Numeric.LinearAlgebra.Util:formatSparse from hmatrix-0.16.1.5

Percentage Accurate: 100.0% → 100.0%

Time: 6.5s

Alternatives: 9

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. Add Preprocessing

3. Taylor expanded in x around -inf 100.0%

   \[\leadsto \color{blue}{\frac{\left|-\left(y + -1 \cdot x\right)\right|}{\left|y\right|}} \]
4. 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%

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

5. Simplified 100.0%

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

6. Final simplification 100.0%

   \[\leadsto \left|1 - \frac{x}{y}\right| \]
7. Add Preprocessing

Alternative 2: 83.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{-43} \lor \neg \left(y \leq 1.25 \cdot 10^{-36}\right):\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -4.5e-43) (not (<= y 1.25e-36)))
   (- 1.0 (/ x y))
   (fabs (/ x y))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -4.5e-43) || !(y <= 1.25e-36)) {
		tmp = 1.0 - (x / y);
	} else {
		tmp = fabs((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 <= (-4.5d-43)) .or. (.not. (y <= 1.25d-36))) then
        tmp = 1.0d0 - (x / y)
    else
        tmp = abs((x / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -4.5e-43) || !(y <= 1.25e-36)) {
		tmp = 1.0 - (x / y);
	} else {
		tmp = Math.abs((x / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -4.5e-43) or not (y <= 1.25e-36):
		tmp = 1.0 - (x / y)
	else:
		tmp = math.fabs((x / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -4.5e-43) || !(y <= 1.25e-36))
		tmp = Float64(1.0 - Float64(x / y));
	else
		tmp = abs(Float64(x / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -4.5e-43) || ~((y <= 1.25e-36)))
		tmp = 1.0 - (x / y);
	else
		tmp = abs((x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -4.5e-43], N[Not[LessEqual[y, 1.25e-36]], $MachinePrecision]], N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision], N[Abs[N[(x / y), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.50000000000000025e-43 or 1.25000000000000001e-36 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around -inf 100.0%

      \[\leadsto \color{blue}{\frac{\left|-\left(y + -1 \cdot x\right)\right|}{\left|y\right|}} \]
   4. 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 10.8%

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

      7. fabs-sqr 10.8%

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

      8. rem-square-sqrt 12.2%

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

      9. div-sub 12.2%

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

      10. sub-neg 12.2%

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

      11. *-inverses 12.2%

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

      12. metadata-eval 12.2%

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

      13. +-commutative 12.2%

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

   5. Simplified 12.2%

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

      1. +-commutative 12.2%

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

      2. metadata-eval 12.2%

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

      3. sub-neg 12.2%

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

      4. *-inverses 12.2%

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

      5. div-sub 12.2%

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

      6. clear-num 12.1%

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

   7. Applied egg-rr 12.1%

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

      1. remove-double-div 12.1%

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

      2. frac-2neg 12.1%

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

      3. metadata-eval 12.1%

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

      4. associate-/r/ 12.1%

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

      5. metadata-eval 12.1%

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

      6. clear-num 12.1%

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

      7. distribute-neg-frac 12.1%

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

      8. metadata-eval 12.1%

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

      9. div-sub 12.1%

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

      10. *-inverses 12.1%

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

      11. sub-neg 12.1%

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

      12. metadata-eval 12.1%

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

   9. Applied egg-rr 12.1%

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

       1. neg-mul-1 12.1%

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

       2. distribute-neg-frac2 12.1%

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

       3. +-commutative 12.1%

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

       4. distribute-neg-in 12.1%

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

       5. metadata-eval 12.1%

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

       6. sub-neg 12.1%

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

       7. rem-square-sqrt 1.3%

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

       8. fabs-sqr 1.3%

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

       9. rem-square-sqrt 1.4%

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

       10. fabs-sub 1.4%

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

       11. sub-neg 1.4%

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

       12. metadata-eval 1.4%

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

       13. rem-square-sqrt 0.1%

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

       14. fabs-sqr 0.1%

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

       15. rem-square-sqrt 89.2%

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

       16. +-commutative 89.2%

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

   11. Simplified 89.2%

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

   12. Taylor expanded in x around 0 89.2%

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

       1. mul-1-neg 89.2%

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

       2. sub-neg 89.2%

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

   14. Simplified 89.2%

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

   if -4.50000000000000025e-43 < y < 1.25000000000000001e-36

   1. Initial program 100.0%

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

   3. Taylor expanded in x around -inf 100.0%

      \[\leadsto \color{blue}{\frac{\left|-\left(y + -1 \cdot x\right)\right|}{\left|y\right|}} \]
   4. 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%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf 82.3%

