Numeric.LinearAlgebra.Util:formatSparse from hmatrix-0.16.1.5

Percentage Accurate: 100.0% → 100.0%

Time: 5.0s

Alternatives: 5

Speedup: 1.1×

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, 1.1× 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. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. neg-fabs N/A

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

   3. div-fabs N/A

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

   4. lower-fabs.f64 N/A

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

   5. lift--.f64 N/A

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

   6. sub-neg N/A

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

   7. +-commutative N/A

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

   8. distribute-neg-in N/A

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

   9. remove-double-neg N/A

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

   10. sub-neg N/A

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

   11. div-sub N/A

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

   12. *-inverses N/A

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

   13. lower--.f64 N/A

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

   14. lower-/.f64 100.0

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

4. Applied egg-rr 100.0%

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

Alternative 2: 98.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left|x - y\right|}{\left|y\right|} \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (fabs (- x y)) (fabs y)) 2.0) 1.0 (fabs (/ x y))))

C

double code(double x, double y) {
	double tmp;
	if ((fabs((x - y)) / fabs(y)) <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = fabs((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 ((abs((x - y)) / abs(y)) <= 2.0d0) then
        tmp = 1.0d0
    else
        tmp = abs((x / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((Math.abs((x - y)) / Math.abs(y)) <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = Math.abs((x / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (math.fabs((x - y)) / math.fabs(y)) <= 2.0:
		tmp = 1.0
	else:
		tmp = math.fabs((x / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(abs(Float64(x - y)) / abs(y)) <= 2.0)
		tmp = 1.0;
	else
		tmp = abs(Float64(x / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((abs((x - y)) / abs(y)) <= 2.0)
		tmp = 1.0;
	else
		tmp = abs((x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Abs[N[(x - y), $MachinePrecision]], $MachinePrecision] / N[Abs[y], $MachinePrecision]), $MachinePrecision], 2.0], 1.0, N[Abs[N[(x / y), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (fabs.f64 (-.f64 x y)) (fabs.f64 y)) < 2

   1. Initial program 100.0%

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

      1. lift--.f64 N/A

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

      2. neg-fabs N/A

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

      3. div-fabs N/A

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

      4. lower-fabs.f64 N/A

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

      5. lift--.f64 N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. remove-double-neg N/A

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

      10. sub-neg N/A

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

      11. div-sub N/A

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

      12. *-inverses N/A

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

      13. lower--.f64 N/A

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

      14. lower-/.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \left|\color{blue}{1}\right| \]
   6. Step-by-step derivation

   7. Simplified 96.9%

      \[\leadsto \left|\color{blue}{1}\right| \]
   8. Step-by-step derivation

      1. metadata-eval 96.9

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

   9. Applied egg-rr 96.9%

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

   if 2 < (/.f64 (fabs.f64 (-.f64 x y)) (fabs.f64 y))

   1. Initial program 100.0%

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

      1. lift--.f64 N/A

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

      2. neg-fabs N/A

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

      3. div-fabs N/A

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

      4. lower-fabs.f64 N/A

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

      5. lift--.f64 N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. remove-double-neg N/A

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

      10. sub-neg N/A

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

      11. div-sub N/A

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

      12. *-inverses N/A

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

      13. lower--.f64 N/A

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

      14. lower-/.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 99.7

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

   7. Simplified 99.7%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left|x - y\right|}{\left|y\right|} \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left|x - y\right|}{\left|y\right|} \leq 500000000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (fabs (- x y)) (fabs y)) 500000000000.0) 1.0 (/ (- x) y)))

C

double code(double x, double y) {
	double tmp;
	if ((fabs((x - y)) / fabs(y)) <= 500000000000.0) {
		tmp = 1.0;
	} else {
		tmp = -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 ((abs((x - y)) / abs(y)) <= 500000000000.0d0) then
        tmp = 1.0d0
    else
        tmp = -x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((Math.abs((x - y)) / Math.abs(y)) <= 500000000000.0) {
		tmp = 1.0;
	} else {
		tmp = -x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (math.fabs((x - y)) / math.fabs(y)) <= 500000000000.0:
		tmp = 1.0
	else:
		tmp = -x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(abs(Float64(x - y)) / abs(y)) <= 500000000000.0)
		tmp = 1.0;
	else
		tmp = Float64(Float64(-x) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((abs((x - y)) / abs(y)) <= 500000000000.0)
		tmp = 1.0;
	else
		tmp = -x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Abs[N[(x - y), $MachinePrecision]], $MachinePrecision] / N[Abs[y], $MachinePrecision]), $MachinePrecision], 500000000000.0], 1.0, N[((-x) / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (fabs.f64 (-.f64 x y)) (fabs.f64 y)) < 5e11

