Numeric.LinearAlgebra.Util:formatSparse from hmatrix-0.16.1.5

Percentage Accurate: 100.0% → 100.0%

Time: 6.3s

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

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

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: 74.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.8 \cdot 10^{-73} \lor \neg \left(x \leq 9.5 \cdot 10^{+71}\right):\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -8.8e-73) (not (<= x 9.5e+71))) (fabs (/ x y)) 1.0))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -8.8e-73) || !(x <= 9.5e+71)) {
		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 <= (-8.8d-73)) .or. (.not. (x <= 9.5d+71))) 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 <= -8.8e-73) || !(x <= 9.5e+71)) {
		tmp = Math.abs((x / y));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -8.8e-73) or not (x <= 9.5e+71):
		tmp = math.fabs((x / y))
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -8.8e-73) || !(x <= 9.5e+71))
		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 <= -8.8e-73) || ~((x <= 9.5e+71)))
		tmp = abs((x / y));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -8.8e-73], N[Not[LessEqual[x, 9.5e+71]], $MachinePrecision]], N[Abs[N[(x / y), $MachinePrecision]], $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < -8.8000000000000001e-73 or 9.50000000000000015e71 < 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 80.2%

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

   if -8.8000000000000001e-73 < x < 9.50000000000000015e71

   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 22.5%

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

   8. Applied egg-rr 79.7%

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

      1. *-inverses 79.7%

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

   10. Simplified 79.7%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.8 \cdot 10^{-73} \lor \neg \left(x \leq 9.5 \cdot 10^{+71}\right):\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 61.2% accurate, 10.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.4 \cdot 10^{+44}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{+50}:\\ \;\;\;\;\left(x + -1\right) \cdot \left(x + \left(x + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3.4e+44)
   1.0
   (if (<= y 1.45e+50) (* (+ x -1.0) (+ x (+ x -1.0))) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3.4e+44) {
		tmp = 1.0;
	} else if (y <= 1.45e+50) {
		tmp = (x + -1.0) * (x + (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 <= (-3.4d+44)) then
        tmp = 1.0d0
    else if (y <= 1.45d+50) then
        tmp = (x + (-1.0d0)) * (x + (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 <= -3.4e+44) {
		tmp = 1.0;
	} else if (y <= 1.45e+50) {
		tmp = (x + -1.0) * (x + (x + -1.0));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -3.4e+44:
		tmp = 1.0
	elif y <= 1.45e+50:
		tmp = (x + -1.0) * (x + (x + -1.0))
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3.4e+44)
		tmp = 1.0;
	elseif (y <= 1.45e+50)
		tmp = Float64(Float64(x + -1.0) * Float64(x + Float64(x + -1.0)));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -3.4e+44)
		tmp = 1.0;
	elseif (y <= 1.45e+50)
		tmp = (x + -1.0) * (x + (x + -1.0));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3.4e+44], 1.0, If[LessEqual[y, 1.45e+50], N[(N[(x + -1.0), $MachinePrecision] * N[(x + N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -3.4e44 or 1.45e50 < y

   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%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf 19.5%

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

   8. Applied egg-rr 82.0%

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

      1. *-inverses 82.0%

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

   10. Simplified 82.0%

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

   if -3.4e44 < y < 1.45e50

   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. Step-by-step derivation

      1. metadata-eval 100.0%

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

      2. *-inverses 100.0%

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

      3. sub-neg 100.0%

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

      4. sub-div 100.0%

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

      5. add-sqr-sqrt 34.1%

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

      6. fabs-sqr 34.1%

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

      7. add-sqr-sqrt 34.8%

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

   7. Applied egg-rr 34.8%

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

   9. Applied egg-rr 49.5%

      \[\leadsto \color{blue}{\left(x + \left(x + -1\right)\right) \cdot \left(x + -1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.4 \cdot 10^{+44}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{+50}:\\ \;\;\;\;\left(x + -1\right) \cdot \left(x + \left(x + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 57.0% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{+128}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq -1.1 \cdot 10^{+21}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 4.9 \cdot 10^{+156}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.4e+128)
   (* x x)
   (if (<= x -1.1e+21) (/ x y) (if (<= x 4.9e+156) 1.0 (* x x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.4e+128) {
		tmp = x * x;
	} else if (x <= -1.1e+21) {
		tmp = x / y;
	} else if (x <= 4.9e+156) {
		tmp = 1.0;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.4d+128)) then
        tmp = x * x
    else if (x <= (-1.1d+21)) then
        tmp = x / y
    else if (x <= 4.9d+156) then
        tmp = 1.0d0
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.4e+128) {
		tmp = x * x;
	} else if (x <= -1.1e+21) {
		tmp = x / y;
	} else if (x <= 4.9e+156) {
		tmp = 1.0;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.4e+128:
		tmp = x * x
	elif x <= -1.1e+21:
		tmp = x / y
	elif x <= 4.9e+156:
		tmp = 1.0
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.4e+128)
		tmp = Float64(x * x);
	elseif (x <= -1.1e+21)
		tmp = Float64(x / y);
	elseif (x <= 4.9e+156)
		tmp = 1.0;
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.4e+128)
		tmp = x * x;
	elseif (x <= -1.1e+21)
		tmp = x / y;
	elseif (x <= 4.9e+156)
		tmp = 1.0;
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.4e+128], N[(x * x), $MachinePrecision], If[LessEqual[x, -1.1e+21], N[(x / y), $MachinePrecision], If[LessEqual[x, 4.9e+156], 1.0, N[(x * x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.39999999999999991e128 or 4.89999999999999969e156 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around -inf 99.9%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf 83.2%

