Numeric.LinearAlgebra.Util:formatSparse from hmatrix-0.16.1.5

Percentage Accurate: 100.0% → 100.0%

Time: 4.6s

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

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

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: 59.6% accurate, 13.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{x + y}\\ \mathbf{if}\;y \leq -3.6 \cdot 10^{-35}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -7.2 \cdot 10^{-265}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{-136}:\\ \;\;\;\;\frac{\frac{x}{y}}{\frac{y - x}{x}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (+ x y))))
   (if (<= y -3.6e-35)
     t_0
     (if (<= y -7.2e-265)
       (/ x y)
       (if (<= y 1.15e-136) (/ (/ x y) (/ (- y x) x)) t_0)))))

C

double code(double x, double y) {
	double t_0 = y / (x + y);
	double tmp;
	if (y <= -3.6e-35) {
		tmp = t_0;
	} else if (y <= -7.2e-265) {
		tmp = x / y;
	} else if (y <= 1.15e-136) {
		tmp = (x / y) / ((y - x) / x);
	} 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 <= (-3.6d-35)) then
        tmp = t_0
    else if (y <= (-7.2d-265)) then
        tmp = x / y
    else if (y <= 1.15d-136) then
        tmp = (x / y) / ((y - x) / x)
    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 <= -3.6e-35) {
		tmp = t_0;
	} else if (y <= -7.2e-265) {
		tmp = x / y;
	} else if (y <= 1.15e-136) {
		tmp = (x / y) / ((y - x) / x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y / (x + y)
	tmp = 0
	if y <= -3.6e-35:
		tmp = t_0
	elif y <= -7.2e-265:
		tmp = x / y
	elif y <= 1.15e-136:
		tmp = (x / y) / ((y - x) / x)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y / Float64(x + y))
	tmp = 0.0
	if (y <= -3.6e-35)
		tmp = t_0;
	elseif (y <= -7.2e-265)
		tmp = Float64(x / y);
	elseif (y <= 1.15e-136)
		tmp = Float64(Float64(x / y) / Float64(Float64(y - x) / x));
	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 <= -3.6e-35)
		tmp = t_0;
	elseif (y <= -7.2e-265)
		tmp = x / y;
	elseif (y <= 1.15e-136)
		tmp = (x / y) / ((y - x) / x);
	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, -3.6e-35], t$95$0, If[LessEqual[y, -7.2e-265], N[(x / y), $MachinePrecision], If[LessEqual[y, 1.15e-136], N[(N[(x / y), $MachinePrecision] / N[(N[(y - x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.60000000000000019e-35 or 1.14999999999999999e-136 < 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 41.8%

