Numeric.LinearAlgebra.Util:formatSparse from hmatrix-0.16.1.5

Percentage Accurate: 100.0% → 100.0%

Time: 4.1s

Alternatives: 6

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in x around -inf 100.0%

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

   1. fabs-neg 100.0%

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

   2. mul-1-neg 100.0%

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

   3. sub-neg 100.0%

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

   4. fabs-div 100.0%

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

   5. div-sub 100.0%

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

   6. *-inverses 100.0%

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

4. Simplified 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 58.7% accurate, 13.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{x + y}\\ \mathbf{if}\;y \leq -1.3 \cdot 10^{-22}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -4.4 \cdot 10^{-267}:\\ \;\;\;\;\frac{x}{y} + -1\\ \mathbf{elif}\;y \leq 4.1 \cdot 10^{-112}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y - x}\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{-79}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (+ x y))))
   (if (<= y -1.3e-22)
     t_0
     (if (<= y -4.4e-267)
       (+ (/ x y) -1.0)
       (if (<= y 4.1e-112)
         (* (/ x y) (/ x (- y x)))
         (if (<= y 1.1e-79) (/ x y) t_0))))))

C

double code(double x, double y) {
	double t_0 = y / (x + y);
	double tmp;
	if (y <= -1.3e-22) {
		tmp = t_0;
	} else if (y <= -4.4e-267) {
		tmp = (x / y) + -1.0;
	} else if (y <= 4.1e-112) {
		tmp = (x / y) * (x / (y - x));
	} else if (y <= 1.1e-79) {
		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 = y / (x + y)
    if (y <= (-1.3d-22)) then
        tmp = t_0
    else if (y <= (-4.4d-267)) then
        tmp = (x / y) + (-1.0d0)
    else if (y <= 4.1d-112) then
        tmp = (x / y) * (x / (y - x))
    else if (y <= 1.1d-79) 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 = y / (x + y);
	double tmp;
	if (y <= -1.3e-22) {
		tmp = t_0;
	} else if (y <= -4.4e-267) {
		tmp = (x / y) + -1.0;
	} else if (y <= 4.1e-112) {
		tmp = (x / y) * (x / (y - x));
	} else if (y <= 1.1e-79) {
		tmp = x / y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y / (x + y)
	tmp = 0
	if y <= -1.3e-22:
		tmp = t_0
	elif y <= -4.4e-267:
		tmp = (x / y) + -1.0
	elif y <= 4.1e-112:
		tmp = (x / y) * (x / (y - x))
	elif y <= 1.1e-79:
		tmp = x / y
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y / Float64(x + y))
	tmp = 0.0
	if (y <= -1.3e-22)
		tmp = t_0;
	elseif (y <= -4.4e-267)
		tmp = Float64(Float64(x / y) + -1.0);
	elseif (y <= 4.1e-112)
		tmp = Float64(Float64(x / y) * Float64(x / Float64(y - x)));
	elseif (y <= 1.1e-79)
		tmp = Float64(x / y);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y / (x + y);
	tmp = 0.0;
	if (y <= -1.3e-22)
		tmp = t_0;
	elseif (y <= -4.4e-267)
		tmp = (x / y) + -1.0;
	elseif (y <= 4.1e-112)
		tmp = (x / y) * (x / (y - x));
	elseif (y <= 1.1e-79)
		tmp = x / y;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.3e-22], t$95$0, If[LessEqual[y, -4.4e-267], N[(N[(x / y), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[y, 4.1e-112], N[(N[(x / y), $MachinePrecision] * N[(x / N[(y - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.1e-79], N[(x / y), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -1.3e-22 or 1.0999999999999999e-79 < y