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

      1. associate-*r/ 82.3%

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

      2. neg-mul-1 82.3%

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

   8. Simplified 82.3%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{-43} \lor \neg \left(y \leq 1.25 \cdot 10^{-36}\right):\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 58.4% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{-y}\\ \mathbf{if}\;x \leq -8.5 \cdot 10^{+145}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 7.4 \cdot 10^{-7}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{+114}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{+133} \lor \neg \left(x \leq 1.8 \cdot 10^{+211}\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (- y))))
   (if (<= x -8.5e+145)
     t_0
     (if (<= x 7.4e-7)
       1.0
       (if (<= x 6.2e+114)
         (/ x y)
         (if (or (<= x 1.9e+133) (not (<= x 1.8e+211)))
           t_0
           (+ (/ x y) -1.0)))))))

C

double code(double x, double y) {
	double t_0 = x / -y;
	double tmp;
	if (x <= -8.5e+145) {
		tmp = t_0;
	} else if (x <= 7.4e-7) {
		tmp = 1.0;
	} else if (x <= 6.2e+114) {
		tmp = x / y;
	} else if ((x <= 1.9e+133) || !(x <= 1.8e+211)) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = x / -y
    if (x <= (-8.5d+145)) then
        tmp = t_0
    else if (x <= 7.4d-7) then
        tmp = 1.0d0
    else if (x <= 6.2d+114) then
        tmp = x / y
    else if ((x <= 1.9d+133) .or. (.not. (x <= 1.8d+211))) then
        tmp = t_0
    else
        tmp = (x / y) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x / -y;
	double tmp;
	if (x <= -8.5e+145) {
		tmp = t_0;
	} else if (x <= 7.4e-7) {
		tmp = 1.0;
	} else if (x <= 6.2e+114) {
		tmp = x / y;
	} else if ((x <= 1.9e+133) || !(x <= 1.8e+211)) {
		tmp = t_0;
	} else {
		tmp = (x / y) + -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / -y
	tmp = 0
	if x <= -8.5e+145:
		tmp = t_0
	elif x <= 7.4e-7:
		tmp = 1.0
	elif x <= 6.2e+114:
		tmp = x / y
	elif (x <= 1.9e+133) or not (x <= 1.8e+211):
		tmp = t_0
	else:
		tmp = (x / y) + -1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(-y))
	tmp = 0.0
	if (x <= -8.5e+145)
		tmp = t_0;
	elseif (x <= 7.4e-7)
		tmp = 1.0;
	elseif (x <= 6.2e+114)
		tmp = Float64(x / y);
	elseif ((x <= 1.9e+133) || !(x <= 1.8e+211))
		tmp = t_0;
	else
		tmp = Float64(Float64(x / y) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / -y;
	tmp = 0.0;
	if (x <= -8.5e+145)
		tmp = t_0;
	elseif (x <= 7.4e-7)
		tmp = 1.0;
	elseif (x <= 6.2e+114)
		tmp = x / y;
	elseif ((x <= 1.9e+133) || ~((x <= 1.8e+211)))
		tmp = t_0;
	else
		tmp = (x / y) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / (-y)), $MachinePrecision]}, If[LessEqual[x, -8.5e+145], t$95$0, If[LessEqual[x, 7.4e-7], 1.0, If[LessEqual[x, 6.2e+114], N[(x / y), $MachinePrecision], If[Or[LessEqual[x, 1.9e+133], N[Not[LessEqual[x, 1.8e+211]], $MachinePrecision]], t$95$0, N[(N[(x / y), $MachinePrecision] + -1.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -8.49999999999999977e145 or 6.2000000000000001e114 < x < 1.9000000000000001e133 or 1.80000000000000001e211 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around -inf 99.9%

