Numeric.LinearAlgebra.Util:formatSparse from hmatrix-0.16.1.5

Percentage Accurate: 100.0% → 100.0%

Time: 4.4s

Alternatives: 8

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. Step-by-step derivation

   1. fabs-sub 100.0%

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

   2. div-fabs 100.0%

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

4. Applied egg-rr 100.0%

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

   1. div-sub 100.0%

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

   2. *-inverses 100.0%

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

6. Applied egg-rr 100.0%

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

Alternative 2: 83.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.8 \cdot 10^{+89} \lor \neg \left(x \leq 6.5 \cdot 10^{+93}\right):\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -7.8e+89) (not (<= x 6.5e+93))) (fabs (/ x y)) (- 1.0 (/ x y))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -7.8e+89) || !(x <= 6.5e+93)) {
		tmp = fabs((x / y));
	} else {
		tmp = 1.0 - (x / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-7.8d+89)) .or. (.not. (x <= 6.5d+93))) then
        tmp = abs((x / y))
    else
        tmp = 1.0d0 - (x / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -7.8e+89) || !(x <= 6.5e+93)) {
		tmp = Math.abs((x / y));
	} else {
		tmp = 1.0 - (x / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -7.8e+89) or not (x <= 6.5e+93):
		tmp = math.fabs((x / y))
	else:
		tmp = 1.0 - (x / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -7.8e+89) || !(x <= 6.5e+93))
		tmp = abs(Float64(x / y));
	else
		tmp = Float64(1.0 - Float64(x / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -7.8e+89) || ~((x <= 6.5e+93)))
		tmp = abs((x / y));
	else
		tmp = 1.0 - (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -7.8e+89], N[Not[LessEqual[x, 6.5e+93]], $MachinePrecision]], N[Abs[N[(x / y), $MachinePrecision]], $MachinePrecision], N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -7.80000000000000021e89 or 6.4999999999999998e93 < x

   1. Initial program 100.0%

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

      1. fabs-sub 100.0%

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

      2. div-fabs 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in y around 0 85.5%