   1. Initial program 100.0%

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

      1. div-inv 99.7%

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

      2. add-sqr-sqrt 50.1%

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

      3. fabs-sqr 50.1%

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

      4. add-sqr-sqrt 51.2%

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

      5. *-commutative 51.2%

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

      6. add-sqr-sqrt 9.3%

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

      7. fabs-sqr 9.3%

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

      8. add-sqr-sqrt 15.4%

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

   3. Applied egg-rr 15.4%

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

      1. flip-- 4.6%

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

      2. associate-*r/ 4.6%

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

      3. +-commutative 4.6%

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

   5. Applied egg-rr 4.6%

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

   6. Taylor expanded in y around inf 2.0%

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

      1. neg-mul-1 2.0%

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

   8. Simplified 2.0%

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

      1. *-un-lft-identity 2.0%

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

      2. lft-mult-inverse 2.0%

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

      3. associate-*r* 1.5%

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

      4. *-commutative 1.5%

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

      5. *-commutative 1.5%

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

      6. associate-/l* 1.5%

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

      7. div-inv 1.5%

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

      8. div-inv 1.5%

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

      9. metadata-eval 1.5%

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

      10. add-sqr-sqrt 0.7%

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

      11. sqrt-unprod 17.3%

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

      12. sqr-neg 17.3%

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

      13. sqrt-unprod 16.6%

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

      14. add-sqr-sqrt 35.8%

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

      15. frac-times 37.5%

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

      16. inv-pow 37.5%

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

      17. inv-pow 37.5%

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

      18. pow-prod-up 37.5%

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

      19. metadata-eval 37.5%

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

   10. Applied egg-rr 37.5%

       \[\leadsto \color{blue}{\frac{1}{y} \cdot \frac{1}{\left(y + x\right) \cdot {y}^{-2}}} \]
   11. Step-by-step derivation

       1. *-commutative 37.5%

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

       2. *-commutative 37.5%

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

       3. associate-/r* 35.8%

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

       4. metadata-eval 35.8%

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

       5. pow-sqr 35.7%

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

       6. unpow-1 35.7%

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

       7. unpow-1 35.7%

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

       8. associate-/l/ 35.8%

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

       9. remove-double-div 35.9%

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

       10. associate-/l* 35.9%

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

       11. /-rgt-identity 35.9%

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

       12. associate-/l* 71.9%

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

       13. associate-*l/ 71.9%

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

       14. /-rgt-identity 71.9%

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

       15. associate-/r/ 71.9%

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

       16. remove-double-div 72.0%

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

       17. *-inverses 72.0%

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

       18. associate-/l* 72.0%

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

       19. *-lft-identity 72.0%

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

   12. Simplified 72.0%

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

   if -1.3e-22 < y < -4.39999999999999976e-267

   1. Initial program 100.0%

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

   2. Taylor expanded in x around -inf 100.0%

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

      1. fabs-neg 100.0%

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

      2. mul-1-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. fabs-sub 100.0%

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

      5. fabs-div 100.0%

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

      6. rem-square-sqrt 55.4%

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

      7. fabs-sqr 55.4%

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

      8. rem-square-sqrt 56.0%

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

      9. div-sub 56.0%

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

      10. sub-neg 56.0%

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

      11. *-inverses 56.0%

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

      12. metadata-eval 56.0%

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

      13. +-commutative 56.0%

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

   4. Simplified 56.0%

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

   if -4.39999999999999976e-267 < y < 4.09999999999999996e-112

   1. Initial program 100.0%

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

      1. div-inv 99.8%

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

      2. add-sqr-sqrt 45.5%

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

      3. fabs-sqr 45.5%

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

      4. add-sqr-sqrt 45.9%

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

      5. *-commutative 45.9%

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

      6. add-sqr-sqrt 32.8%

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

      7. fabs-sqr 32.8%

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

      8. add-sqr-sqrt 37.2%

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

   3. Applied egg-rr 37.2%

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

      1. flip-- 30.9%

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

      2. associate-*r/ 26.9%

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

      3. +-commutative 26.9%

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

   5. Applied egg-rr 26.9%

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

   6. Taylor expanded in y around 0 27.1%

      \[\leadsto \frac{\color{blue}{\frac{{x}^{2}}{y}}}{y + x} \]
   7. Step-by-step derivation

      1. unpow2 27.1%

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

      2. associate-/l* 33.5%

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

      3. associate-/r/ 33.5%

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

   8. Simplified 33.5%

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

      1. associate-/r/ 33.5%

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

      2. expm1-log1p-u 32.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{x}{\frac{y}{x}}}{y + x}\right)\right)} \]

