Numeric.LinearAlgebra.Util:formatSparse from hmatrix-0.16.1.5

Percentage Accurate: 100.0% → 100.0%

Time: 6.6s

Alternatives: 5

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: 57.8% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.6 \cdot 10^{+21}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -2.4 \cdot 10^{-213}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{-237}:\\ \;\;\;\;\frac{-x}{y}\\ \mathbf{elif}\;y \leq 3 \cdot 10^{-86}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -5.6e+21)
   1.0
   (if (<= y -2.4e-213)
     (/ x y)
     (if (<= y 2.7e-237) (/ (- x) y) (if (<= y 3e-86) (/ x y) 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -5.6e+21) {
		tmp = 1.0;
	} else if (y <= -2.4e-213) {
		tmp = x / y;
	} else if (y <= 2.7e-237) {
		tmp = -x / y;
	} else if (y <= 3e-86) {
		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 <= (-5.6d+21)) then
        tmp = 1.0d0
    else if (y <= (-2.4d-213)) then
        tmp = x / y
    else if (y <= 2.7d-237) then
        tmp = -x / y
    else if (y <= 3d-86) 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 <= -5.6e+21) {
		tmp = 1.0;
	} else if (y <= -2.4e-213) {
		tmp = x / y;
	} else if (y <= 2.7e-237) {
		tmp = -x / y;
	} else if (y <= 3e-86) {
		tmp = x / y;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -5.6e+21:
		tmp = 1.0
	elif y <= -2.4e-213:
		tmp = x / y
	elif y <= 2.7e-237:
		tmp = -x / y
	elif y <= 3e-86:
		tmp = x / y
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -5.6e+21)
		tmp = 1.0;
	elseif (y <= -2.4e-213)
		tmp = Float64(x / y);
	elseif (y <= 2.7e-237)
		tmp = Float64(Float64(-x) / y);
	elseif (y <= 3e-86)
		tmp = Float64(x / y);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -5.6e+21)
		tmp = 1.0;
	elseif (y <= -2.4e-213)
		tmp = x / y;
	elseif (y <= 2.7e-237)
		tmp = -x / y;
	elseif (y <= 3e-86)
		tmp = x / y;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -5.6e+21], 1.0, If[LessEqual[y, -2.4e-213], N[(x / y), $MachinePrecision], If[LessEqual[y, 2.7e-237], N[((-x) / y), $MachinePrecision], If[LessEqual[y, 3e-86], N[(x / y), $MachinePrecision], 1.0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.6e21 or 3.0000000000000001e-86 < 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. Step-by-step derivation

      1. metadata-eval 100.0%

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

      2. sub-neg 100.0%

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

      3. fabs-sub 100.0%

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

      4. *-inverses 100.0%

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

      5. div-sub 100.0%

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

      6. add-sqr-sqrt 90.0%

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

      7. fabs-sqr 90.0%

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

      8. add-sqr-sqrt 90.3%

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

      9. div-sub 90.3%

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

      10. *-inverses 90.3%

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

   7. Applied egg-rr 90.3%

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

   8. Taylor expanded in x around 0 72.0%

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

   if -5.6e21 < y < -2.39999999999999996e-213 or 2.69999999999999984e-237 < y < 3.0000000000000001e-86