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

      1. fabs-neg 99.9%

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

      2. mul-1-neg 99.9%

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

      3. sub-neg 99.9%

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

      4. fabs-sub 99.9%

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

      5. fabs-div 99.9%

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

      6. rem-square-sqrt 22.8%

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

      7. fabs-sqr 22.8%

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

      8. rem-square-sqrt 23.5%

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

      9. div-sub 23.5%

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

      10. sub-neg 23.5%

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

      11. *-inverses 23.5%

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

      12. metadata-eval 23.5%

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

      13. +-commutative 23.5%

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

   5. Simplified 23.5%

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

      1. +-commutative 23.5%

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

      2. metadata-eval 23.5%

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

      3. sub-neg 23.5%

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

      4. *-inverses 23.5%

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

      5. div-sub 23.5%

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

      6. clear-num 23.3%

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

   7. Applied egg-rr 23.3%

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

      1. remove-double-div 23.3%

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

      2. frac-2neg 23.3%

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

      3. metadata-eval 23.3%

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

      4. associate-/r/ 23.3%

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

      5. metadata-eval 23.3%

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

      6. clear-num 23.4%

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

      7. distribute-neg-frac 23.4%

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

      8. metadata-eval 23.4%

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

      9. div-sub 23.4%

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

      10. *-inverses 23.4%

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

      11. sub-neg 23.4%

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

      12. metadata-eval 23.4%

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

   9. Applied egg-rr 23.4%

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

       1. neg-mul-1 23.4%

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

       2. distribute-neg-frac2 23.4%

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

       3. +-commutative 23.4%

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

       4. distribute-neg-in 23.4%

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

       5. metadata-eval 23.4%

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

       6. sub-neg 23.4%

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

       7. rem-square-sqrt 0.5%

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

       8. fabs-sqr 0.5%

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

       9. rem-square-sqrt 0.7%

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

       10. fabs-sub 0.7%

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

       11. sub-neg 0.7%

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

       12. metadata-eval 0.7%

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

       13. rem-square-sqrt 0.1%

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

       14. fabs-sqr 0.1%

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

       15. rem-square-sqrt 77.0%

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

       16. +-commutative 77.0%

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

   11. Simplified 77.0%

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

   12. Taylor expanded in x around inf 61.4%

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

       1. mul-1-neg 61.4%

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

       2. distribute-frac-neg2 61.4%

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

   14. Simplified 61.4%

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

   if -8.49999999999999977e145 < x < 7.40000000000000009e-7

   1. Initial program 100.0%

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

   3. Taylor expanded in x around -inf 100.0%

      \[\leadsto \color{blue}{\frac{\left|-\left(y + -1 \cdot x\right)\right|}{\left|y\right|}} \]
   4. 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 20.8%

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

      7. fabs-sqr 20.8%

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

      8. rem-square-sqrt 22.1%

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

      9. div-sub 22.1%

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

      10. sub-neg 22.1%

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

      11. *-inverses 22.1%

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

      12. metadata-eval 22.1%

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

      13. +-commutative 22.1%

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

   5. Simplified 22.1%

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

      1. +-commutative 22.1%

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

      2. metadata-eval 22.1%

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

      3. sub-neg 22.1%

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

      4. *-inverses 22.1%

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

      5. div-sub 22.1%

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

      6. clear-num 22.0%

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

   7. Applied egg-rr 22.0%

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

      1. remove-double-div 22.0%

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

      2. frac-2neg 22.0%

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

      3. metadata-eval 22.0%

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

      4. associate-/r/ 22.0%

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

      5. metadata-eval 22.0%

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

      6. clear-num 22.0%

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

      7. distribute-neg-frac 22.0%

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

      8. metadata-eval 22.0%

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

      9. div-sub 22.0%

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

      10. *-inverses 22.0%

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

      11. sub-neg 22.0%

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

      12. metadata-eval 22.0%

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

   9. Applied egg-rr 22.0%

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

       1. neg-mul-1 22.0%

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

       2. distribute-neg-frac2 22.0%

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

       3. +-commutative 22.0%

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

       4. distribute-neg-in 22.0%

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

       5. metadata-eval 22.0%

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

       6. sub-neg 22.0%

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

       7. rem-square-sqrt 1.1%

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

       8. fabs-sqr 1.1%

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

       9. rem-square-sqrt 1.3%

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

       10. fabs-sub 1.3%

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

       11. sub-neg 1.3%

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

       12. metadata-eval 1.3%

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

       13. rem-square-sqrt 0.2%

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

       14. fabs-sqr 0.2%

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

       15. rem-square-sqrt 79.2%

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

       16. +-commutative 79.2%

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

   11. Simplified 79.2%

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

   12. Taylor expanded in x around 0 64.9%

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

   if 7.40000000000000009e-7 < x < 6.2000000000000001e114

   1. Initial program 100.0%

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

   3. Taylor expanded in x around -inf 100.0%

      \[\leadsto \color{blue}{\frac{\left|-\left(y + -1 \cdot x\right)\right|}{\left|y\right|}} \]
   4. 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 59.8%

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

      7. fabs-sqr 59.8%

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

      8. rem-square-sqrt 60.5%

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

      9. div-sub 60.5%

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

      10. sub-neg 60.5%

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

      11. *-inverses 60.5%

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

      12. metadata-eval 60.5%

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

      13. +-commutative 60.5%

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

   5. Simplified 60.5%

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

   6. Taylor expanded in x around inf 60.6%

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

   if 1.9000000000000001e133 < x < 1.80000000000000001e211

   1. Initial program 100.0%

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

   3. Taylor expanded in x around -inf 100.0%

      \[\leadsto \color{blue}{\frac{\left|-\left(y + -1 \cdot x\right)\right|}{\left|y\right|}} \]
   4. 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 68.2%

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

      7. fabs-sqr 68.2%

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

      8. rem-square-sqrt 68.9%

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

      9. div-sub 68.9%

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

      10. sub-neg 68.9%

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

      11. *-inverses 68.9%

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

      12. metadata-eval 68.9%

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

      13. +-commutative 68.9%

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

   5. Simplified 68.9%

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

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{+145}:\\ \;\;\;\;\frac{x}{-y}\\ \mathbf{elif}\;x \leq 7.4 \cdot 10^{-7}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{+114}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{+133} \lor \neg \left(x \leq 1.8 \cdot 10^{+211}\right):\\ \;\;\;\;\frac{x}{-y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 58.4% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{-y}\\ \mathbf{if}\;x \leq -8.2 \cdot 10^{+145}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 8.6 \cdot 10^{-7}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{+114} \lor \neg \left(x \leq 2.2 \cdot 10^{+132}\right) \land x \leq 8 \cdot 10^{+208}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (- y))))
   (if (<= x -8.2e+145)
     t_0
     (if (<= x 8.6e-7)
       1.0
       (if (or (<= x 1.6e+114) (and (not (<= x 2.2e+132)) (<= x 8e+208)))
         (/ x y)
         t_0)))))