      3. expm1-udef 32.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{x}{\frac{y}{x}}}{y + x}\right)} - 1} \]

   10. Applied egg-rr 44.7%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{x \cdot x}{y}}{y - x}\right)} - 1} \]
   11. Step-by-step derivation

       1. expm1-def 44.8%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{x \cdot x}{y}}{y - x}\right)\right)} \]

       2. expm1-log1p 46.9%

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

       3. associate-/l/ 44.1%

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

       4. times-frac 52.9%

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

   12. Simplified 52.9%

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

   if 4.09999999999999996e-112 < y < 1.0999999999999999e-79

   1. Initial program 100.0%

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

      1. div-inv 99.0%

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

      2. add-sqr-sqrt 74.1%

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

      3. fabs-sqr 74.1%

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

      4. add-sqr-sqrt 74.6%

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

      5. *-commutative 74.6%

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

      6. add-sqr-sqrt 74.3%

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

      7. fabs-sqr 74.3%

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

      8. add-sqr-sqrt 74.6%

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

   3. Applied egg-rr 74.6%

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

   4. Taylor expanded in y around 0 75.3%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{-22}:\\ \;\;\;\;\frac{y}{x + y}\\ \mathbf{elif}\;y \leq -4.4 \cdot 10^{-267}:\\ \;\;\;\;\frac{x}{y} + -1\\ \mathbf{elif}\;y \leq 4.1 \cdot 10^{-112}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y - x}\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{-79}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x + y}\\ \end{array} \]

Alternative 3: 57.5% accurate, 15.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{x + y}\\ \mathbf{if}\;y \leq -2.7 \cdot 10^{-23}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -6 \cdot 10^{-302}:\\ \;\;\;\;\frac{x}{y} + -1\\ \mathbf{elif}\;y \leq 7 \cdot 10^{-116}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y}\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{-79}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (+ x y))))
   (if (<= y -2.7e-23)
     t_0
     (if (<= y -6e-302)
       (+ (/ x y) -1.0)
       (if (<= y 7e-116)
         (* (/ x y) (/ x y))
         (if (<= y 1.05e-79) (/ x y) t_0))))))

C

double code(double x, double y) {
	double t_0 = y / (x + y);
	double tmp;
	if (y <= -2.7e-23) {
		tmp = t_0;
	} else if (y <= -6e-302) {
		tmp = (x / y) + -1.0;
	} else if (y <= 7e-116) {
		tmp = (x / y) * (x / y);
	} else if (y <= 1.05e-79) {
		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 = y / (x + y)
    if (y <= (-2.7d-23)) then
        tmp = t_0
    else if (y <= (-6d-302)) then
        tmp = (x / y) + (-1.0d0)
    else if (y <= 7d-116) then
        tmp = (x / y) * (x / y)
    else if (y <= 1.05d-79) 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 = y / (x + y);
	double tmp;
	if (y <= -2.7e-23) {
		tmp = t_0;
	} else if (y <= -6e-302) {
		tmp = (x / y) + -1.0;
	} else if (y <= 7e-116) {
		tmp = (x / y) * (x / y);
	} else if (y <= 1.05e-79) {
		tmp = x / y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y / (x + y)
	tmp = 0
	if y <= -2.7e-23:
		tmp = t_0
	elif y <= -6e-302:
		tmp = (x / y) + -1.0
	elif y <= 7e-116:
		tmp = (x / y) * (x / y)
	elif y <= 1.05e-79:
		tmp = x / y
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y / Float64(x + y))
	tmp = 0.0
	if (y <= -2.7e-23)
		tmp = t_0;
	elseif (y <= -6e-302)
		tmp = Float64(Float64(x / y) + -1.0);
	elseif (y <= 7e-116)
		tmp = Float64(Float64(x / y) * Float64(x / y));
	elseif (y <= 1.05e-79)
		tmp = Float64(x / y);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y / (x + y);
	tmp = 0.0;
	if (y <= -2.7e-23)
		tmp = t_0;
	elseif (y <= -6e-302)
		tmp = (x / y) + -1.0;
	elseif (y <= 7e-116)
		tmp = (x / y) * (x / y);
	elseif (y <= 1.05e-79)
		tmp = x / y;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.7e-23], t$95$0, If[LessEqual[y, -6e-302], N[(N[(x / y), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[y, 7e-116], N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.05e-79], N[(x / y), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -2.69999999999999985e-23 or 1.05e-79 < y