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

      3. fabs-sqr 41.8%

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

      4. add-sqr-sqrt 42.8%

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

      5. *-commutative 42.8%

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

      6. add-sqr-sqrt 8.1%

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

      7. fabs-sqr 8.1%

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

      8. add-sqr-sqrt 13.7%

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

   3. Applied egg-rr 13.7%

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

      1. flip-- 5.5%

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

      2. associate-*r/ 4.9%

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

      3. +-commutative 4.9%

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

   5. Applied egg-rr 4.9%

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

   6. Taylor expanded in y around 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. *-un-lft-identity 1.5%

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

      7. times-frac 1.5%

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

      8. *-un-lft-identity 1.5%

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

      9. *-un-lft-identity 1.5%

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

      10. add-sqr-sqrt 0.5%

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

      11. sqrt-unprod 24.5%

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

      12. sqr-neg 24.5%

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

      13. sqrt-unprod 23.8%

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

      14. add-sqr-sqrt 37.4%

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

   10. Applied egg-rr 37.4%

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

       1. /-rgt-identity 37.4%

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

       2. *-commutative 37.4%

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

       3. associate-*l/ 37.4%

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

       4. /-rgt-identity 37.4%

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

       5. associate-/r/ 37.4%

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

       6. remove-double-div 37.5%

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

       7. associate-/l* 73.8%

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

       8. *-inverses 73.8%

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

       9. /-rgt-identity 73.8%

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

   12. Simplified 73.8%

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

   if -3.60000000000000019e-35 < y < -7.2000000000000004e-265

   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 48.7%

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

      3. fabs-sqr 48.7%

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

      4. add-sqr-sqrt 49.1%

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

      5. *-commutative 49.1%

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

      6. add-sqr-sqrt 0.0%

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

      7. fabs-sqr 0.0%

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

      8. add-sqr-sqrt 51.2%

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

   3. Applied egg-rr 51.2%

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

   4. Taylor expanded in y around 0 51.6%

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

   if -7.2000000000000004e-265 < y < 1.14999999999999999e-136

   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 45.0%

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

      3. fabs-sqr 45.0%

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

      4. add-sqr-sqrt 45.5%

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

      5. *-commutative 45.5%

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

      6. add-sqr-sqrt 28.7%

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

      7. fabs-sqr 28.7%

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

      8. add-sqr-sqrt 31.2%

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

   3. Applied egg-rr 31.2%

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

      1. flip-- 24.5%

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

      2. associate-*r/ 22.3%

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

      3. +-commutative 22.3%

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

   5. Applied egg-rr 22.3%

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

   6. Taylor expanded in x around inf 22.4%

      \[\leadsto \frac{\frac{1}{y} \cdot \color{blue}{{x}^{2}}}{y + x} \]
   7. Step-by-step derivation

      1. unpow2 22.4%

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

   8. Simplified 22.4%

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

      1. expm1-log1p-u 21.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{1}{y} \cdot \left(x \cdot x\right)}{y + x}\right)\right)} \]

      2. expm1-udef 21.6%

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

   10. Applied egg-rr 56.9%

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

       1. expm1-def 57.1%

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

       2. expm1-log1p 60.3%

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

       3. associate-/r/ 60.4%

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

       4. associate-/l* 62.5%

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

   12. Simplified 62.5%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.6 \cdot 10^{-35}:\\ \;\;\;\;\frac{y}{x + y}\\ \mathbf{elif}\;y \leq -7.2 \cdot 10^{-265}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{-136}:\\ \;\;\;\;\frac{\frac{x}{y}}{\frac{y - x}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x + y}\\ \end{array} \]

Alternative 3: 58.2% accurate, 15.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{x + y}\\ \mathbf{if}\;y \leq -4.5 \cdot 10^{-35}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -7.2 \cdot 10^{-235}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{-187}:\\ \;\;\;\;\frac{x \cdot x}{y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (+ x y))))
   (if (<= y -4.5e-35)
     t_0
     (if (<= y -7.2e-235)
       (/ x y)
       (if (<= y 4.4e-187) (/ (* x x) (* y y)) t_0)))))

C

double code(double x, double y) {
	double t_0 = y / (x + y);
	double tmp;
	if (y <= -4.5e-35) {
		tmp = t_0;
	} else if (y <= -7.2e-235) {
		tmp = x / y;
	} else if (y <= 4.4e-187) {
		tmp = (x * x) / (y * 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 <= (-4.5d-35)) then
        tmp = t_0
    else if (y <= (-7.2d-235)) then
        tmp = x / y
    else if (y <= 4.4d-187) then
        tmp = (x * x) / (y * 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 <= -4.5e-35) {
		tmp = t_0;
	} else if (y <= -7.2e-235) {
		tmp = x / y;
	} else if (y <= 4.4e-187) {
		tmp = (x * x) / (y * y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y / (x + y)
	tmp = 0
	if y <= -4.5e-35:
		tmp = t_0
	elif y <= -7.2e-235:
		tmp = x / y
	elif y <= 4.4e-187:
		tmp = (x * x) / (y * y)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y / Float64(x + y))
	tmp = 0.0
	if (y <= -4.5e-35)
		tmp = t_0;
	elseif (y <= -7.2e-235)
		tmp = Float64(x / y);
	elseif (y <= 4.4e-187)
		tmp = Float64(Float64(x * x) / Float64(y * 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 <= -4.5e-35)
		tmp = t_0;
	elseif (y <= -7.2e-235)
		tmp = x / y;
	elseif (y <= 4.4e-187)
		tmp = (x * x) / (y * 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, -4.5e-35], t$95$0, If[LessEqual[y, -7.2e-235], N[(x / y), $MachinePrecision], If[LessEqual[y, 4.4e-187], N[(N[(x * x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.5000000000000001e-35 or 4.40000000000000016e-187 < 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 40.5%