C

double code(double x, double y) {
	double t_0 = x / -y;
	double tmp;
	if (x <= -8.2e+145) {
		tmp = t_0;
	} else if (x <= 8.6e-7) {
		tmp = 1.0;
	} else if ((x <= 1.6e+114) || (!(x <= 2.2e+132) && (x <= 8e+208))) {
		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 = x / -y
    if (x <= (-8.2d+145)) then
        tmp = t_0
    else if (x <= 8.6d-7) then
        tmp = 1.0d0
    else if ((x <= 1.6d+114) .or. (.not. (x <= 2.2d+132)) .and. (x <= 8d+208)) 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 = x / -y;
	double tmp;
	if (x <= -8.2e+145) {
		tmp = t_0;
	} else if (x <= 8.6e-7) {
		tmp = 1.0;
	} else if ((x <= 1.6e+114) || (!(x <= 2.2e+132) && (x <= 8e+208))) {
		tmp = x / y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / -y
	tmp = 0
	if x <= -8.2e+145:
		tmp = t_0
	elif x <= 8.6e-7:
		tmp = 1.0
	elif (x <= 1.6e+114) or (not (x <= 2.2e+132) and (x <= 8e+208)):
		tmp = x / y
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(-y))
	tmp = 0.0
	if (x <= -8.2e+145)
		tmp = t_0;
	elseif (x <= 8.6e-7)
		tmp = 1.0;
	elseif ((x <= 1.6e+114) || (!(x <= 2.2e+132) && (x <= 8e+208)))
		tmp = Float64(x / y);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / -y;
	tmp = 0.0;
	if (x <= -8.2e+145)
		tmp = t_0;
	elseif (x <= 8.6e-7)
		tmp = 1.0;
	elseif ((x <= 1.6e+114) || (~((x <= 2.2e+132)) && (x <= 8e+208)))
		tmp = x / y;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / (-y)), $MachinePrecision]}, If[LessEqual[x, -8.2e+145], t$95$0, If[LessEqual[x, 8.6e-7], 1.0, If[Or[LessEqual[x, 1.6e+114], And[N[Not[LessEqual[x, 2.2e+132]], $MachinePrecision], LessEqual[x, 8e+208]]], N[(x / y), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.2000000000000003e145 or 1.6e114 < x < 2.19999999999999989e132 or 7.9999999999999999e208 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around -inf 100.0%

      \[\leadsto \color{blue}{\frac{\left|-\left(y + -1 \cdot x\right)\right|}{\left|y\right|}} \]
   4. 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 23.3%

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

      7. fabs-sqr 23.3%

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

      8. rem-square-sqrt 23.9%

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

      9. div-sub 23.9%

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

      10. sub-neg 23.9%

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

      11. *-inverses 23.9%

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

      12. metadata-eval 23.9%

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

      13. +-commutative 23.9%

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

   5. Simplified 23.9%

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

      1. +-commutative 23.9%

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

      2. metadata-eval 23.9%

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

      3. sub-neg 23.9%

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

      4. *-inverses 23.9%

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

      5. div-sub 23.9%

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

      6. clear-num 23.8%

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

   7. Applied egg-rr 23.8%

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

      1. remove-double-div 23.8%

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

      2. frac-2neg 23.8%

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

      3. metadata-eval 23.8%

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

      4. associate-/r/ 23.8%

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

      5. metadata-eval 23.8%

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

      6. clear-num 23.9%

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

      7. distribute-neg-frac 23.9%

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

      8. metadata-eval 23.9%

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

      9. div-sub 23.9%

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

      10. *-inverses 23.9%

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

      11. sub-neg 23.9%

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

      12. metadata-eval 23.9%

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

   9. Applied egg-rr 23.9%

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

       1. neg-mul-1 23.9%

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

       2. distribute-neg-frac2 23.9%

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

       3. +-commutative 23.9%

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

       4. distribute-neg-in 23.9%

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

       5. metadata-eval 23.9%

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

       6. sub-neg 23.9%

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

       7. rem-square-sqrt 0.5%

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

       8. fabs-sqr 0.5%

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

       9. rem-square-sqrt 0.7%

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

       10. fabs-sub 0.7%

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

       11. sub-neg 0.7%

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

       12. metadata-eval 0.7%

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

       13. rem-square-sqrt 0.1%

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

       14. fabs-sqr 0.1%

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

       15. rem-square-sqrt 76.5%

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

       16. +-commutative 76.5%

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

   11. Simplified 76.5%

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

   12. Taylor expanded in x around inf 62.7%

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

       1. mul-1-neg 62.7%

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

       2. distribute-frac-neg2 62.7%

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

   14. Simplified 62.7%

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

   if -8.2000000000000003e145 < x < 8.6000000000000002e-7

   1. Initial program 100.0%

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

   3. Taylor expanded in x around -inf 100.0%

      \[\leadsto \color{blue}{\frac{\left|-\left(y + -1 \cdot x\right)\right|}{\left|y\right|}} \]
   4. 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 20.8%

