.5e-236) || (~((x <= 8.6e-103)) && (x <= 3.4e+85)))))
		tmp = 1.0 - (x / y);
	else
		tmp = (x + y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -2.1e+114], And[N[Not[LessEqual[x, -780000000.0]], $MachinePrecision], Or[LessEqual[x, -6.5e-236], And[N[Not[LessEqual[x, 8.6e-103]], $MachinePrecision], LessEqual[x, 3.4e+85]]]]], N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision], N[(N[(x + y), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.1e114 or -7.8e8 < x < -6.5000000000000001e-236 or 8.60000000000000045e-103 < x < 3.4000000000000003e85

   1. Initial program 99.9%

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

      1. add-sqr-sqrt 32.3%

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

      2. fabs-sqr 32.3%

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

      3. add-sqr-sqrt 7.1%

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

      4. fabs-sqr 7.1%

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

      5. add-sqr-sqrt 7.7%

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

      6. add-sqr-sqrt 22.3%

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

      7. clear-num 22.3%

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

      8. associate-/r/ 22.3%

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

   3. Applied egg-rr 22.3%

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

      1. associate-*l/ 22.3%

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

      2. *-un-lft-identity 22.3%

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

      3. frac-2neg 22.3%

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

      4. sub-neg 22.3%

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

      5. distribute-neg-in 22.3%

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

      6. remove-double-neg 22.3%

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

      7. add-sqr-sqrt 14.6%

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

      8. sqrt-unprod 45.0%

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

      9. sqr-neg 45.0%

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

      10. sqrt-unprod 53.0%

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

      11. add-sqr-sqrt 78.7%

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

   5. Applied egg-rr 78.7%

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

   6. Taylor expanded in x around 0 78.7%

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

      1. mul-1-neg 78.7%

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

      2. unsub-neg 78.7%

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

   8. Simplified 78.7%

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

   if -2.1e114 < x < -7.8e8 or -6.5000000000000001e-236 < x < 8.60000000000000045e-103 or 3.4000000000000003e85 < x

   1. Initial program 100.0%

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

      1. add-sqr-sqrt 54.4%

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

      2. fabs-sqr 54.4%

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

      3. add-sqr-sqrt 27.5%

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

      4. fabs-sqr 27.5%

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

      5. add-sqr-sqrt 28.1%

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

      6. add-sqr-sqrt 38.5%

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

      7. clear-num 38.4%

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

      8. associate-/r/ 38.4%

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

   3. Applied egg-rr 38.4%

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

      1. associate-*l/ 38.5%

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

      2. *-un-lft-identity 38.5%

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

      3. sub-neg 38.5%

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

      4. add-sqr-sqrt 10.3%

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

      5. sqrt-unprod 50.7%

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

      6. sqr-neg 50.7%

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

      7. sqrt-unprod 58.9%

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

      8. add-sqr-sqrt 85.1%

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

   5. Applied egg-rr 85.1%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{+114} \lor \neg \left(x \leq -780000000\right) \land \left(x \leq -6.5 \cdot 10^{-236} \lor \neg \left(x \leq 8.6 \cdot 10^{-103}\right) \land x \leq 3.4 \cdot 10^{+85}\right):\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y}{y}\\ \end{array} \]

Alternative 3: 58.1% accurate, 18.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{y}\\ \mathbf{if}\;x \leq -2.95 \cdot 10^{+118}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -440000000:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -3.25 \cdot 10^{-12}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+89}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- x) y)))
   (if (<= x -2.95e+118)
     t_0
     (if (<= x -440000000.0)
       (/ x y)
       (if (<= x -3.25e-12) t_0 (if (<= x 4e+89) 1.0 (/ x y)))))))