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

      1. neg-mul-1 85.5%

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

      2. distribute-neg-frac2 85.5%

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

   7. Simplified 85.5%

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

   if -7.80000000000000021e89 < x < 6.4999999999999998e93

   1. Initial program 100.0%

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

      1. add-sqr-sqrt 47.3%

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

      2. fabs-sqr 47.3%

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

      3. add-sqr-sqrt 6.9%

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

      4. fabs-sqr 6.9%

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

      5. add-sqr-sqrt 7.5%

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

      6. add-sqr-sqrt 16.3%

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

      7. clear-num 16.3%

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

   4. Applied egg-rr 16.3%

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

      1. div-inv 16.3%

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

   6. Applied egg-rr 16.3%

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

      1. *-commutative 16.3%

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

      2. associate-/r/ 16.3%

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

      3. metadata-eval 16.3%

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

      4. div-sub 16.3%

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

      5. *-inverses 16.3%

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

      6. sub-neg 16.3%

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

      7. metadata-eval 16.3%

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

      8. associate-/r* 16.3%

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

      9. +-commutative 16.3%

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

      10. distribute-lft-in 16.3%

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

      11. metadata-eval 16.3%

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

      12. neg-mul-1 16.3%

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

      13. sub-neg 16.3%

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

      14. rem-square-sqrt 1.2%

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

      15. fabs-sqr 1.2%

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

      16. rem-square-sqrt 1.3%

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

      17. fabs-sub 1.3%

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

      18. sub-neg 1.3%

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

      19. metadata-eval 1.3%

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

      20. rem-square-sqrt 0.1%

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

      21. fabs-sqr 0.1%

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

      22. rem-square-sqrt 84.9%

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

   8. Simplified 84.9%

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

   9. Taylor expanded in y around 0 85.0%

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

       1. neg-mul-1 85.0%

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

       2. sub-neg 85.0%

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

       3. div-sub 85.0%

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

       4. *-inverses 85.0%

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

   11. Simplified 85.0%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.8 \cdot 10^{+89} \lor \neg \left(x \leq 6.5 \cdot 10^{+93}\right):\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 59.0% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{-y}\\ t_1 := \frac{x}{y} + -1\\ \mathbf{if}\;x \leq -7 \cdot 10^{+113}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{+16}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{+43}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{+111}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{+182}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (- y))) (t_1 (+ (/ x y) -1.0)))
   (if (<= x -7e+113)
     t_1
     (if (<= x 1.6e+16)
       1.0
       (if (<= x 1.4e+43)
         t_0
         (if (<= x 2.7e+111) 1.0 (if (<= x 3.5e+182) t_1 t_0)))))))

C

double code(double x, double y) {
	double t_0 = x / -y;
	double t_1 = (x / y) + -1.0;
	double tmp;
	if (x <= -7e+113) {
		tmp = t_1;
	} else if (x <= 1.6e+16) {
		tmp = 1.0;
	} else if (x <= 1.4e+43) {
		tmp = t_0;
	} else if (x <= 2.7e+111) {
		tmp = 1.0;
	} else if (x <= 3.5e+182) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_0 = x / -y
    t_1 = (x / y) + (-1.0d0)
    if (x <= (-7d+113)) then
        tmp = t_1
    else if (x <= 1.6d+16) then
        tmp = 1.0d0
    else if (x <= 1.4d+43) then
        tmp = t_0
    else if (x <= 2.7d+111) then
        tmp = 1.0d0
    else if (x <= 3.5d+182) then
        tmp = t_1
    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 t_1 = (x / y) + -1.0;
	double tmp;
	if (x <= -7e+113) {
		tmp = t_1;
	} else if (x <= 1.6e+16) {
		tmp = 1.0;
	} else if (x <= 1.4e+43) {
		tmp = t_0;
	} else if (x <= 2.7e+111) {
		tmp = 1.0;
	} else if (x <= 3.5e+182) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / -y
	t_1 = (x / y) + -1.0
	tmp = 0
	if x <= -7e+113:
		tmp = t_1
	elif x <= 1.6e+16:
		tmp = 1.0
	elif x <= 1.4e+43:
		tmp = t_0
	elif x <= 2.7e+111:
		tmp = 1.0
	elif x <= 3.5e+182:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(-y))
	t_1 = Float64(Float64(x / y) + -1.0)
	tmp = 0.0
	if (x <= -7e+113)
		tmp = t_1;
	elseif (x <= 1.6e+16)
		tmp = 1.0;
	elseif (x <= 1.4e+43)
		tmp = t_0;
	elseif (x <= 2.7e+111)
		tmp = 1.0;
	elseif (x <= 3.5e+182)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / -y;
	t_1 = (x / y) + -1.0;
	tmp = 0.0;
	if (x <= -7e+113)
		tmp = t_1;
	elseif (x <= 1.6e+16)
		tmp = 1.0;
	elseif (x <= 1.4e+43)
		tmp = t_0;
	elseif (x <= 2.7e+111)
		tmp = 1.0;
	elseif (x <= 3.5e+182)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / (-y)), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x / y), $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[x, -7e+113], t$95$1, If[LessEqual[x, 1.6e+16], 1.0, If[LessEqual[x, 1.4e+43], t$95$0, If[LessEqual[x, 2.7e+111], 1.0, If[LessEqual[x, 3.5e+182], t$95$1, t$95$0]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.0000000000000001e113 or 2.6999999999999999e111 < x < 3.50000000000000023e182

   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 54.3%

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

      7. fabs-sqr 54.3%

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

      8. rem-square-sqrt 55.0%

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

      9. div-sub 55.1%

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

      10. sub-neg 55.1%

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

      11. *-inverses 55.1%

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

      12. metadata-eval 55.1%

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

      13. +-commutative 55.1%

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

   5. Simplified 55.1%

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

   if -7.0000000000000001e113 < x < 1.6e16 or 1.40000000000000009e43 < x < 2.6999999999999999e111