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

      7. fabs-sqr 20.8%

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

      8. rem-square-sqrt 22.1%

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

      9. div-sub 22.1%

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

      10. sub-neg 22.1%

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

      11. *-inverses 22.1%

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

      12. metadata-eval 22.1%

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

      13. +-commutative 22.1%

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

   5. Simplified 22.1%

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

      1. +-commutative 22.1%

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

      2. metadata-eval 22.1%

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

      3. sub-neg 22.1%

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

      4. *-inverses 22.1%

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

      5. div-sub 22.1%

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

      6. clear-num 22.0%

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

   7. Applied egg-rr 22.0%

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

      1. remove-double-div 22.0%

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

      2. frac-2neg 22.0%

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

      3. metadata-eval 22.0%

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

      4. associate-/r/ 22.0%

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

      5. metadata-eval 22.0%

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

      6. clear-num 22.0%

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

      7. distribute-neg-frac 22.0%

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

      8. metadata-eval 22.0%

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

      9. div-sub 22.0%

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

      10. *-inverses 22.0%

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

      11. sub-neg 22.0%

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

      12. metadata-eval 22.0%

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

   9. Applied egg-rr 22.0%

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

       1. neg-mul-1 22.0%

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

       2. distribute-neg-frac2 22.0%

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

       3. +-commutative 22.0%

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

       4. distribute-neg-in 22.0%

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

       5. metadata-eval 22.0%

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

       6. sub-neg 22.0%

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

       7. rem-square-sqrt 1.1%

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

       8. fabs-sqr 1.1%

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

       9. rem-square-sqrt 1.3%

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

       10. fabs-sub 1.3%

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

       11. sub-neg 1.3%

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

       12. metadata-eval 1.3%

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

       13. rem-square-sqrt 0.2%

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

       14. fabs-sqr 0.2%

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

       15. rem-square-sqrt 79.2%

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

       16. +-commutative 79.2%

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

   11. Simplified 79.2%

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

   12. Taylor expanded in x around 0 64.9%

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

   if 8.6000000000000002e-7 < x < 1.6e114 or 2.19999999999999989e132 < x < 7.9999999999999999e208

   1. Initial program 100.0%

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

   3. Taylor expanded in x around -inf 100.0%

      \[\leadsto \color{blue}{\frac{\left|-\left(y + -1 \cdot x\right)\right|}{\left|y\right|}} \]
   4. 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 61.8%

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

      7. fabs-sqr 61.8%

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

      8. rem-square-sqrt 62.6%

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

      9. div-sub 62.6%

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

      10. sub-neg 62.6%

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

      11. *-inverses 62.6%

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

      12. metadata-eval 62.6%

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

      13. +-commutative 62.6%

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

   5. Simplified 62.6%

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

   6. Taylor expanded in x around inf 61.3%

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

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.2 \cdot 10^{+145}:\\ \;\;\;\;\frac{x}{-y}\\ \mathbf{elif}\;x \leq 8.6 \cdot 10^{-7}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{+114} \lor \neg \left(x \leq 2.2 \cdot 10^{+132}\right) \land x \leq 8 \cdot 10^{+208}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.0% accurate, 13.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.9 \cdot 10^{-209} \lor \neg \left(y \leq 4 \cdot 10^{-242}\right):\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.9e-209) (not (<= y 4e-242)))
   (- 1.0 (/ x y))
   (+ (/ x y) -1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.9e-209) || !(y <= 4e-242)) {
		tmp = 1.0 - (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 <= (-1.9d-209)) .or. (.not. (y <= 4d-242))) then
        tmp = 1.0d0 - (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 <= -1.9e-209) || !(y <= 4e-242)) {
		tmp = 1.0 - (x / y);
	} else {
		tmp = (x / y) + -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.9e-209) or not (y <= 4e-242):
		tmp = 1.0 - (x / y)
	else:
		tmp = (x / y) + -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.9e-209) || !(y <= 4e-242))
		tmp = Float64(1.0 - 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 <= -1.9e-209) || ~((y <= 4e-242)))
		tmp = 1.0 - (x / y);
	else
		tmp = (x / y) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.9e-209], N[Not[LessEqual[y, 4e-242]], $MachinePrecision]], N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.8999999999999999e-209 or 4e-242 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around -inf 100.0%

      \[\leadsto \color{blue}{\frac{\left|-\left(y + -1 \cdot x\right)\right|}{\left|y\right|}} \]
   4. 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 19.4%