C

double code(double x, double y) {
	double t_0 = -x / y;
	double tmp;
	if (x <= -2.95e+118) {
		tmp = t_0;
	} else if (x <= -440000000.0) {
		tmp = x / y;
	} else if (x <= -3.25e-12) {
		tmp = t_0;
	} else if (x <= 4e+89) {
		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) :: t_0
    real(8) :: tmp
    t_0 = -x / y
    if (x <= (-2.95d+118)) then
        tmp = t_0
    else if (x <= (-440000000.0d0)) then
        tmp = x / y
    else if (x <= (-3.25d-12)) then
        tmp = t_0
    else if (x <= 4d+89) then
        tmp = 1.0d0
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = -x / y;
	double tmp;
	if (x <= -2.95e+118) {
		tmp = t_0;
	} else if (x <= -440000000.0) {
		tmp = x / y;
	} else if (x <= -3.25e-12) {
		tmp = t_0;
	} else if (x <= 4e+89) {
		tmp = 1.0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = -x / y
	tmp = 0
	if x <= -2.95e+118:
		tmp = t_0
	elif x <= -440000000.0:
		tmp = x / y
	elif x <= -3.25e-12:
		tmp = t_0
	elif x <= 4e+89:
		tmp = 1.0
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(-x) / y)
	tmp = 0.0
	if (x <= -2.95e+118)
		tmp = t_0;
	elseif (x <= -440000000.0)
		tmp = Float64(x / y);
	elseif (x <= -3.25e-12)
		tmp = t_0;
	elseif (x <= 4e+89)
		tmp = 1.0;
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = -x / y;
	tmp = 0.0;
	if (x <= -2.95e+118)
		tmp = t_0;
	elseif (x <= -440000000.0)
		tmp = x / y;
	elseif (x <= -3.25e-12)
		tmp = t_0;
	elseif (x <= 4e+89)
		tmp = 1.0;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[((-x) / y), $MachinePrecision]}, If[LessEqual[x, -2.95e+118], t$95$0, If[LessEqual[x, -440000000.0], N[(x / y), $MachinePrecision], If[LessEqual[x, -3.25e-12], t$95$0, If[LessEqual[x, 4e+89], 1.0, N[(x / y), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.9499999999999999e118 or -4.4e8 < x < -3.2500000000000001e-12