   1. Initial program 100.0%

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

      1. div-inv 99.7%

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

      2. add-sqr-sqrt 50.1%

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

      3. fabs-sqr 50.1%

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

      4. add-sqr-sqrt 51.2%

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

      5. *-commutative 51.2%

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

      6. add-sqr-sqrt 9.3%

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

      7. fabs-sqr 9.3%

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

      8. add-sqr-sqrt 15.4%

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

   3. Applied egg-rr 15.4%

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

      1. flip-- 4.6%

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

      2. associate-*r/ 4.6%

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

      3. +-commutative 4.6%

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

   5. Applied egg-rr 4.6%

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

   6. Taylor expanded in y around inf 2.0%

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

      1. neg-mul-1 2.0%

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

   8. Simplified 2.0%

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

      1. *-un-lft-identity 2.0%

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

      2. lft-mult-inverse 2.0%

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

      3. associate-*r* 1.5%

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

      4. *-commutative 1.5%

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

      5. *-commutative 1.5%

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

      6. associate-/l* 1.5%

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

      7. div-inv 1.5%

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

      8. div-inv 1.5%

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

      9. metadata-eval 1.5%

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

      10. add-sqr-sqrt 0.7%

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

      11. sqrt-unprod 17.3%

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

      12. sqr-neg 17.3%

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

      13. sqrt-unprod 16.6%

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

      14. add-sqr-sqrt 35.8%

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

      15. frac-times 37.5%

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

      16. inv-pow 37.5%

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

      17. inv-pow 37.5%

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

      18. pow-prod-up 37.5%

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

      19. metadata-eval 37.5%

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

   10. Applied egg-rr 37.5%

       \[\leadsto \color{blue}{\frac{1}{y} \cdot \frac{1}{\left(y + x\right) \cdot {y}^{-2}}} \]
   11. Step-by-step derivation

       1. *-commutative 37.5%

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

       2. *-commutative 37.5%

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

       3. associate-/r* 35.8%

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

       4. metadata-eval 35.8%

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

       5. pow-sqr 35.7%

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

       6. unpow-1 35.7%

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

       7. unpow-1 35.7%

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

       8. associate-/l/ 35.8%

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

       9. remove-double-div 35.9%

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

       10. associate-/l* 35.9%

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

       11. /-rgt-identity 35.9%

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

       12. associate-/l* 71.9%

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

       13. associate-*l/ 71.9%

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

       14. /-rgt-identity 71.9%

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

       15. associate-/r/ 71.9%

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

       16. remove-double-div 72.0%

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

       17. *-inverses 72.0%

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

       18. associate-/l* 72.0%

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

       19. *-lft-identity 72.0%

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

   12. Simplified 72.0%

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

   if -2.69999999999999985e-23 < y < -5.99999999999999978e-302

   1. Initial program 100.0%

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

   2. Taylor expanded in x around -inf 100.0%

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

      1. fabs-neg 100.0%

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

      2. mul-1-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. fabs-sub 100.0%

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

      5. fabs-div 100.0%

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

      6. rem-square-sqrt 53.2%

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

      7. fabs-sqr 53.2%

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

      8. rem-square-sqrt 53.7%

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

      9. div-sub 53.7%

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

      10. sub-neg 53.7%

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

      11. *-inverses 53.7%

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

      12. metadata-eval 53.7%

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

      13. +-commutative 53.7%

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

   4. Simplified 53.7%

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

   if -5.99999999999999978e-302 < y < 6.99999999999999968e-116

   1. Initial program 100.0%

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

      1. div-inv 99.8%

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

      2. add-sqr-sqrt 42.3%

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

      3. fabs-sqr 42.3%

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

      4. add-sqr-sqrt 42.7%

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

      5. *-commutative 42.7%

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

      6. add-sqr-sqrt 37.7%

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

      7. fabs-sqr 37.7%

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

      8. add-sqr-sqrt 37.8%

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

   3. Applied egg-rr 37.8%

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

      1. flip-- 35.4%

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

      2. associate-*r/ 30.8%

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

      3. +-commutative 30.8%

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

   5. Applied egg-rr 30.8%

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

   6. Taylor expanded in y around 0 31.0%

      \[\leadsto \frac{\color{blue}{\frac{{x}^{2}}{y}}}{y + x} \]
   7. Step-by-step derivation