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

   8. Applied egg-rr 44.4%

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

   if -1.39999999999999991e128 < x < -1.1e21

   1. Initial program 100.0%

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

      1. div-inv 99.6%

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

      2. add-sqr-sqrt 7.9%

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

      3. fabs-sqr 7.9%

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

      4. add-sqr-sqrt 8.4%

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

      5. *-commutative 8.4%

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

      6. add-sqr-sqrt 0.3%

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

      7. fabs-sqr 0.3%

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

      8. add-sqr-sqrt 52.2%

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

   4. Applied egg-rr 52.2%

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

   5. Taylor expanded in x around inf 52.5%

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

   6. Taylor expanded in y around 0 52.7%

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

   if -1.1e21 < x < 4.89999999999999969e156

   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 31.7%

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

   8. Applied egg-rr 70.9%

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

      1. *-inverses 70.9%

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

   10. Simplified 70.9%

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

Alternative 5: 61.2% accurate, 12.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.4 \cdot 10^{+44}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{+50}:\\ \;\;\;\;\left(x + -1\right) \cdot \left(x + -1\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3.4e+44) 1.0 (if (<= y 1.4e+50) (* (+ x -1.0) (+ x -1.0)) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3.4e+44) {
		tmp = 1.0;
	} else if (y <= 1.4e+50) {
		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 <= (-3.4d+44)) then
        tmp = 1.0d0
    else if (y <= 1.4d+50) 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 <= -3.4e+44) {
		tmp = 1.0;
	} else if (y <= 1.4e+50) {
		tmp = (x + -1.0) * (x + -1.0);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -3.4e+44:
		tmp = 1.0
	elif y <= 1.4e+50:
		tmp = (x + -1.0) * (x + -1.0)
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3.4e+44)
		tmp = 1.0;
	elseif (y <= 1.4e+50)
		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 <= -3.4e+44)
		tmp = 1.0;
	elseif (y <= 1.4e+50)
		tmp = (x + -1.0) * (x + -1.0);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3.4e+44], 1.0, If[LessEqual[y, 1.4e+50], N[(N[(x + -1.0), $MachinePrecision] * N[(x + -1.0), $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -3.4e44 or 1.3999999999999999e50 < y

   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%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf 19.5%

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

   8. Applied egg-rr 82.0%

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

      1. *-inverses 82.0%

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

   10. Simplified 82.0%

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

   if -3.4e44 < y < 1.3999999999999999e50

   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. Step-by-step derivation

      1. metadata-eval 100.0%

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

      2. *-inverses 100.0%

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

      3. sub-neg 100.0%

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

      4. sub-div 100.0%

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

      5. add-sqr-sqrt 34.1%

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

      6. fabs-sqr 34.1%

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

      7. add-sqr-sqrt 34.8%

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

   7. Applied egg-rr 34.8%

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

   9. Applied egg-rr 49.5%

      \[\leadsto \color{blue}{\left(x + -1\right) \cdot \left(x + -1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 58.1% accurate, 15.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{+164} \lor \neg \left(x \leq 4.6 \cdot 10^{+157}\right):\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -4e+164) (not (<= x 4.6e+157))) (* x x) 1.0))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -4e+164) || !(x <= 4.6e+157)) {
		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 <= (-4d+164)) .or. (.not. (x <= 4.6d+157))) 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 <= -4e+164) || !(x <= 4.6e+157)) {
		tmp = x * x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -4e+164) or not (x <= 4.6e+157):
		tmp = x * x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -4e+164) || !(x <= 4.6e+157))
		tmp = Float64(x * x);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -4e+164) || ~((x <= 4.6e+157)))
		tmp = x * x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -4e+164], N[Not[LessEqual[x, 4.6e+157]], $MachinePrecision]], N[(x * x), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < -4e164 or 4.60000000000000008e157 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around -inf 99.9%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf 85.1%

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

   8. Applied egg-rr 48.2%

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

   if -4e164 < x < 4.60000000000000008e157

   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 40.1%

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

   8. Applied egg-rr 62.3%

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

      1. *-inverses 62.3%

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

   10. Simplified 62.3%

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

4. Final simplification 58.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{+164} \lor \neg \left(x \leq 4.6 \cdot 10^{+157}\right):\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 52.1% 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 52.1%

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

8. Applied egg-rr 50.1%

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

   1. *-inverses 50.1%

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

10. Simplified 50.1%

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

Reproduce

herbie shell --seed 2024141 
(FPCore (x y)
  :name "Numeric.LinearAlgebra.Util:formatSparse from hmatrix-0.16.1.5"
  :precision binary64
  (/ (fabs (- x y)) (fabs y)))