   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 0 100.0%

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

      1. sub-neg 100.0%

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

      2. metadata-eval 100.0%

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

      3. rem-square-sqrt 48.4%

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

      4. fabs-sqr 48.4%

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

      5. rem-square-sqrt 49.2%

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

   8. Simplified 49.2%

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

   9. Taylor expanded in x around inf 49.5%

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

   if -2.39999999999999996e-213 < y < 2.69999999999999984e-237

   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. sub-neg 100.0%

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

      3. fabs-sub 100.0%

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

      4. *-inverses 100.0%

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

      5. div-sub 100.0%

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

      6. add-sqr-sqrt 66.5%

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

      7. fabs-sqr 66.5%

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

      8. add-sqr-sqrt 66.8%

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

      9. div-sub 66.7%

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

      10. *-inverses 66.7%

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

   7. Applied egg-rr 66.7%

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

   8. Taylor expanded in x around inf 62.1%

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

      1. mul-1-neg 62.1%

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

      2. distribute-frac-neg 62.1%

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

   10. Simplified 62.1%

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

Alternative 3: 58.0% accurate, 15.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.76 \cdot 10^{+25}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 3 \cdot 10^{-88}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.76e+25) 1.0 (if (<= y 3e-88) (/ x y) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.76e+25) {
		tmp = 1.0;
	} else if (y <= 3e-88) {
		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 <= (-1.76d+25)) then
        tmp = 1.0d0
    else if (y <= 3d-88) 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 <= -1.76e+25) {
		tmp = 1.0;
	} else if (y <= 3e-88) {
		tmp = x / y;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.76e+25:
		tmp = 1.0
	elif y <= 3e-88:
		tmp = x / y
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.76e+25)
		tmp = 1.0;
	elseif (y <= 3e-88)
		tmp = Float64(x / y);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.76e+25)
		tmp = 1.0;
	elseif (y <= 3e-88)
		tmp = x / y;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.76e+25], 1.0, If[LessEqual[y, 3e-88], N[(x / y), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -1.76000000000000001e25 or 2.9999999999999999e-88 < 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. Step-by-step derivation

      1. metadata-eval 100.0%

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

      2. sub-neg 100.0%

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

      3. fabs-sub 100.0%

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

      4. *-inverses 100.0%

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

      5. div-sub 100.0%

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

      6. add-sqr-sqrt 90.0%

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

      7. fabs-sqr 90.0%

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

      8. add-sqr-sqrt 90.3%

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

      9. div-sub 90.3%

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

      10. *-inverses 90.3%

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

   7. Applied egg-rr 90.3%

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

   8. Taylor expanded in x around 0 72.0%

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

   if -1.76000000000000001e25 < y < 2.9999999999999999e-88

   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 0 100.0%

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

      1. sub-neg 100.0%

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

      2. metadata-eval 100.0%

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

      3. rem-square-sqrt 43.2%

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

      4. fabs-sqr 43.2%

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

      5. rem-square-sqrt 43.8%

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

   8. Simplified 43.8%

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

   9. Taylor expanded in x around inf 44.1%

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

Alternative 4: 74.0% accurate, 20.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{+214}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.4e+214) (/ x y) (- 1.0 (/ x y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.4e+214) {
		tmp = 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 <= (-1.4d+214)) then
        tmp = 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 <= -1.4e+214) {
		tmp = x / y;
	} else {
		tmp = 1.0 - (x / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.4e+214:
		tmp = x / y
	else:
		tmp = 1.0 - (x / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.4e+214)
		tmp = 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 <= -1.4e+214)
		tmp = x / y;
	else
		tmp = 1.0 - (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.4e+214], N[(x / y), $MachinePrecision], N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.3999999999999999e214

   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 0 100.0%

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

      1. sub-neg 100.0%

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

      2. metadata-eval 100.0%

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

      3. rem-square-sqrt 63.4%

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

      4. fabs-sqr 63.4%

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

      5. rem-square-sqrt 63.8%

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

   8. Simplified 63.8%

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

   9. Taylor expanded in x around inf 64.0%

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

   if -1.3999999999999999e214 < 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. Step-by-step derivation

      1. metadata-eval 100.0%

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

      2. sub-neg 100.0%

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

      3. fabs-sub 100.0%

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

      4. *-inverses 100.0%

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

      5. div-sub 100.0%

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

      6. add-sqr-sqrt 78.9%

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

      7. fabs-sqr 78.9%

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

      8. add-sqr-sqrt 79.2%

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

      9. div-sub 79.2%

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

      10. *-inverses 79.2%

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

   7. Applied egg-rr 79.2%

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

Alternative 5: 50.4% 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. Step-by-step derivation

   1. metadata-eval 100.0%

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

   2. sub-neg 100.0%

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

   3. fabs-sub 100.0%

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

   4. *-inverses 100.0%

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

   5. div-sub 100.0%

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

   6. add-sqr-sqrt 75.2%

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

   7. fabs-sqr 75.2%

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

   8. add-sqr-sqrt 75.5%

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

   9. div-sub 75.5%

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

   10. *-inverses 75.5%

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

7. Applied egg-rr 75.5%

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

8. Taylor expanded in x around 0 48.4%

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

Reproduce

herbie shell --seed 2024144 
(FPCore (x y)
  :name "Numeric.LinearAlgebra.Util:formatSparse from hmatrix-0.16.1.5"
  :precision binary64
  (/ (fabs (- x y)) (fabs y)))