   1. Initial program 100.0%

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

      1. add-sqr-sqrt 46.1%

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

      2. fabs-sqr 46.1%

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

      3. add-sqr-sqrt 5.8%

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

      4. fabs-sqr 5.8%

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

      5. add-sqr-sqrt 6.4%

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

      6. add-sqr-sqrt 15.9%

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

      7. clear-num 15.9%

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

   4. Applied egg-rr 15.9%

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

      1. div-inv 15.9%

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

   6. Applied egg-rr 15.9%

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

      1. *-commutative 15.9%

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

      2. associate-/r/ 15.9%

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

      3. metadata-eval 15.9%

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

      4. div-sub 15.9%

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

      5. *-inverses 15.9%

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

      6. sub-neg 15.9%

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

      7. metadata-eval 15.9%

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

      8. associate-/r* 15.9%

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

      9. +-commutative 15.9%

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

      10. distribute-lft-in 15.9%

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

      11. metadata-eval 15.9%

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

      12. neg-mul-1 15.9%

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

      13. sub-neg 15.9%

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

      14. rem-square-sqrt 1.2%

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

      15. fabs-sqr 1.2%

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

      16. rem-square-sqrt 1.3%

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

      17. fabs-sub 1.3%

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

      18. sub-neg 1.3%

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

      19. metadata-eval 1.3%

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

      20. rem-square-sqrt 0.1%

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

      21. fabs-sqr 0.1%

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

      22. rem-square-sqrt 85.3%

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

   8. Simplified 85.3%

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

   9. Taylor expanded in y around inf 71.6%

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

   if 1.6e16 < x < 1.40000000000000009e43 or 3.50000000000000023e182 < x

   1. Initial program 100.0%

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

      1. add-sqr-sqrt 93.6%

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

      2. fabs-sqr 93.6%

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

      3. add-sqr-sqrt 27.0%

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

      4. fabs-sqr 27.0%

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

      5. add-sqr-sqrt 27.3%

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

      6. add-sqr-sqrt 27.6%

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

      7. clear-num 27.6%

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

   4. Applied egg-rr 27.6%

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

      1. div-inv 27.6%

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

   6. Applied egg-rr 27.6%

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

      1. *-commutative 27.6%

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

      2. associate-/r/ 27.6%

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

      3. metadata-eval 27.6%

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

      4. div-sub 27.6%

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

      5. *-inverses 27.6%

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

      6. sub-neg 27.6%

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

      7. metadata-eval 27.6%

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

      8. associate-/r* 27.6%

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

      9. +-commutative 27.6%

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

      10. distribute-lft-in 27.6%

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

      11. metadata-eval 27.6%

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

      12. neg-mul-1 27.6%

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

      13. sub-neg 27.6%

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

      14. rem-square-sqrt 0.3%

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

      15. fabs-sqr 0.3%

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

      16. rem-square-sqrt 0.4%

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

      17. fabs-sub 0.4%

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

      18. sub-neg 0.4%

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

      19. metadata-eval 0.4%

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

      20. rem-square-sqrt 0.1%

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

      21. fabs-sqr 0.1%

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

      22. rem-square-sqrt 72.7%

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

   8. Simplified 72.7%

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

   9. Taylor expanded in y around 0 66.8%

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

       1. associate-*r/ 66.8%

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

       2. neg-mul-1 66.8%

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

   11. Simplified 66.8%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{+113}:\\ \;\;\;\;\frac{x}{y} + -1\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{+16}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{+43}:\\ \;\;\;\;\frac{x}{-y}\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{+111}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{+182}:\\ \;\;\;\;\frac{x}{y} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 59.0% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{-y}\\ \mathbf{if}\;x \leq -8.8 \cdot 10^{+111}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{+16}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{+43}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+111}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+182}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (- y))))
   (if (<= x -8.8e+111)
     (/ x y)
     (if (<= x 1.6e+16)
       1.0
       (if (<= x 4.8e+43)
         t_0
         (if (<= x 1.35e+111) 1.0 (if (<= x 4e+182) (/ x y) t_0)))))))