      1. unpow2 31.0%

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

      2. associate-/l* 33.5%

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

      3. associate-/r/ 33.5%

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

   8. Simplified 33.5%

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

   9. Taylor expanded in x around 0 48.0%

      \[\leadsto \color{blue}{\frac{{x}^{2}}{{y}^{2}}} \]
   10. Step-by-step derivation

       1. unpow2 48.0%

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

       2. unpow2 48.0%

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

       3. times-frac 48.5%

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

   11. Simplified 48.5%

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

   if 6.99999999999999968e-116 < y < 1.05e-79

   1. Initial program 100.0%

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

      1. div-inv 99.0%

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

      2. add-sqr-sqrt 74.1%

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

      3. fabs-sqr 74.1%

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

      4. add-sqr-sqrt 74.6%

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

      5. *-commutative 74.6%

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

      6. add-sqr-sqrt 74.3%

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

      7. fabs-sqr 74.3%

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

      8. add-sqr-sqrt 74.6%

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

   3. Applied egg-rr 74.6%

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

   4. Taylor expanded in y around 0 75.3%

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

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{-23}:\\ \;\;\;\;\frac{y}{x + y}\\ \mathbf{elif}\;y \leq -6 \cdot 10^{-302}:\\ \;\;\;\;\frac{x}{y} + -1\\ \mathbf{elif}\;y \leq 7 \cdot 10^{-116}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y}\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{-79}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x + y}\\ \end{array} \]

Alternative 4: 57.9% accurate, 22.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.8 \cdot 10^{-28} \lor \neg \left(y \leq 9.8 \cdot 10^{-80}\right):\\ \;\;\;\;\frac{y}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -5.8e-28) (not (<= y 9.8e-80))) (/ y (+ x y)) (+ (/ x y) -1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -5.8e-28) || !(y <= 9.8e-80)) {
		tmp = y / (x + y);
	} else {
		tmp = (x / y) + -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 ((y <= (-5.8d-28)) .or. (.not. (y <= 9.8d-80))) then
        tmp = y / (x + y)
    else
        tmp = (x / y) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -5.8e-28) || !(y <= 9.8e-80)) {
		tmp = y / (x + y);
	} else {
		tmp = (x / y) + -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -5.8e-28) or not (y <= 9.8e-80):
		tmp = y / (x + y)
	else:
		tmp = (x / y) + -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -5.8e-28) || !(y <= 9.8e-80))
		tmp = Float64(y / Float64(x + y));
	else
		tmp = Float64(Float64(x / y) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -5.8e-28) || ~((y <= 9.8e-80)))
		tmp = y / (x + y);
	else
		tmp = (x / y) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -5.8e-28], N[Not[LessEqual[y, 9.8e-80]], $MachinePrecision]], N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5.80000000000000026e-28 or 9.79999999999999981e-80 < y