   1. Initial program 99.9%

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

      1. add-sqr-sqrt 6.2%

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

      2. fabs-sqr 6.2%

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

      3. add-sqr-sqrt 0.0%

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

      4. fabs-sqr 0.0%

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

      5. add-sqr-sqrt 0.4%

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

      6. add-sqr-sqrt 25.5%

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

      7. clear-num 25.4%

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

      8. associate-/r/ 25.4%

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

   3. Applied egg-rr 25.4%

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

      1. associate-*l/ 25.5%

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

      2. *-un-lft-identity 25.5%

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

      3. frac-2neg 25.5%

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

      4. sub-neg 25.5%

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

      5. distribute-neg-in 25.5%

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

      6. remove-double-neg 25.5%

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

      7. add-sqr-sqrt 25.0%

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

      8. sqrt-unprod 78.1%

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

      9. sqr-neg 78.1%

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

      10. sqrt-unprod 68.4%

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

      11. add-sqr-sqrt 74.9%

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

   5. Applied egg-rr 74.9%

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

   6. Taylor expanded in x around inf 60.5%

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

      1. associate-*r/ 60.5%

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

      2. mul-1-neg 60.5%

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

   8. Simplified 60.5%

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

   if -2.9499999999999999e118 < x < -4.4e8 or 3.99999999999999998e89 < x

   1. Initial program 100.0%

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

      1. add-sqr-sqrt 61.2%

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

      2. fabs-sqr 61.2%

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

      3. add-sqr-sqrt 38.4%

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

      4. fabs-sqr 38.4%

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

      5. add-sqr-sqrt 38.8%

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

      6. add-sqr-sqrt 55.1%

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

      7. clear-num 55.0%

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

      8. associate-/r/ 54.9%

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

   3. Applied egg-rr 54.9%

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

   4. Taylor expanded in y around 0 55.4%

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

   if -3.2500000000000001e-12 < x < 3.99999999999999998e89

   1. Initial program 100.0%

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

      1. add-sqr-sqrt 47.1%

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

      2. fabs-sqr 47.1%

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

      3. add-sqr-sqrt 11.9%

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

      4. fabs-sqr 11.9%

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

      5. add-sqr-sqrt 12.7%

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

      6. add-sqr-sqrt 18.5%

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

      7. clear-num 18.5%

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

      8. associate-/r/ 18.5%

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

   3. Applied egg-rr 18.5%

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

      1. associate-*l/ 18.5%

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

      2. *-un-lft-identity 18.5%

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

      3. sub-neg 18.5%

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

      4. add-sqr-sqrt 5.7%

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

      5. sqrt-unprod 39.5%

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

      6. sqr-neg 39.5%

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

      7. sqrt-unprod 53.9%

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

      8. add-sqr-sqrt 82.6%

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

   5. Applied egg-rr 82.6%

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

   6. Taylor expanded in x around 0 68.8%

      \[\leadsto \color{blue}{1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.95 \cdot 10^{+118}:\\ \;\;\;\;\frac{-x}{y}\\ \mathbf{elif}\;x \leq -440000000:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -3.25 \cdot 10^{-12}:\\ \;\;\;\;\frac{-x}{y}\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+89}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 4: 58.3% accurate, 28.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{+79} \lor \neg \left(x \leq 2.2 \cdot 10^{+87}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -2.9e+79) (not (<= x 2.2e+87))) (/ x y) 1.0))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -2.9e+79) || !(x <= 2.2e+87)) {
		tmp = x / y;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-2.9d+79)) .or. (.not. (x <= 2.2d+87))) then
        tmp = x / y
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -2.9e+79) || !(x <= 2.2e+87)) {
		tmp = x / y;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -2.9e+79) or not (x <= 2.2e+87):
		tmp = x / y
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -2.9e+79) || !(x <= 2.2e+87))
		tmp = Float64(x / y);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -2.9e+79) || ~((x <= 2.2e+87)))
		tmp = x / y;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -2.9e+79], N[Not[LessEqual[x, 2.2e+87]], $MachinePrecision]], N[(x / y), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < -2.89999999999999992e79 or 2.2000000000000001e87 < x

   1. Initial program 99.9%

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

      1. add-sqr-sqrt 48.2%

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

      2. fabs-sqr 48.2%

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

      3. add-sqr-sqrt 30.3%

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

      4. fabs-sqr 30.3%

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

      5. add-sqr-sqrt 30.6%

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

      6. add-sqr-sqrt 47.7%

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

      7. clear-num 47.6%

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

      8. associate-/r/ 47.6%

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

   3. Applied egg-rr 47.6%

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

   4. Taylor expanded in y around 0 48.3%

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

   if -2.89999999999999992e79 < x < 2.2000000000000001e87

   1. Initial program 100.0%

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

      1. add-sqr-sqrt 40.7%

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

      2. fabs-sqr 40.7%

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

      3. add-sqr-sqrt 9.9%

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

      4. fabs-sqr 9.9%

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

      5. add-sqr-sqrt 10.6%

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

      6. add-sqr-sqrt 20.4%

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

      7. clear-num 20.4%

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

      8. associate-/r/ 20.3%

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

   3. Applied egg-rr 20.3%

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

      1. associate-*l/ 20.4%

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

      2. *-un-lft-identity 20.4%

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

      3. sub-neg 20.4%

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

      4. add-sqr-sqrt 9.7%

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

      5. sqrt-unprod 39.0%

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

      6. sqr-neg 39.0%

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

      7. sqrt-unprod 48.2%

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

      8. add-sqr-sqrt 78.6%

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

   5. Applied egg-rr 78.6%

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

   6. Taylor expanded in x around 0 62.9%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{+79} \lor \neg \left(x \leq 2.2 \cdot 10^{+87}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 5: 74.9% accurate, 29.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.1 \cdot 10^{+165}:\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x 3.1e+165) (- 1.0 (/ x y)) (/ x y)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 3.1e+165) {
		tmp = 1.0 - (x / y);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 3.1d+165) then
        tmp = 1.0d0 - (x / y)
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 3.1e+165) {
		tmp = 1.0 - (x / y);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 3.1e+165:
		tmp = 1.0 - (x / y)
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 3.1e+165)
		tmp = Float64(1.0 - Float64(x / y));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 3.1e+165)
		tmp = 1.0 - (x / y);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 3.1e+165], N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.1000000000000002e165