C

double code(double x, double y) {
	double t_0 = x / -y;
	double tmp;
	if (x <= -8.8e+111) {
		tmp = x / y;
	} else if (x <= 1.6e+16) {
		tmp = 1.0;
	} else if (x <= 4.8e+43) {
		tmp = t_0;
	} else if (x <= 1.35e+111) {
		tmp = 1.0;
	} else if (x <= 4e+182) {
		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 <= (-8.8d+111)) then
        tmp = x / y
    else if (x <= 1.6d+16) then
        tmp = 1.0d0
    else if (x <= 4.8d+43) then
        tmp = t_0
    else if (x <= 1.35d+111) then
        tmp = 1.0d0
    else if (x <= 4d+182) 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 <= -8.8e+111) {
		tmp = x / y;
	} else if (x <= 1.6e+16) {
		tmp = 1.0;
	} else if (x <= 4.8e+43) {
		tmp = t_0;
	} else if (x <= 1.35e+111) {
		tmp = 1.0;
	} else if (x <= 4e+182) {
		tmp = x / y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / -y
	tmp = 0
	if x <= -8.8e+111:
		tmp = x / y
	elif x <= 1.6e+16:
		tmp = 1.0
	elif x <= 4.8e+43:
		tmp = t_0
	elif x <= 1.35e+111:
		tmp = 1.0
	elif x <= 4e+182:
		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 <= -8.8e+111)
		tmp = Float64(x / y);
	elseif (x <= 1.6e+16)
		tmp = 1.0;
	elseif (x <= 4.8e+43)
		tmp = t_0;
	elseif (x <= 1.35e+111)
		tmp = 1.0;
	elseif (x <= 4e+182)
		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 <= -8.8e+111)
		tmp = x / y;
	elseif (x <= 1.6e+16)
		tmp = 1.0;
	elseif (x <= 4.8e+43)
		tmp = t_0;
	elseif (x <= 1.35e+111)
		tmp = 1.0;
	elseif (x <= 4e+182)
		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, -8.8e+111], N[(x / y), $MachinePrecision], If[LessEqual[x, 1.6e+16], 1.0, If[LessEqual[x, 4.8e+43], t$95$0, If[LessEqual[x, 1.35e+111], 1.0, If[LessEqual[x, 4e+182], N[(x / y), $MachinePrecision], t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.79999999999999994e111 or 1.3499999999999999e111 < x < 4.0000000000000003e182

   1. Initial program 99.9%

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

      1. add-sqr-sqrt 39.4%

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

      2. fabs-sqr 39.4%

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

      3. add-sqr-sqrt 20.5%

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

      4. fabs-sqr 20.5%

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

      5. add-sqr-sqrt 20.8%

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

      6. add-sqr-sqrt 55.0%

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

   4. Applied egg-rr 55.0%

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

   5. Taylor expanded in x around inf 53.7%

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

   if -8.79999999999999994e111 < x < 1.6e16 or 4.80000000000000046e43 < x < 1.3499999999999999e111