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

      3. fabs-sqr 40.5%

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

      4. add-sqr-sqrt 41.5%

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

      5. *-commutative 41.5%

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

      6. add-sqr-sqrt 8.4%

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

      7. fabs-sqr 8.4%

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

      8. add-sqr-sqrt 13.6%

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

   3. Applied egg-rr 13.6%

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

      1. flip-- 5.9%

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

      2. associate-*r/ 5.3%

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

      3. +-commutative 5.3%

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

   5. Applied egg-rr 5.3%

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

   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. *-un-lft-identity 1.5%

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

      7. times-frac 1.5%

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

      8. *-un-lft-identity 1.5%

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

      9. *-un-lft-identity 1.5%

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

      10. add-sqr-sqrt 0.5%

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

      11. sqrt-unprod 23.7%

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

      12. sqr-neg 23.7%

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

      13. sqrt-unprod 23.0%

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

      14. add-sqr-sqrt 36.0%

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

   10. Applied egg-rr 36.0%

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

       1. /-rgt-identity 36.0%

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

       2. *-commutative 36.0%

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

       3. associate-*l/ 35.9%

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

       4. /-rgt-identity 35.9%

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

       5. associate-/r/ 36.0%

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

       6. remove-double-div 36.1%

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

       7. associate-/l* 72.1%

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

       8. *-inverses 72.1%

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

       9. /-rgt-identity 72.1%

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

   12. Simplified 72.1%

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

   if -4.5000000000000001e-35 < y < -7.19999999999999998e-235

   1. Initial program 100.0%

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

      1. div-inv 99.6%

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

      2. add-sqr-sqrt 51.8%

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

      3. fabs-sqr 51.8%

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

      4. add-sqr-sqrt 52.3%

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

      5. *-commutative 52.3%

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

      6. add-sqr-sqrt 0.0%

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

      7. fabs-sqr 0.0%

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

      8. add-sqr-sqrt 48.0%

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

   3. Applied egg-rr 48.0%

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

   4. Taylor expanded in y around 0 48.3%

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

   if -7.19999999999999998e-235 < y < 4.40000000000000016e-187

   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 48.6%

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

      3. fabs-sqr 48.6%

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

      4. add-sqr-sqrt 48.9%

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

      5. *-commutative 48.9%

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

      6. add-sqr-sqrt 25.6%

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

      7. fabs-sqr 25.6%

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

      8. add-sqr-sqrt 42.0%

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

   3. Applied egg-rr 42.0%

      \[\leadsto \color{blue}{\frac{1}{y} \cdot \left(x - y\right)} \]
   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/ 33.3%

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

      3. +-commutative 33.3%

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

   5. Applied egg-rr 33.3%

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

   6. Taylor expanded in x around inf 33.3%

      \[\leadsto \frac{\frac{1}{y} \cdot \color{blue}{{x}^{2}}}{y + x} \]
   7. Step-by-step derivation

      1. unpow2 33.3%

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

   8. Simplified 33.3%

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

   9. Taylor expanded in y around inf 55.3%

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

       1. unpow2 55.3%

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

       2. unpow2 55.3%

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

   11. Simplified 55.3%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{-35}:\\ \;\;\;\;\frac{y}{x + y}\\ \mathbf{elif}\;y \leq -7.2 \cdot 10^{-235}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{-187}:\\ \;\;\;\;\frac{x \cdot x}{y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x + y}\\ \end{array} \]

Alternative 4: 58.5% accurate, 15.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{x + y}\\ \mathbf{if}\;y \leq -6 \cdot 10^{-35}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -8.8 \cdot 10^{-235}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{-187}:\\ \;\;\;\;\frac{\frac{x}{\frac{y}{x}}}{y}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (+ x y))))
   (if (<= y -6e-35)
     t_0
     (if (<= y -8.8e-235)
       (/ x y)
       (if (<= y 8.2e-187) (/ (/ x (/ y x)) y) t_0)))))