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

      7. fabs-sqr 19.4%

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

      8. rem-square-sqrt 20.6%

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

      9. div-sub 20.6%

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

      10. sub-neg 20.6%

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

      11. *-inverses 20.6%

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

      12. metadata-eval 20.6%

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

      13. +-commutative 20.6%

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

   5. Simplified 20.6%

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

      1. +-commutative 20.6%

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

      2. metadata-eval 20.6%

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

      3. sub-neg 20.6%

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

      4. *-inverses 20.6%

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

      5. div-sub 20.6%

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

      6. clear-num 20.5%

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

   7. Applied egg-rr 20.5%

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

      1. remove-double-div 20.5%

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

      2. frac-2neg 20.5%

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

      3. metadata-eval 20.5%

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

      4. associate-/r/ 20.5%

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

      5. metadata-eval 20.5%

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

      6. clear-num 20.5%

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

      7. distribute-neg-frac 20.5%

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

      8. metadata-eval 20.5%

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

      9. div-sub 20.6%

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

      10. *-inverses 20.6%

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

      11. sub-neg 20.6%

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

      12. metadata-eval 20.6%

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

   9. Applied egg-rr 20.6%

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

       1. neg-mul-1 20.6%

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

       2. distribute-neg-frac2 20.6%

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

       3. +-commutative 20.6%

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

       4. distribute-neg-in 20.6%

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

       5. metadata-eval 20.6%

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

       6. sub-neg 20.6%

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

       7. rem-square-sqrt 1.1%

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

       8. fabs-sqr 1.1%

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

       9. rem-square-sqrt 1.2%

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

       10. fabs-sub 1.2%

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

       11. sub-neg 1.2%

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

       12. metadata-eval 1.2%

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

       13. rem-square-sqrt 0.2%

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

       14. fabs-sqr 0.2%

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

       15. rem-square-sqrt 80.6%

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

       16. +-commutative 80.6%

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

   11. Simplified 80.6%

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

   12. Taylor expanded in x around 0 80.6%

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

       1. mul-1-neg 80.6%

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

       2. sub-neg 80.6%

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

   14. Simplified 80.6%

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

   if -1.8999999999999999e-209 < y < 4e-242

   1. Initial program 100.0%

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

   3. Taylor expanded in x around -inf 100.0%

      \[\leadsto \color{blue}{\frac{\left|-\left(y + -1 \cdot x\right)\right|}{\left|y\right|}} \]
   4. 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 68.1%

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

      7. fabs-sqr 68.1%

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

      8. rem-square-sqrt 68.5%

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

      9. div-sub 68.4%

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

      10. sub-neg 68.4%

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

      11. *-inverses 68.4%

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

      12. metadata-eval 68.4%

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

      13. +-commutative 68.4%

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

   5. Simplified 68.4%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.9 \cdot 10^{-209} \lor \neg \left(y \leq 4 \cdot 10^{-242}\right):\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.0% accurate, 13.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.85 \cdot 10^{-209} \lor \neg \left(y \leq 3.4 \cdot 10^{-242}\right):\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.85e-209) (not (<= y 3.4e-242)))
   (- 1.0 (/ x y))
   (/ (- x y) y)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.85e-209) || !(y <= 3.4e-242)) {
		tmp = 1.0 - (x / y);
	} else {
		tmp = (x - y) / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-1.85d-209)) .or. (.not. (y <= 3.4d-242))) then
        tmp = 1.0d0 - (x / y)
    else
        tmp = (x - y) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.85e-209) || !(y <= 3.4e-242)) {
		tmp = 1.0 - (x / y);
	} else {
		tmp = (x - y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.85e-209) or not (y <= 3.4e-242):
		tmp = 1.0 - (x / y)
	else:
		tmp = (x - y) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.85e-209) || !(y <= 3.4e-242))
		tmp = Float64(1.0 - Float64(x / y));
	else
		tmp = Float64(Float64(x - y) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.85e-209) || ~((y <= 3.4e-242)))
		tmp = 1.0 - (x / y);
	else
		tmp = (x - y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.85e-209], N[Not[LessEqual[y, 3.4e-242]], $MachinePrecision]], N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision], N[(N[(x - y), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.8499999999999999e-209 or 3.4000000000000001e-242 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around -inf 100.0%

      \[\leadsto \color{blue}{\frac{\left|-\left(y + -1 \cdot x\right)\right|}{\left|y\right|}} \]
   4. 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 19.4%

