Numeric.LinearAlgebra.Util:formatSparse from hmatrix-0.16.1.5

Percentage Accurate: 100.0% → 100.0%

Time: 5.5s

Alternatives: 6

Speedup: 2.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around -inf 100.0%

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

5. Simplified 100.0%

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

Alternative 2: 74.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{-77}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.06 \cdot 10^{+34}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3.2e-77) 1.0 (if (<= y 1.06e+34) (fabs (/ x y)) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3.2e-77) {
		tmp = 1.0;
	} else if (y <= 1.06e+34) {
		tmp = fabs((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 <= (-3.2d-77)) then
        tmp = 1.0d0
    else if (y <= 1.06d+34) then
        tmp = abs((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 <= -3.2e-77) {
		tmp = 1.0;
	} else if (y <= 1.06e+34) {
		tmp = Math.abs((x / y));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -3.2e-77:
		tmp = 1.0
	elif y <= 1.06e+34:
		tmp = math.fabs((x / y))
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3.2e-77)
		tmp = 1.0;
	elseif (y <= 1.06e+34)
		tmp = abs(Float64(x / y));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -3.2e-77)
		tmp = 1.0;
	elseif (y <= 1.06e+34)
		tmp = abs((x / y));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3.2e-77], 1.0, If[LessEqual[y, 1.06e+34], N[Abs[N[(x / y), $MachinePrecision]], $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -3.2e-77 or 1.06000000000000005e34 < 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%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 80.9%

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

   8. Applied egg-rr 80.9%

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

   if -3.2e-77 < y < 1.06000000000000005e34

   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%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf 78.7%

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

Alternative 3: 59.5% accurate, 9.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{-123}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{-301}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 1.06 \cdot 10^{+34}:\\ \;\;\;\;\left(x + -1\right) \cdot \left(x + -1\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -4.5e-123)
   1.0
   (if (<= y 3.1e-301)
     (/ x y)
     (if (<= y 1.06e+34) (* (+ x -1.0) (+ x -1.0)) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -4.5e-123) {
		tmp = 1.0;
	} else if (y <= 3.1e-301) {
		tmp = x / y;
	} else if (y <= 1.06e+34) {
		tmp = (x + -1.0) * (x + -1.0);
	} 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 <= (-4.5d-123)) then
        tmp = 1.0d0
    else if (y <= 3.1d-301) then
        tmp = x / y
    else if (y <= 1.06d+34) then
        tmp = (x + (-1.0d0)) * (x + (-1.0d0))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -4.5e-123) {
		tmp = 1.0;
	} else if (y <= 3.1e-301) {
		tmp = x / y;
	} else if (y <= 1.06e+34) {
		tmp = (x + -1.0) * (x + -1.0);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -4.5e-123:
		tmp = 1.0
	elif y <= 3.1e-301:
		tmp = x / y
	elif y <= 1.06e+34:
		tmp = (x + -1.0) * (x + -1.0)
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -4.5e-123)
		tmp = 1.0;
	elseif (y <= 3.1e-301)
		tmp = Float64(x / y);
	elseif (y <= 1.06e+34)
		tmp = Float64(Float64(x + -1.0) * Float64(x + -1.0));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -4.5e-123)
		tmp = 1.0;
	elseif (y <= 3.1e-301)
		tmp = x / y;
	elseif (y <= 1.06e+34)
		tmp = (x + -1.0) * (x + -1.0);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -4.5e-123], 1.0, If[LessEqual[y, 3.1e-301], N[(x / y), $MachinePrecision], If[LessEqual[y, 1.06e+34], N[(N[(x + -1.0), $MachinePrecision] * N[(x + -1.0), $MachinePrecision]), $MachinePrecision], 1.0]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.49999999999999993e-123 or 1.06000000000000005e34 < 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%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 78.0%