C

double code(double x, double y) {
	double t_0 = y / (x + y);
	double tmp;
	if (y <= -6e-35) {
		tmp = t_0;
	} else if (y <= -8.8e-235) {
		tmp = x / y;
	} else if (y <= 8.2e-187) {
		tmp = (x / (y / 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 <= (-6d-35)) then
        tmp = t_0
    else if (y <= (-8.8d-235)) then
        tmp = x / y
    else if (y <= 8.2d-187) then
        tmp = (x / (y / 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 <= -6e-35) {
		tmp = t_0;
	} else if (y <= -8.8e-235) {
		tmp = x / y;
	} else if (y <= 8.2e-187) {
		tmp = (x / (y / x)) / y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y / (x + y)
	tmp = 0
	if y <= -6e-35:
		tmp = t_0
	elif y <= -8.8e-235:
		tmp = x / y
	elif y <= 8.2e-187:
		tmp = (x / (y / 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 <= -6e-35)
		tmp = t_0;
	elseif (y <= -8.8e-235)
		tmp = Float64(x / y);
	elseif (y <= 8.2e-187)
		tmp = Float64(Float64(x / Float64(y / 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 <= -6e-35)
		tmp = t_0;
	elseif (y <= -8.8e-235)
		tmp = x / y;
	elseif (y <= 8.2e-187)
		tmp = (x / (y / 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, -6e-35], t$95$0, If[LessEqual[y, -8.8e-235], N[(x / y), $MachinePrecision], If[LessEqual[y, 8.2e-187], N[(N[(x / N[(y / x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.99999999999999978e-35 or 8.2000000000000004e-187 < 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 40.5%

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

      3. fabs-sqr 40.5%

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

      4. add-sqr-sqrt 41.5%

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

      5. *-commutative 41.5%

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

      6. add-sqr-sqrt 8.4%

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

      7. fabs-sqr 8.4%

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

      8. add-sqr-sqrt 13.6%

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

   3. Applied egg-rr 13.6%

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

      1. flip-- 5.9%

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

      2. associate-*r/ 5.3%

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

      3. +-commutative 5.3%

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

   5. Applied egg-rr 5.3%

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

   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. *-un-lft-identity 1.5%

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

      7. times-frac 1.5%

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

      8. *-un-lft-identity 1.5%

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

      9. *-un-lft-identity 1.5%

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

      10. add-sqr-sqrt 0.5%

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

      11. sqrt-unprod 23.7%

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

      12. sqr-neg 23.7%

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

      13. sqrt-unprod 23.0%

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

      14. add-sqr-sqrt 36.0%

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

   10. Applied egg-rr 36.0%

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

       1. /-rgt-identity 36.0%

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

       2. *-commutative 36.0%

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

       3. associate-*l/ 35.9%

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

       4. /-rgt-identity 35.9%

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

       5. associate-/r/ 36.0%

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

       6. remove-double-div 36.1%

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

       7. associate-/l* 72.1%

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

       8. *-inverses 72.1%

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

       9. /-rgt-identity 72.1%

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

   12. Simplified 72.1%

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

   if -5.99999999999999978e-35 < y < -8.79999999999999935e-235

   1. Initial program 100.0%

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

      1. div-inv 99.6%

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

      2. add-sqr-sqrt 51.8%

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

      3. fabs-sqr 51.8%

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

      4. add-sqr-sqrt 52.3%

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

      5. *-commutative 52.3%

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

      6. add-sqr-sqrt 0.0%

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

      7. fabs-sqr 0.0%

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

      8. add-sqr-sqrt 48.0%

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

   3. Applied egg-rr 48.0%

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

   4. Taylor expanded in y around 0 48.3%

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

   if -8.79999999999999935e-235 < y < 8.2000000000000004e-187

   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 48.6%

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

      3. fabs-sqr 48.6%

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

      4. add-sqr-sqrt 48.9%

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

      5. *-commutative 48.9%

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

      6. add-sqr-sqrt 25.6%

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

      7. fabs-sqr 25.6%

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

      8. add-sqr-sqrt 42.0%

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

   3. Applied egg-rr 42.0%

      \[\leadsto \color{blue}{\frac{1}{y} \cdot \left(x - y\right)} \]
   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/ 33.3%

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

      3. +-commutative 33.3%

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

   5. Applied egg-rr 33.3%

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

   6. Taylor expanded in x around inf 33.3%

      \[\leadsto \frac{\frac{1}{y} \cdot \color{blue}{{x}^{2}}}{y + x} \]
   7. Step-by-step derivation

      1. unpow2 33.3%

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

   8. Simplified 33.3%

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

   9. Taylor expanded in y around inf 55.3%

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

       1. unpow2 55.3%

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

       2. unpow2 55.3%

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

       3. associate-/r* 55.9%

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

       4. associate-/l* 56.8%

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

   11. Simplified 56.8%

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

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{-35}:\\ \;\;\;\;\frac{y}{x + y}\\ \mathbf{elif}\;y \leq -8.8 \cdot 10^{-235}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{-187}:\\ \;\;\;\;\frac{\frac{x}{\frac{y}{x}}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x + y}\\ \end{array} \]