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

      7. fabs-sqr 19.4%

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

      8. rem-square-sqrt 20.6%

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

      9. div-sub 20.6%

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

      10. sub-neg 20.6%

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

      11. *-inverses 20.6%

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

      12. metadata-eval 20.6%

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

      13. +-commutative 20.6%

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

   5. Simplified 20.6%

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

      1. +-commutative 20.6%

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

      2. metadata-eval 20.6%

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

      3. sub-neg 20.6%

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

      4. *-inverses 20.6%

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

      5. div-sub 20.6%

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

      6. clear-num 20.5%

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

   7. Applied egg-rr 20.5%

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

      1. remove-double-div 20.5%

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

      2. frac-2neg 20.5%

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

      3. metadata-eval 20.5%

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

      4. associate-/r/ 20.5%

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

      5. metadata-eval 20.5%

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

      6. clear-num 20.5%

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

      7. distribute-neg-frac 20.5%

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

      8. metadata-eval 20.5%

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

      9. div-sub 20.6%

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

      10. *-inverses 20.6%

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

      11. sub-neg 20.6%

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

      12. metadata-eval 20.6%

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

   9. Applied egg-rr 20.6%

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

       1. neg-mul-1 20.6%

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

       2. distribute-neg-frac2 20.6%

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

       3. +-commutative 20.6%

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

       4. distribute-neg-in 20.6%

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

       5. metadata-eval 20.6%

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

       6. sub-neg 20.6%

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

       7. rem-square-sqrt 1.1%

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

       8. fabs-sqr 1.1%

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

       9. rem-square-sqrt 1.2%

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

       10. fabs-sub 1.2%

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

       11. sub-neg 1.2%

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

       12. metadata-eval 1.2%

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

       13. rem-square-sqrt 0.2%

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

       14. fabs-sqr 0.2%

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

       15. rem-square-sqrt 80.6%

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

       16. +-commutative 80.6%

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

   11. Simplified 80.6%

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

   12. Taylor expanded in x around 0 80.6%

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

       1. mul-1-neg 80.6%

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

       2. sub-neg 80.6%

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

   14. Simplified 80.6%

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

   if -1.8499999999999999e-209 < y < 3.4000000000000001e-242

   1. Initial program 100.0%

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

   3. Taylor expanded in x around -inf 100.0%

      \[\leadsto \color{blue}{\frac{\left|-\left(y + -1 \cdot x\right)\right|}{\left|y\right|}} \]
   4. 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 68.1%

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

      7. fabs-sqr 68.1%

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

      8. rem-square-sqrt 68.5%

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

      9. div-sub 68.4%

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

      10. sub-neg 68.4%

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

      11. *-inverses 68.4%

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

      12. metadata-eval 68.4%

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

      13. +-commutative 68.4%

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

   5. Simplified 68.4%

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

      1. +-commutative 68.4%

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

      2. metadata-eval 68.4%

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

      3. sub-neg 68.4%

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

      4. *-inverses 68.4%

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

      5. div-sub 68.5%

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

   7. Applied egg-rr 68.5%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.85 \cdot 10^{-209} \lor \neg \left(y \leq 3.4 \cdot 10^{-242}\right):\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 59.6% accurate, 15.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{-48}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-98}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -5.5e-48) 1.0 (if (<= y 4.2e-98) (/ x y) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -5.5e-48) {
		tmp = 1.0;
	} else if (y <= 4.2e-98) {
		tmp = x / y;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-5.5d-48)) then
        tmp = 1.0d0
    else if (y <= 4.2d-98) then
        tmp = x / y
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -5.5e-48) {
		tmp = 1.0;
	} else if (y <= 4.2e-98) {
		tmp = x / y;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -5.5e-48:
		tmp = 1.0
	elif y <= 4.2e-98:
		tmp = x / y
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -5.5e-48)
		tmp = 1.0;
	elseif (y <= 4.2e-98)
		tmp = Float64(x / y);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -5.5e-48)
		tmp = 1.0;
	elseif (y <= 4.2e-98)
		tmp = x / y;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -5.5e-48], 1.0, If[LessEqual[y, 4.2e-98], N[(x / y), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -5.50000000000000047e-48 or 4.19999999999999984e-98 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around -inf 100.0%

      \[\leadsto \color{blue}{\frac{\left|-\left(y + -1 \cdot x\right)\right|}{\left|y\right|}} \]
   4. 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 13.3%

