Numeric.LinearAlgebra.Util:formatSparse from hmatrix-0.16.1.5

Percentage Accurate: 100.0% → 100.0%

Time: 5.6s

Alternatives: 7

Speedup: 2.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\frac{\left|x - y\right|}{\left|y\right|} \]
2. 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. Final simplification 100.0%

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

Alternative 2: 75.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{-81} \lor \neg \left(x \leq 1.62 \cdot 10^{+19}\right):\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -3.6e-81) (not (<= x 1.62e+19))) (fabs (/ x y)) 1.0))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -3.6e-81) || !(x <= 1.62e+19)) {
		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 ((x <= (-3.6d-81)) .or. (.not. (x <= 1.62d+19))) 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 ((x <= -3.6e-81) || !(x <= 1.62e+19)) {
		tmp = Math.abs((x / y));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -3.6e-81) or not (x <= 1.62e+19):
		tmp = math.fabs((x / y))
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -3.6e-81) || !(x <= 1.62e+19))
		tmp = abs(Float64(x / y));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -3.6e-81) || ~((x <= 1.62e+19)))
		tmp = abs((x / y));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -3.6e-81], N[Not[LessEqual[x, 1.62e+19]], $MachinePrecision]], N[Abs[N[(x / y), $MachinePrecision]], $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < -3.5999999999999999e-81 or 1.62e19 < 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%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf 75.9%

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

   if -3.5999999999999999e-81 < x < 1.62e19

   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 24.3%

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

   8. Applied egg-rr 78.0%

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

      1. *-inverses 78.0%

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

   10. Simplified 78.0%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{-81} \lor \neg \left(x \leq 1.62 \cdot 10^{+19}\right):\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 58.6% accurate, 13.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.1 \cdot 10^{-66}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{-99}:\\ \;\;\;\;\frac{x - y}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -4.1e-66) 1.0 (if (<= y 3.4e-99) (/ (- x y) y) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -4.1e-66) {
		tmp = 1.0;
	} else if (y <= 3.4e-99) {
		tmp = (x - y) / 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 <= (-4.1d-66)) then
        tmp = 1.0d0
    else if (y <= 3.4d-99) then
        tmp = (x - y) / y
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -4.1e-66) {
		tmp = 1.0;
	} else if (y <= 3.4e-99) {
		tmp = (x - y) / y;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -4.1e-66:
		tmp = 1.0
	elif y <= 3.4e-99:
		tmp = (x - y) / y
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -4.1e-66)
		tmp = 1.0;
	elseif (y <= 3.4e-99)
		tmp = Float64(Float64(x - y) / y);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -4.1e-66)
		tmp = 1.0;
	elseif (y <= 3.4e-99)
		tmp = (x - y) / y;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -4.1e-66], 1.0, If[LessEqual[y, 3.4e-99], N[(N[(x - y), $MachinePrecision] / y), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -4.09999999999999998e-66 or 3.40000000000000007e-99 < 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 inf 36.6%

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

   8. Applied egg-rr 67.6%

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

      1. *-inverses 67.6%

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

   10. Simplified 67.6%

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

   if -4.09999999999999998e-66 < y < 3.40000000000000007e-99

   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 52.5%

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

      2. *-un-lft-identity 52.5%

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

      3. log-prod 52.5%

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

      4. metadata-eval 52.5%

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

      5. add-log-exp 100.0%

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

      6. add-sqr-sqrt 52.8%

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

      7. fabs-sqr 52.8%

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

      8. add-sqr-sqrt 23.2%

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

      9. fabs-sqr 23.2%

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

      10. add-sqr-sqrt 23.5%

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

      11. add-sqr-sqrt 54.7%

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

   4. Applied egg-rr 54.7%

      \[\leadsto \color{blue}{0 + \frac{x - y}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.1 \cdot 10^{-66}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{-99}:\\ \;\;\;\;\frac{x - y}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 58.5% accurate, 13.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.2 \cdot 10^{-65}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{-96}:\\ \;\;\;\;\frac{x}{y} + -1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -4.2e-65) 1.0 (if (<= y 2.9e-96) (+ (/ x y) -1.0) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -4.2e-65) {
		tmp = 1.0;
	} else if (y <= 2.9e-96) {
		tmp = (x / y) + -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 <= (-4.2d-65)) then
        tmp = 1.0d0
    else if (y <= 2.9d-96) then
        tmp = (x / y) + (-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 <= -4.2e-65) {
		tmp = 1.0;
	} else if (y <= 2.9e-96) {
		tmp = (x / y) + -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -4.2e-65:
		tmp = 1.0
	elif y <= 2.9e-96:
		tmp = (x / y) + -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -4.2e-65)
		tmp = 1.0;
	elseif (y <= 2.9e-96)
		tmp = Float64(Float64(x / y) + -1.0);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -4.2e-65)
		tmp = 1.0;
	elseif (y <= 2.9e-96)
		tmp = (x / y) + -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -4.2e-65], 1.0, If[LessEqual[y, 2.9e-96], N[(N[(x / y), $MachinePrecision] + -1.0), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -4.20000000000000006e-65 or 2.89999999999999994e-96 < 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 inf 36.6%