Alternative 5: 57.4% accurate, 22.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.2 \cdot 10^{-35} \lor \neg \left(y \leq 5.8 \cdot 10^{-216}\right):\\ \;\;\;\;\frac{y}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -6.2e-35) (not (<= y 5.8e-216))) (/ y (+ x y)) (/ x y)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -6.2e-35) || !(y <= 5.8e-216)) {
		tmp = y / (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 ((y <= (-6.2d-35)) .or. (.not. (y <= 5.8d-216))) then
        tmp = y / (x + y)
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -6.2e-35) || !(y <= 5.8e-216)) {
		tmp = y / (x + y);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -6.2e-35) or not (y <= 5.8e-216):
		tmp = y / (x + y)
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -6.2e-35) || !(y <= 5.8e-216))
		tmp = Float64(y / Float64(x + y));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -6.2e-35) || ~((y <= 5.8e-216)))
		tmp = y / (x + y);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -6.2e-35], N[Not[LessEqual[y, 5.8e-216]], $MachinePrecision]], N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -6.20000000000000024e-35 or 5.8000000000000001e-216 < 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 39.1%

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

      3. fabs-sqr 39.1%

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

      4. add-sqr-sqrt 40.0%

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

      5. *-commutative 40.0%

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

      6. add-sqr-sqrt 8.1%

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

      7. fabs-sqr 8.1%

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

      8. add-sqr-sqrt 13.2%

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

   3. Applied egg-rr 13.2%

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

      1. flip-- 5.7%

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

      2. associate-*r/ 5.1%

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

      3. +-commutative 5.1%

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

   5. Applied egg-rr 5.1%

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

   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. *-un-lft-identity 1.5%

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

      7. times-frac 1.5%

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

      8. *-un-lft-identity 1.5%

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

      9. *-un-lft-identity 1.5%

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

      10. add-sqr-sqrt 0.5%

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

      11. sqrt-unprod 22.9%

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

      12. sqr-neg 22.9%

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

      13. sqrt-unprod 22.3%

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

      14. add-sqr-sqrt 34.8%

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

   10. Applied egg-rr 34.8%

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

       1. /-rgt-identity 34.8%

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

       2. *-commutative 34.8%

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

       3. associate-*l/ 34.8%

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

       4. /-rgt-identity 34.8%

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

       5. associate-/r/ 34.8%

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

       6. remove-double-div 34.9%

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

       7. associate-/l* 69.7%

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

       8. *-inverses 69.7%

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

       9. /-rgt-identity 69.7%

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

   12. Simplified 69.7%

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

   if -6.20000000000000024e-35 < y < 5.8000000000000001e-216

   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 54.0%

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

      3. fabs-sqr 54.0%

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

      4. add-sqr-sqrt 54.4%

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

      5. *-commutative 54.4%

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

      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 48.4%

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

   3. Applied egg-rr 48.4%

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

   4. Taylor expanded in y around 0 48.6%

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

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.2 \cdot 10^{-35} \lor \neg \left(y \leq 5.8 \cdot 10^{-216}\right):\\ \;\;\;\;\frac{y}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 6: 25.9% 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 43.8%

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

   3. fabs-sqr 43.8%

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

   4. add-sqr-sqrt 44.6%

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

   5. *-commutative 44.6%

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

   6. add-sqr-sqrt 9.8%

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

   7. fabs-sqr 9.8%

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

   8. add-sqr-sqrt 24.3%

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

3. Applied egg-rr 24.3%

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

4. Taylor expanded in y around 0 25.1%

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

5. Final simplification 25.1%

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

Alternative 7: 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 43.8%

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

   3. fabs-sqr 43.8%

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

   4. add-sqr-sqrt 44.6%

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

   5. *-commutative 44.6%

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

   6. add-sqr-sqrt 9.8%

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

   7. fabs-sqr 9.8%

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

   8. add-sqr-sqrt 24.3%

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

3. Applied egg-rr 24.3%

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

4. Taylor expanded in y around inf 1.3%

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

5. Final simplification 1.3%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023229 
(FPCore (x y)
  :name "Numeric.LinearAlgebra.Util:formatSparse from hmatrix-0.16.1.5"
  :precision binary64
  (/ (fabs (- x y)) (fabs y)))