   1. Initial program 100.0%

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

      1. add-sqr-sqrt 46.1%

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

      2. fabs-sqr 46.1%

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

      3. add-sqr-sqrt 5.8%

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

      4. fabs-sqr 5.8%

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

      5. add-sqr-sqrt 6.4%

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

      6. add-sqr-sqrt 15.9%

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

      7. clear-num 15.9%

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

   4. Applied egg-rr 15.9%

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

      1. div-inv 15.9%

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

   6. Applied egg-rr 15.9%

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

      1. *-commutative 15.9%

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

      2. associate-/r/ 15.9%

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

      3. metadata-eval 15.9%

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

      4. div-sub 15.9%

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

      5. *-inverses 15.9%

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

      6. sub-neg 15.9%

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

      7. metadata-eval 15.9%

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

      8. associate-/r* 15.9%

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

      9. +-commutative 15.9%

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

      10. distribute-lft-in 15.9%

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

      11. metadata-eval 15.9%

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

      12. neg-mul-1 15.9%

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

      13. sub-neg 15.9%

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

      14. rem-square-sqrt 1.2%

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

      15. fabs-sqr 1.2%

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

      16. rem-square-sqrt 1.3%

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

      17. fabs-sub 1.3%

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

      18. sub-neg 1.3%

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

      19. metadata-eval 1.3%

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

      20. rem-square-sqrt 0.1%

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

      21. fabs-sqr 0.1%

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

      22. rem-square-sqrt 85.3%

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

   8. Simplified 85.3%

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

   9. Taylor expanded in y around inf 71.6%

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

   if 1.6e16 < x < 4.80000000000000046e43 or 4.0000000000000003e182 < x

   1. Initial program 100.0%

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

      1. add-sqr-sqrt 93.6%

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

      2. fabs-sqr 93.6%

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

      3. add-sqr-sqrt 27.0%

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

      4. fabs-sqr 27.0%

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

      5. add-sqr-sqrt 27.3%

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

      6. add-sqr-sqrt 27.6%

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

      7. clear-num 27.6%

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

   4. Applied egg-rr 27.6%

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

      1. div-inv 27.6%

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

   6. Applied egg-rr 27.6%

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

      1. *-commutative 27.6%

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

      2. associate-/r/ 27.6%

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

      3. metadata-eval 27.6%

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

      4. div-sub 27.6%

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

      5. *-inverses 27.6%

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

      6. sub-neg 27.6%

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

      7. metadata-eval 27.6%

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

      8. associate-/r* 27.6%

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

      9. +-commutative 27.6%

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

      10. distribute-lft-in 27.6%

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

      11. metadata-eval 27.6%

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

      12. neg-mul-1 27.6%

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

      13. sub-neg 27.6%

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

      14. rem-square-sqrt 0.3%

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

      15. fabs-sqr 0.3%

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

      16. rem-square-sqrt 0.4%

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

      17. fabs-sub 0.4%

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

      18. sub-neg 0.4%

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

      19. metadata-eval 0.4%

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

      20. rem-square-sqrt 0.1%

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

      21. fabs-sqr 0.1%

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

      22. rem-square-sqrt 72.7%

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

   8. Simplified 72.7%

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

   9. Taylor expanded in y around 0 66.8%

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

       1. associate-*r/ 66.8%

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

       2. neg-mul-1 66.8%

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

   11. Simplified 66.8%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.8 \cdot 10^{+111}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{+16}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{+43}:\\ \;\;\;\;\frac{x}{-y}\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+111}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+182}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.5% accurate, 13.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{-252} \lor \neg \left(y \leq 2.25 \cdot 10^{-287}\right):\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -3.5e-252) (not (<= y 2.25e-287)))
   (- 1.0 (/ x y))
   (+ (/ x y) -1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -3.5e-252) || !(y <= 2.25e-287)) {
		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 <= (-3.5d-252)) .or. (.not. (y <= 2.25d-287))) 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 <= -3.5e-252) || !(y <= 2.25e-287)) {
		tmp = 1.0 - (x / y);
	} else {
		tmp = (x / y) + -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -3.5e-252) or not (y <= 2.25e-287):
		tmp = 1.0 - (x / y)
	else:
		tmp = (x / y) + -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -3.5e-252) || !(y <= 2.25e-287))
		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 <= -3.5e-252) || ~((y <= 2.25e-287)))
		tmp = 1.0 - (x / y);
	else
		tmp = (x / y) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -3.5e-252], N[Not[LessEqual[y, 2.25e-287]], $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 < -3.49999999999999986e-252 or 2.25000000000000008e-287 < y