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

   8. Applied egg-rr 78.0%

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

   if -4.49999999999999993e-123 < y < 3.10000000000000014e-301

   1. Initial program 100.0%

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

      1. div-inv 99.7%

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

      2. add-sqr-sqrt 47.5%

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

      3. fabs-sqr 47.5%

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

      4. add-sqr-sqrt 47.9%

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

      5. *-commutative 47.9%

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

      6. add-sqr-sqrt 2.3%

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

      7. fabs-sqr 2.3%

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

      8. add-sqr-sqrt 52.4%

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

   4. Applied egg-rr 52.4%

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

   5. Taylor expanded in x around inf 52.5%

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

   6. Taylor expanded in y around 0 52.6%

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

   if 3.10000000000000014e-301 < y < 1.06000000000000005e34

   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 30.6%

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

      2. fabs-sqr 30.6%

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

      3. add-sqr-sqrt 30.6%

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

      4. fabs-sqr 30.6%

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

      5. add-sqr-sqrt 30.6%

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

      6. add-sqr-sqrt 31.4%

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

      7. div-sub 31.4%

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

   4. Applied egg-rr 31.4%

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

   5. Taylor expanded in y around 0 31.4%

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

   7. Applied egg-rr 48.8%

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

      1. +-commutative 48.8%

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

      2. +-commutative 48.8%

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

   9. Simplified 48.8%

      \[\leadsto \color{blue}{\left(x + -1\right) \cdot \left(x + -1\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 57.2% accurate, 15.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{+157} \lor \neg \left(x \leq 6.2 \cdot 10^{+149}\right):\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -4.5e+157) (not (<= x 6.2e+149))) (* x x) 1.0))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -4.5e+157) || !(x <= 6.2e+149)) {
		tmp = x * x;
	} 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 <= (-4.5d+157)) .or. (.not. (x <= 6.2d+149))) then
        tmp = x * x
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -4.5e+157) || !(x <= 6.2e+149)) {
		tmp = x * x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -4.5e+157) or not (x <= 6.2e+149):
		tmp = x * x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -4.5e+157) || !(x <= 6.2e+149))
		tmp = Float64(x * x);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -4.5e+157) || ~((x <= 6.2e+149)))
		tmp = x * x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -4.5e+157], N[Not[LessEqual[x, 6.2e+149]], $MachinePrecision]], N[(x * x), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < -4.49999999999999985e157 or 6.19999999999999974e149 < x

   1. Initial program 100.0%

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

      1. add-log-exp 51.6%

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

      2. *-un-lft-identity 51.6%

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

      3. log-prod 51.6%

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

      4. metadata-eval 51.6%

         \[\leadsto \color{blue}{0} + \log \left(e^{\frac{\left|x - y\right|}{\left|y\right|}}\right) \]

      5. add-log-exp 100.0%

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

      6. add-sqr-sqrt 48.7%

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

      7. fabs-sqr 48.7%

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

      8. add-sqr-sqrt 16.9%

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

      9. fabs-sqr 16.9%

         \[\leadsto 0 + \frac{\sqrt{x - y} \cdot \sqrt{x - y}}{\color{blue}{\sqrt{y} \cdot \sqrt{y}}} \]

      10. add-sqr-sqrt 17.2%

          \[\leadsto 0 + \frac{\sqrt{x - y} \cdot \sqrt{x - y}}{\color{blue}{y}} \]

      11. add-sqr-sqrt 36.5%

          \[\leadsto 0 + \frac{\color{blue}{x - y}}{y} \]

   4. Applied egg-rr 36.5%

      \[\leadsto \color{blue}{0 + \frac{x - y}{y}} \]

   5. Taylor expanded in x around inf 36.8%

      \[\leadsto 0 + \frac{\color{blue}{x}}{y} \]

   7. Applied egg-rr 43.4%

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

   if -4.49999999999999985e157 < x < 6.19999999999999974e149

   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%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 61.5%

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

   8. Applied egg-rr 61.5%

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

4. Final simplification 58.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{+157} \lor \neg \left(x \leq 6.2 \cdot 10^{+149}\right):\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 58.4% accurate, 15.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{-126}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{-179}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -7.2e-126) 1.0 (if (<= y 3.1e-179) (/ x y) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -7.2e-126) {
		tmp = 1.0;
	} else if (y <= 3.1e-179) {
		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 <= (-7.2d-126)) then
        tmp = 1.0d0
    else if (y <= 3.1d-179) 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 <= -7.2e-126) {
		tmp = 1.0;
	} else if (y <= 3.1e-179) {
		tmp = x / y;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -7.2e-126:
		tmp = 1.0
	elif y <= 3.1e-179:
		tmp = x / y
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -7.2e-126)
		tmp = 1.0;
	elseif (y <= 3.1e-179)
		tmp = Float64(x / y);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -7.2e-126)
		tmp = 1.0;
	elseif (y <= 3.1e-179)
		tmp = x / y;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -7.2e-126], 1.0, If[LessEqual[y, 3.1e-179], N[(x / y), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -7.1999999999999999e-126 or 3.1000000000000002e-179 < 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%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 67.9%

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

   8. Applied egg-rr 67.9%

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

   if -7.1999999999999999e-126 < y < 3.1000000000000002e-179

   1. Initial program 100.0%

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

      1. div-inv 99.7%

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

      2. add-sqr-sqrt 46.2%

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

      3. fabs-sqr 46.2%

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

      4. add-sqr-sqrt 46.7%

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

      5. *-commutative 46.7%

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

      6. add-sqr-sqrt 18.4%

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

      7. fabs-sqr 18.4%

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

      8. add-sqr-sqrt 49.4%

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

   4. Applied egg-rr 49.4%

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

   5. Taylor expanded in x around inf 49.6%

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

   6. Taylor expanded in y around 0 49.8%

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

Alternative 6: 50.7% 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%

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

5. Simplified 100.0%

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

6. Taylor expanded in x around 0 52.2%

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

8. Applied egg-rr 52.2%

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

Reproduce

herbie shell --seed 2024163 
(FPCore (x y)
  :name "Numeric.LinearAlgebra.Util:formatSparse from hmatrix-0.16.1.5"
  :precision binary64
  (/ (fabs (- x y)) (fabs y)))