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

   8. Applied egg-rr 67.6%

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

      1. *-inverses 67.6%

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

   10. Simplified 67.6%

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

   if -4.20000000000000006e-65 < y < 2.89999999999999994e-96

   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 52.8%

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

      2. fabs-sqr 52.8%

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

      3. add-sqr-sqrt 23.2%

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

      4. fabs-sqr 23.2%

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

      5. add-sqr-sqrt 23.5%

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

      6. add-sqr-sqrt 54.7%

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

      7. div-sub 54.7%

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

   4. Applied egg-rr 54.7%

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

   5. Taylor expanded in y around 0 54.7%

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

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.2 \cdot 10^{-65}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{-96}:\\ \;\;\;\;\frac{x}{y} + -1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 56.7% accurate, 15.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.55 \cdot 10^{+178} \lor \neg \left(x \leq 2.7 \cdot 10^{+218}\right):\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -2.55e+178) (not (<= x 2.7e+218))) (* x x) 1.0))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -2.55e+178) || !(x <= 2.7e+218)) {
		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 <= (-2.55d+178)) .or. (.not. (x <= 2.7d+218))) 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 <= -2.55e+178) || !(x <= 2.7e+218)) {
		tmp = x * x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -2.55e+178) or not (x <= 2.7e+218):
		tmp = x * x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -2.55e+178) || !(x <= 2.7e+218))
		tmp = Float64(x * x);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -2.55e+178) || ~((x <= 2.7e+218)))
		tmp = x * x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -2.55e+178], N[Not[LessEqual[x, 2.7e+218]], $MachinePrecision]], N[(x * x), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < -2.5499999999999998e178 or 2.70000000000000013e218 < 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%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf 91.4%

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

   8. Applied egg-rr 48.0%

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

   if -2.5499999999999998e178 < x < 2.70000000000000013e218

   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 44.1%

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

   8. Applied egg-rr 59.2%

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

      1. *-inverses 59.2%

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

   10. Simplified 59.2%

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

4. Final simplification 56.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.55 \cdot 10^{+178} \lor \neg \left(x \leq 2.7 \cdot 10^{+218}\right):\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 58.5% accurate, 15.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.15 \cdot 10^{-66}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 4 \cdot 10^{-98}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.15e-66) 1.0 (if (<= y 4e-98) (/ x y) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.15e-66) {
		tmp = 1.0;
	} else if (y <= 4e-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 <= (-2.15d-66)) then
        tmp = 1.0d0
    else if (y <= 4d-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 <= -2.15e-66) {
		tmp = 1.0;
	} else if (y <= 4e-98) {
		tmp = x / y;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.15e-66:
		tmp = 1.0
	elif y <= 4e-98:
		tmp = x / y
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.15e-66)
		tmp = 1.0;
	elseif (y <= 4e-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 <= -2.15e-66)
		tmp = 1.0;
	elseif (y <= 4e-98)
		tmp = x / y;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.15e-66], 1.0, If[LessEqual[y, 4e-98], N[(x / y), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -2.15000000000000007e-66 or 3.99999999999999976e-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%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf 36.6%

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

   8. Applied egg-rr 67.6%

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

      1. *-inverses 67.6%

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

   10. Simplified 67.6%

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

   if -2.15000000000000007e-66 < y < 3.99999999999999976e-98

   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 52.8%

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

      3. fabs-sqr 52.8%

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

      4. add-sqr-sqrt 53.3%

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

      5. *-commutative 53.3%

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

      6. add-sqr-sqrt 23.4%

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

      7. fabs-sqr 23.4%

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

      8. add-sqr-sqrt 54.5%

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

   4. Applied egg-rr 54.5%

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

   5. Taylor expanded in x around inf 53.6%

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

   6. Taylor expanded in y around 0 53.7%

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

Alternative 7: 50.8% 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 inf 53.9%

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

8. Applied egg-rr 49.4%

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

   1. *-inverses 49.4%

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

10. Simplified 49.4%

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

Reproduce

herbie shell --seed 2024116 
(FPCore (x y)
  :name "Numeric.LinearAlgebra.Util:formatSparse from hmatrix-0.16.1.5"
  :precision binary64
  (/ (fabs (- x y)) (fabs y)))