Numeric.LinearAlgebra.Util:formatSparse from hmatrix-0.16.1.5

Percentage Accurate: 100.0% → 100.0%

Time: 4.9s

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: 74.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.5 \cdot 10^{-26}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{+33}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -7.5e-26) 1.0 (if (<= y 1.8e+33) (fabs (/ x y)) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -7.5e-26) {
		tmp = 1.0;
	} else if (y <= 1.8e+33) {
		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 <= (-7.5d-26)) then
        tmp = 1.0d0
    else if (y <= 1.8d+33) 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 <= -7.5e-26) {
		tmp = 1.0;
	} else if (y <= 1.8e+33) {
		tmp = Math.abs((x / y));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -7.5e-26:
		tmp = 1.0
	elif y <= 1.8e+33:
		tmp = math.fabs((x / y))
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -7.5e-26)
		tmp = 1.0;
	elseif (y <= 1.8e+33)
		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 <= -7.5e-26)
		tmp = 1.0;
	elseif (y <= 1.8e+33)
		tmp = abs((x / y));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -7.5e-26], 1.0, If[LessEqual[y, 1.8e+33], N[Abs[N[(x / y), $MachinePrecision]], $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -7.4999999999999994e-26 or 1.8000000000000001e33 < 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 78.9%

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

   8. Applied egg-rr 78.9%

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

   if -7.4999999999999994e-26 < y < 1.8000000000000001e33

   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 79.5%

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

Alternative 3: 60.4% accurate, 10.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -900000000:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -4.2 \cdot 10^{-218}:\\ \;\;\;\;\left(x + -1\right) \cdot \left(x + \left(x + -1\right)\right)\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{-69}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -900000000.0)
   1.0
   (if (<= y -4.2e-218)
     (* (+ x -1.0) (+ x (+ x -1.0)))
     (if (<= y 3.6e-69) (/ x y) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -900000000.0) {
		tmp = 1.0;
	} else if (y <= -4.2e-218) {
		tmp = (x + -1.0) * (x + (x + -1.0));
	} else if (y <= 3.6e-69) {
		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 <= (-900000000.0d0)) then
        tmp = 1.0d0
    else if (y <= (-4.2d-218)) then
        tmp = (x + (-1.0d0)) * (x + (x + (-1.0d0)))
    else if (y <= 3.6d-69) 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 <= -900000000.0) {
		tmp = 1.0;
	} else if (y <= -4.2e-218) {
		tmp = (x + -1.0) * (x + (x + -1.0));
	} else if (y <= 3.6e-69) {
		tmp = x / y;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -900000000.0:
		tmp = 1.0
	elif y <= -4.2e-218:
		tmp = (x + -1.0) * (x + (x + -1.0))
	elif y <= 3.6e-69:
		tmp = x / y
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -900000000.0)
		tmp = 1.0;
	elseif (y <= -4.2e-218)
		tmp = Float64(Float64(x + -1.0) * Float64(x + Float64(x + -1.0)));
	elseif (y <= 3.6e-69)
		tmp = Float64(x / y);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -900000000.0)
		tmp = 1.0;
	elseif (y <= -4.2e-218)
		tmp = (x + -1.0) * (x + (x + -1.0));
	elseif (y <= 3.6e-69)
		tmp = x / y;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -900000000.0], 1.0, If[LessEqual[y, -4.2e-218], N[(N[(x + -1.0), $MachinePrecision] * N[(x + N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.6e-69], N[(x / y), $MachinePrecision], 1.0]]]

Derivation

1. Split input into 3 regimes

2. if y < -9e8 or 3.60000000000000018e-69 < 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 75.1%