   1. Initial program 100.0%

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

      1. add-sqr-sqrt 52.2%

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

      2. fabs-sqr 52.2%

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

      3. add-sqr-sqrt 11.8%

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

      4. fabs-sqr 11.8%

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

      5. add-sqr-sqrt 12.4%

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

      6. add-sqr-sqrt 21.9%

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

      7. clear-num 21.9%

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

   4. Applied egg-rr 21.9%

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

      1. div-inv 21.9%

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

   6. Applied egg-rr 21.9%

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

      1. *-commutative 21.9%

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

      2. associate-/r/ 21.9%

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

      3. metadata-eval 21.9%

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

      4. div-sub 21.9%

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

      5. *-inverses 21.9%

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

      6. sub-neg 21.9%

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

      7. metadata-eval 21.9%

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

      8. associate-/r* 21.9%

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

      9. +-commutative 21.9%

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

      10. distribute-lft-in 21.9%

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

      11. metadata-eval 21.9%

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

      12. neg-mul-1 21.9%

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

      13. sub-neg 21.9%

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

      14. rem-square-sqrt 1.0%

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

      15. fabs-sqr 1.0%

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

      16. rem-square-sqrt 1.1%

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

      17. fabs-sub 1.1%

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

      18. sub-neg 1.1%

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

      19. metadata-eval 1.1%

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

      20. rem-square-sqrt 0.1%

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

      21. fabs-sqr 0.1%

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

      22. rem-square-sqrt 79.1%

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

   8. Simplified 79.1%

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

   9. Taylor expanded in y around 0 79.2%

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

       1. neg-mul-1 79.2%

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

       2. sub-neg 79.2%

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

       3. div-sub 79.2%

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

       4. *-inverses 79.2%

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

   11. Simplified 79.2%

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

   if -3.49999999999999986e-252 < y < 2.25000000000000008e-287

   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 63.4%

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

      7. fabs-sqr 63.4%

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

      8. rem-square-sqrt 63.9%

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

      9. div-sub 63.9%

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

      10. sub-neg 63.9%

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

      11. *-inverses 63.9%

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

      12. metadata-eval 63.9%

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

      13. +-commutative 63.9%

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

   5. Simplified 63.9%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{-252} \lor \neg \left(y \leq 2.25 \cdot 10^{-287}\right):\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 59.4% accurate, 15.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.9 \cdot 10^{+107} \lor \neg \left(x \leq 1.45 \cdot 10^{+110}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -4.9e+107) (not (<= x 1.45e+110))) (/ x y) 1.0))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -4.9e+107) || !(x <= 1.45e+110)) {
		tmp = x / y;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-4.9d+107)) .or. (.not. (x <= 1.45d+110))) then
        tmp = x / y
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -4.9e+107) || !(x <= 1.45e+110)) {
		tmp = x / y;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -4.9e+107) or not (x <= 1.45e+110):
		tmp = x / y
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -4.9e+107) || !(x <= 1.45e+110))
		tmp = Float64(x / y);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -4.9e+107) || ~((x <= 1.45e+110)))
		tmp = x / y;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -4.9e+107], N[Not[LessEqual[x, 1.45e+110]], $MachinePrecision]], N[(x / y), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < -4.9000000000000001e107 or 1.45e110 < x