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

      7. fabs-sqr 13.3%

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

      8. rem-square-sqrt 14.7%

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

      9. div-sub 14.7%

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

      10. sub-neg 14.7%

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

      11. *-inverses 14.7%

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

      12. metadata-eval 14.7%

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

      13. +-commutative 14.7%

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

   5. Simplified 14.7%

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

      1. +-commutative 14.7%

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

      2. metadata-eval 14.7%

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

      3. sub-neg 14.7%

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

      4. *-inverses 14.7%

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

      5. div-sub 14.7%

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

      6. clear-num 14.6%

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

   7. Applied egg-rr 14.6%

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

      1. remove-double-div 14.6%

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

      2. frac-2neg 14.6%

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

      3. metadata-eval 14.6%

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

      4. associate-/r/ 14.6%

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

      5. metadata-eval 14.6%

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

      6. clear-num 14.7%

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

      7. distribute-neg-frac 14.7%

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

      8. metadata-eval 14.7%

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

      9. div-sub 14.7%

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

      10. *-inverses 14.7%

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

      11. sub-neg 14.7%

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

      12. metadata-eval 14.7%

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

   9. Applied egg-rr 14.7%

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

       1. neg-mul-1 14.7%

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

       2. distribute-neg-frac2 14.7%

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

       3. +-commutative 14.7%

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

       4. distribute-neg-in 14.7%

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

       5. metadata-eval 14.7%

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

       6. sub-neg 14.7%

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

       7. rem-square-sqrt 1.3%

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

       8. fabs-sqr 1.3%

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

       9. rem-square-sqrt 1.4%

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

       10. fabs-sub 1.4%

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

       11. sub-neg 1.4%

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

       12. metadata-eval 1.4%

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

       13. rem-square-sqrt 0.1%

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

       14. fabs-sqr 0.1%

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

       15. rem-square-sqrt 86.6%

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

       16. +-commutative 86.6%

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

   11. Simplified 86.6%

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

   12. Taylor expanded in x around 0 69.3%

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

   if -5.50000000000000047e-48 < y < 4.19999999999999984e-98

   1. Initial program 100.0%

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

   3. Taylor expanded in x around -inf 100.0%

      \[\leadsto \color{blue}{\frac{\left|-\left(y + -1 \cdot x\right)\right|}{\left|y\right|}} \]
   4. 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 48.8%

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

      7. fabs-sqr 48.8%

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

      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%

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

   5. Simplified 49.4%

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

   6. Taylor expanded in x around inf 48.6%

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

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{-48}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-98}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 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. Add Preprocessing

3. Taylor expanded in x around -inf 100.0%

   \[\leadsto \color{blue}{\frac{\left|-\left(y + -1 \cdot x\right)\right|}{\left|y\right|}} \]
4. 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 27.2%

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

   7. fabs-sqr 27.2%

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

   8. rem-square-sqrt 28.3%

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

   9. div-sub 28.3%

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

   10. sub-neg 28.3%

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

   11. *-inverses 28.3%

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

   12. metadata-eval 28.3%

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

   13. +-commutative 28.3%

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

5. Simplified 28.3%

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

6. Taylor expanded in x around 0 1.3%

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

7. Final simplification 1.3%

   \[\leadsto -1 \]
8. Add Preprocessing

Alternative 9: 52.7% 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. Add Preprocessing

3. Taylor expanded in x around -inf 100.0%

   \[\leadsto \color{blue}{\frac{\left|-\left(y + -1 \cdot x\right)\right|}{\left|y\right|}} \]
4. 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 27.2%

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

   7. fabs-sqr 27.2%

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

   8. rem-square-sqrt 28.3%

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

   9. div-sub 28.3%

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

   10. sub-neg 28.3%

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

   11. *-inverses 28.3%

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

   12. metadata-eval 28.3%

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

   13. +-commutative 28.3%

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

5. Simplified 28.3%

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

   1. +-commutative 28.3%

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

   2. metadata-eval 28.3%

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

   3. sub-neg 28.3%

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

   4. *-inverses 28.3%

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

   5. div-sub 28.3%

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

   6. clear-num 28.2%

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

7. Applied egg-rr 28.2%

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

   1. remove-double-div 28.2%

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

   2. frac-2neg 28.2%

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

   3. metadata-eval 28.2%

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

   4. associate-/r/ 28.2%

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

   5. metadata-eval 28.2%

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

   6. clear-num 28.2%

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

   7. distribute-neg-frac 28.2%

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

   8. metadata-eval 28.2%

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

   9. div-sub 28.2%

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

   10. *-inverses 28.2%

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

   11. sub-neg 28.2%

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

   12. metadata-eval 28.2%

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

9. Applied egg-rr 28.2%

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

    1. neg-mul-1 28.2%

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

    2. distribute-neg-frac2 28.2%

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

    3. +-commutative 28.2%

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

    4. distribute-neg-in 28.2%

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

    5. metadata-eval 28.2%

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

    6. sub-neg 28.2%

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

    7. rem-square-sqrt 0.9%

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

    8. fabs-sqr 0.9%

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

    9. rem-square-sqrt 1.1%

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

    10. fabs-sub 1.1%

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

    11. sub-neg 1.1%

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

    12. metadata-eval 1.1%

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

    13. rem-square-sqrt 0.2%

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

    14. fabs-sqr 0.2%

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

    15. rem-square-sqrt 72.8%

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

    16. +-commutative 72.8%

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

11. Simplified 72.8%

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

12. Taylor expanded in x around 0 49.1%

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

13. Final simplification 49.1%

    \[\leadsto 1 \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024053 
(FPCore (x y)
  :name "Numeric.LinearAlgebra.Util:formatSparse from hmatrix-0.16.1.5"
  :precision binary64
  (/ (fabs (- x y)) (fabs y)))