   1. Initial program 100.0%

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

      1. add-sqr-sqrt 36.1%

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

      2. fabs-sqr 36.1%

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

      3. add-sqr-sqrt 10.7%

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

      4. fabs-sqr 10.7%

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

      5. add-sqr-sqrt 11.3%

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

      6. add-sqr-sqrt 25.6%

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

      7. clear-num 25.5%

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

      8. associate-/r/ 25.5%

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

   3. Applied egg-rr 25.5%

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

      1. associate-*l/ 25.6%

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

      2. *-un-lft-identity 25.6%

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

      3. frac-2neg 25.6%

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

      4. sub-neg 25.6%

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

      5. distribute-neg-in 25.6%

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

      6. remove-double-neg 25.6%

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

      7. add-sqr-sqrt 14.2%

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

      8. sqrt-unprod 39.2%

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

      9. sqr-neg 39.2%

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

      10. sqrt-unprod 49.5%

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

      11. add-sqr-sqrt 75.5%

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

   5. Applied egg-rr 75.5%

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

   6. Taylor expanded in x around 0 75.5%

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

      1. mul-1-neg 75.5%

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

      2. unsub-neg 75.5%

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

   8. Simplified 75.5%

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

   if 3.1000000000000002e165 < x

   1. Initial program 100.0%

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

      1. add-sqr-sqrt 93.8%

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

      2. fabs-sqr 93.8%

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

      3. add-sqr-sqrt 63.4%

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

      4. fabs-sqr 63.4%

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

      5. add-sqr-sqrt 63.5%

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

      6. add-sqr-sqrt 63.8%

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

      7. clear-num 63.7%

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

      8. associate-/r/ 63.7%

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

   3. Applied egg-rr 63.7%

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

   4. Taylor expanded in y around 0 64.3%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.1 \cdot 10^{+165}:\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 6: 1.3% accurate, 205.0× 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. Step-by-step derivation

   1. add-sqr-sqrt 43.5%

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

   2. fabs-sqr 43.5%

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

   3. add-sqr-sqrt 17.5%

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

   4. fabs-sqr 17.5%

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

   5. add-sqr-sqrt 18.0%

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

   6. add-sqr-sqrt 30.5%

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

   7. clear-num 30.5%

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

   8. associate-/r/ 30.4%

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

3. Applied egg-rr 30.4%

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

4. Taylor expanded in y around inf 1.2%

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

5. Final simplification 1.2%

   \[\leadsto -1 \]

Alternative 7: 51.9% accurate, 205.0× 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. Step-by-step derivation

   1. add-sqr-sqrt 43.5%

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

   2. fabs-sqr 43.5%

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

   3. add-sqr-sqrt 17.5%

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

   4. fabs-sqr 17.5%

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

   5. add-sqr-sqrt 18.0%

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

   6. add-sqr-sqrt 30.5%

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

   7. clear-num 30.5%

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

   8. associate-/r/ 30.4%

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

3. Applied egg-rr 30.4%

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

   1. associate-*l/ 30.5%

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

   2. *-un-lft-identity 30.5%

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

   3. sub-neg 30.5%

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

   4. add-sqr-sqrt 12.4%

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

   5. sqrt-unprod 42.4%

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

   6. sqr-neg 42.4%

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

   7. sqrt-unprod 45.5%

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

   8. add-sqr-sqrt 72.1%

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

5. Applied egg-rr 72.1%

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

6. Taylor expanded in x around 0 45.9%

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

7. Final simplification 45.9%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023322 
(FPCore (x y)
  :name "Numeric.LinearAlgebra.Util:formatSparse from hmatrix-0.16.1.5"
  :precision binary64
  (/ (fabs (- x y)) (fabs y)))