   1. Initial program 100.0%

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

      1. add-sqr-sqrt 56.9%

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

      2. fabs-sqr 56.9%

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

      3. add-sqr-sqrt 21.8%

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

      4. fabs-sqr 21.8%

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

      5. add-sqr-sqrt 22.1%

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

      6. add-sqr-sqrt 45.7%

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

   4. Applied egg-rr 45.7%

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

   5. Taylor expanded in x around inf 44.9%

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

   if -4.9000000000000001e107 < x < 1.45e110

   1. Initial program 100.0%

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

      1. add-sqr-sqrt 48.2%

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

      2. fabs-sqr 48.2%

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

      3. add-sqr-sqrt 7.2%

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

      4. fabs-sqr 7.2%

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

      5. add-sqr-sqrt 7.8%

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

      6. add-sqr-sqrt 16.8%

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

      7. clear-num 16.8%

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

   4. Applied egg-rr 16.8%

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

      1. div-inv 16.8%

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

   6. Applied egg-rr 16.8%

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

      1. *-commutative 16.8%

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

      2. associate-/r/ 16.8%

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

      3. metadata-eval 16.8%

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

      4. div-sub 16.8%

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

      5. *-inverses 16.8%

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

      6. sub-neg 16.8%

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

      7. metadata-eval 16.8%

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

      8. associate-/r* 16.8%

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

      9. +-commutative 16.8%

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

      10. distribute-lft-in 16.8%

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

      11. metadata-eval 16.8%

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

      12. neg-mul-1 16.8%

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

      13. sub-neg 16.8%

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

      14. rem-square-sqrt 1.2%

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

      15. fabs-sqr 1.2%

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

      16. rem-square-sqrt 1.3%

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

      17. fabs-sub 1.3%

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

      18. sub-neg 1.3%

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

      19. metadata-eval 1.3%

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

      20. rem-square-sqrt 0.1%

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

      21. fabs-sqr 0.1%

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

      22. rem-square-sqrt 84.4%

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

   8. Simplified 84.4%

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

   9. Taylor expanded in y around inf 68.5%

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

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.9 \cdot 10^{+107} \lor \neg \left(x \leq 1.45 \cdot 10^{+110}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 51.7% 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. Step-by-step derivation

   1. add-sqr-sqrt 50.8%

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

   2. fabs-sqr 50.8%

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

   3. add-sqr-sqrt 11.6%

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

   4. fabs-sqr 11.6%

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

   5. add-sqr-sqrt 12.1%

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

   6. add-sqr-sqrt 25.5%

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

   7. clear-num 25.5%

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

4. Applied egg-rr 25.5%

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

   1. div-inv 25.5%

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

6. Applied egg-rr 25.5%

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

   1. *-commutative 25.5%

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

   2. associate-/r/ 25.5%

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

   3. metadata-eval 25.5%

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

   4. div-sub 25.5%

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

   5. *-inverses 25.5%

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

   6. sub-neg 25.5%

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

   7. metadata-eval 25.5%

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

   8. associate-/r* 25.5%

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

   9. +-commutative 25.5%

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

   10. distribute-lft-in 25.5%

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

   11. metadata-eval 25.5%

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

   12. neg-mul-1 25.5%

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

   13. sub-neg 25.5%

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

   14. rem-square-sqrt 0.9%

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

   15. fabs-sqr 0.9%

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

   16. rem-square-sqrt 1.1%

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

   17. fabs-sub 1.1%

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

   18. sub-neg 1.1%

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

   19. metadata-eval 1.1%

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

   20. rem-square-sqrt 0.1%

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

   21. fabs-sqr 0.1%

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

   22. rem-square-sqrt 75.5%

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

8. Simplified 75.5%

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

9. Taylor expanded in y around inf 52.9%

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

   1. add-sqr-sqrt 50.8%

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

   2. fabs-sqr 50.8%

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

   3. add-sqr-sqrt 11.6%

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

   4. fabs-sqr 11.6%

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

   5. add-sqr-sqrt 12.1%

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

   6. add-sqr-sqrt 25.5%

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

4. Applied egg-rr 25.5%

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

5. Taylor expanded in x around 0 1.3%

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

Reproduce

herbie shell --seed 2024096 
(FPCore (x y)
  :name "Numeric.LinearAlgebra.Util:formatSparse from hmatrix-0.16.1.5"
  :precision binary64
  (/ (fabs (- x y)) (fabs y)))