   1. Initial program 100.0%

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

      1. div-inv 99.7%

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

      2. add-sqr-sqrt 50.1%

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

      3. fabs-sqr 50.1%

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

      4. add-sqr-sqrt 51.2%

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

      5. *-commutative 51.2%

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

      6. add-sqr-sqrt 9.3%

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

      7. fabs-sqr 9.3%

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

      8. add-sqr-sqrt 15.4%

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

   3. Applied egg-rr 15.4%

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

      1. flip-- 4.6%

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

      2. associate-*r/ 4.6%

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

      3. +-commutative 4.6%

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

   5. Applied egg-rr 4.6%

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

   6. Taylor expanded in y around inf 2.0%

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

      1. neg-mul-1 2.0%

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

   8. Simplified 2.0%

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

      1. *-un-lft-identity 2.0%

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

      2. lft-mult-inverse 2.0%

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

      3. associate-*r* 1.5%

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

      4. *-commutative 1.5%

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

      5. *-commutative 1.5%

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

      6. associate-/l* 1.5%

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

      7. div-inv 1.5%

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

      8. div-inv 1.5%

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

      9. metadata-eval 1.5%

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

      10. add-sqr-sqrt 0.7%

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

      11. sqrt-unprod 17.3%

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

      12. sqr-neg 17.3%

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

      13. sqrt-unprod 16.6%

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

      14. add-sqr-sqrt 35.8%

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

      15. frac-times 37.5%

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

      16. inv-pow 37.5%

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

      17. inv-pow 37.5%

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

      18. pow-prod-up 37.5%

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

      19. metadata-eval 37.5%

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

   10. Applied egg-rr 37.5%

       \[\leadsto \color{blue}{\frac{1}{y} \cdot \frac{1}{\left(y + x\right) \cdot {y}^{-2}}} \]
   11. Step-by-step derivation

       1. *-commutative 37.5%

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

       2. *-commutative 37.5%

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

       3. associate-/r* 35.8%

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

       4. metadata-eval 35.8%

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

       5. pow-sqr 35.7%

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

       6. unpow-1 35.7%

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

       7. unpow-1 35.7%

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

       8. associate-/l/ 35.8%

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

       9. remove-double-div 35.9%

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

       10. associate-/l* 35.9%

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

       11. /-rgt-identity 35.9%

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

       12. associate-/l* 71.9%

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

       13. associate-*l/ 71.9%

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

       14. /-rgt-identity 71.9%

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

       15. associate-/r/ 71.9%

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

       16. remove-double-div 72.0%

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

       17. *-inverses 72.0%

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

       18. associate-/l* 72.0%

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

       19. *-lft-identity 72.0%

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

   12. Simplified 72.0%

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

   if -5.80000000000000026e-28 < y < 9.79999999999999981e-80

   1. Initial program 100.0%

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

   2. Taylor expanded in x around -inf 100.0%

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

      1. fabs-neg 100.0%

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

      2. mul-1-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. fabs-sub 100.0%

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

      5. fabs-div 100.0%

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

      6. rem-square-sqrt 48.9%

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

      7. fabs-sqr 48.9%

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

      8. rem-square-sqrt 49.4%

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

      9. div-sub 49.4%

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

      10. sub-neg 49.4%

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

      11. *-inverses 49.4%

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

      12. metadata-eval 49.4%

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

      13. +-commutative 49.4%

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

   4. Simplified 49.4%

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

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.8 \cdot 10^{-28} \lor \neg \left(y \leq 9.8 \cdot 10^{-80}\right):\\ \;\;\;\;\frac{y}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + -1\\ \end{array} \]

Alternative 5: 25.7% accurate, 68.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. div-inv 99.7%

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

   2. add-sqr-sqrt 48.8%

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

   3. fabs-sqr 48.8%

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

   4. add-sqr-sqrt 49.6%

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

   5. *-commutative 49.6%

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

   6. add-sqr-sqrt 13.6%

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

   7. fabs-sqr 13.6%

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

   8. add-sqr-sqrt 29.7%

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

3. Applied egg-rr 29.7%

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

4. Taylor expanded in y around 0 30.4%

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

5. Final simplification 30.4%

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

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. div-inv 99.7%

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

   2. add-sqr-sqrt 48.8%

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

   3. fabs-sqr 48.8%

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

   4. add-sqr-sqrt 49.6%

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

   5. *-commutative 49.6%

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

   6. add-sqr-sqrt 13.6%

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

   7. fabs-sqr 13.6%

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

   8. add-sqr-sqrt 29.7%

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

3. Applied egg-rr 29.7%

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

4. Taylor expanded in y around inf 1.3%

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

5. Final simplification 1.3%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023290 
(FPCore (x y)
  :name "Numeric.LinearAlgebra.Util:formatSparse from hmatrix-0.16.1.5"
  :precision binary64
  (/ (fabs (- x y)) (fabs y)))