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

   8. Applied egg-rr 75.1%

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

   if -9e8 < y < -4.19999999999999988e-218

   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 31.7%

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

      6. fabs-sqr 31.7%

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

      7. add-sqr-sqrt 32.6%

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

   7. Applied egg-rr 32.6%

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

   9. Applied egg-rr 57.6%

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

   if -4.19999999999999988e-218 < y < 3.60000000000000018e-69

   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 35.6%

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

      3. fabs-sqr 35.6%

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

      4. add-sqr-sqrt 36.2%

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

      5. *-commutative 36.2%

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

      6. add-sqr-sqrt 24.0%

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

      7. fabs-sqr 24.0%

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

      8. add-sqr-sqrt 46.5%

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

   4. Applied egg-rr 46.5%

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

   5. Taylor expanded in x around inf 47.0%

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

   6. Taylor expanded in y around 0 47.1%

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

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -900000000:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -4.2 \cdot 10^{-218}:\\ \;\;\;\;\left(x + -1\right) \cdot \left(x + \left(x + -1\right)\right)\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{-69}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 58.3% accurate, 15.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.1 \cdot 10^{-143}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-67}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3.1e-143) 1.0 (if (<= y 2.2e-67) (/ x y) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3.1e-143) {
		tmp = 1.0;
	} else if (y <= 2.2e-67) {
		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 <= (-3.1d-143)) then
        tmp = 1.0d0
    else if (y <= 2.2d-67) 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 <= -3.1e-143) {
		tmp = 1.0;
	} else if (y <= 2.2e-67) {
		tmp = x / y;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -3.1e-143:
		tmp = 1.0
	elif y <= 2.2e-67:
		tmp = x / y
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3.1e-143)
		tmp = 1.0;
	elseif (y <= 2.2e-67)
		tmp = Float64(x / y);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -3.1e-143)
		tmp = 1.0;
	elseif (y <= 2.2e-67)
		tmp = x / y;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3.1e-143], 1.0, If[LessEqual[y, 2.2e-67], N[(x / y), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -3.10000000000000007e-143 or 2.2000000000000001e-67 < 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 71.9%

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

   8. Applied egg-rr 71.9%

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

   if -3.10000000000000007e-143 < y < 2.2000000000000001e-67

   1. Initial program 99.9%

      \[\frac{\left|x - y\right|}{\left|y\right|} \]
   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 41.1%

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

      3. fabs-sqr 41.1%

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

      4. add-sqr-sqrt 41.7%

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

      5. *-commutative 41.7%

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

      6. add-sqr-sqrt 18.5%

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

      7. fabs-sqr 18.5%

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

      8. add-sqr-sqrt 45.1%

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

   4. Applied egg-rr 45.1%

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

   5. Taylor expanded in x around inf 45.6%

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

   6. Taylor expanded in y around 0 45.8%

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

Alternative 5: 54.5% accurate, 25.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{+161}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x -2.4e+161) (* x x) 1.0))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.4e+161) {
		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 <= (-2.4d+161)) 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 <= -2.4e+161) {
		tmp = x * x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.4e+161:
		tmp = x * x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.4e+161)
		tmp = Float64(x * x);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.4e+161)
		tmp = x * x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.4e+161], N[(x * x), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < -2.3999999999999999e161

   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 61.2%

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

      2. *-un-lft-identity 61.2%

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

      3. log-prod 61.2%

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

      4. metadata-eval 61.2%

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

      5. add-log-exp 100.0%

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

      6. add-sqr-sqrt 11.1%

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

      7. fabs-sqr 11.1%

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

      8. add-sqr-sqrt 0.0%

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

      9. fabs-sqr 0.0%

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

      10. add-sqr-sqrt 0.2%

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

      11. add-sqr-sqrt 52.2%

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

   4. Applied egg-rr 52.2%

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

   5. Taylor expanded in x around inf 51.1%

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

   7. Applied egg-rr 50.4%

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

   if -2.3999999999999999e161 < 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 0 57.9%

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

   8. Applied egg-rr 57.9%

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

Alternative 6: 51.6% 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 53.4%

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

8. Applied egg-rr 53.4%

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

Reproduce

herbie shell --seed 2024165 
(FPCore (x y)
  :name "Numeric.LinearAlgebra.Util:formatSparse from hmatrix-0.16.1.5"
  :precision binary64
  (/ (fabs (- x y)) (fabs y)))