   1. Initial program 100.0%

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

      1. lift--.f64 N/A

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

      2. neg-fabs N/A

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

      3. div-fabs N/A

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

      4. lower-fabs.f64 N/A

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

      5. lift--.f64 N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. remove-double-neg N/A

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

      10. sub-neg N/A

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

      11. div-sub N/A

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

      12. *-inverses N/A

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

      13. lower--.f64 N/A

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

      14. lower-/.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \left|\color{blue}{1}\right| \]
   6. Step-by-step derivation

   7. Simplified 96.2%

      \[\leadsto \left|\color{blue}{1}\right| \]
   8. Step-by-step derivation

      1. metadata-eval 96.2

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

   9. Applied egg-rr 96.2%

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

   if 5e11 < (/.f64 (fabs.f64 (-.f64 x y)) (fabs.f64 y))

   1. Initial program 100.0%

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

      1. lift--.f64 N/A

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

      2. neg-fabs N/A

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

      3. div-fabs N/A

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

      4. lower-fabs.f64 N/A

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

      5. lift--.f64 N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. remove-double-neg N/A

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

      10. sub-neg N/A

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

      11. div-sub N/A

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

      12. *-inverses N/A

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

      13. lower--.f64 N/A

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

      14. lower-/.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 99.9

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

   7. Simplified 99.9%

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

      1. lift-neg.f64 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. sqr-pow N/A

         \[\leadsto \left|\color{blue}{{\left(\frac{\mathsf{neg}\left(y\right)}{x}\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(\frac{\mathsf{neg}\left(y\right)}{x}\right)}^{\left(\frac{-1}{2}\right)}}\right| \]

      5. fabs-sqr N/A

         \[\leadsto \color{blue}{{\left(\frac{\mathsf{neg}\left(y\right)}{x}\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(\frac{\mathsf{neg}\left(y\right)}{x}\right)}^{\left(\frac{-1}{2}\right)}} \]

      6. sqr-pow N/A

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

      7. inv-pow N/A

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

      8. clear-num N/A

         \[\leadsto \color{blue}{\frac{x}{\mathsf{neg}\left(y\right)}} \]

      9. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(x\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)}} \]

      10. lift-neg.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(x\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)} \]

      11. remove-double-neg N/A

          \[\leadsto \frac{\mathsf{neg}\left(x\right)}{\color{blue}{y}} \]

      12. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(x\right)}{y}} \]

      13. lower-neg.f64 55.3

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

   9. Applied egg-rr 55.3%

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

Alternative 4: 73.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left|x - y\right|}{\left|y\right|} \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (fabs (- x y)) (fabs y)) 2.0) 1.0 (/ x y)))

C

double code(double x, double y) {
	double tmp;
	if ((fabs((x - y)) / fabs(y)) <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = 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 ((abs((x - y)) / abs(y)) <= 2.0d0) then
        tmp = 1.0d0
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((Math.abs((x - y)) / Math.abs(y)) <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (math.fabs((x - y)) / math.fabs(y)) <= 2.0:
		tmp = 1.0
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(abs(Float64(x - y)) / abs(y)) <= 2.0)
		tmp = 1.0;
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((abs((x - y)) / abs(y)) <= 2.0)
		tmp = 1.0;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Abs[N[(x - y), $MachinePrecision]], $MachinePrecision] / N[Abs[y], $MachinePrecision]), $MachinePrecision], 2.0], 1.0, N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (fabs.f64 (-.f64 x y)) (fabs.f64 y)) < 2

   1. Initial program 100.0%

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

      1. lift--.f64 N/A

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

      2. neg-fabs N/A

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

      3. div-fabs N/A

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

      4. lower-fabs.f64 N/A

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

      5. lift--.f64 N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. remove-double-neg N/A

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

      10. sub-neg N/A

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

      11. div-sub N/A

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

      12. *-inverses N/A

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

      13. lower--.f64 N/A

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

      14. lower-/.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \left|\color{blue}{1}\right| \]
   6. Step-by-step derivation

   7. Simplified 96.9%

      \[\leadsto \left|\color{blue}{1}\right| \]
   8. Step-by-step derivation

      1. metadata-eval 96.9

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

   9. Applied egg-rr 96.9%

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

   if 2 < (/.f64 (fabs.f64 (-.f64 x y)) (fabs.f64 y))

   1. Initial program 100.0%

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

      1. lift--.f64 N/A

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

      2. neg-fabs N/A

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

      3. div-fabs N/A

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

      4. lower-fabs.f64 N/A

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

      5. lift--.f64 N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. remove-double-neg N/A

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

      10. sub-neg N/A

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

      11. div-sub N/A

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

      12. *-inverses N/A

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

      13. lower--.f64 N/A

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

      14. lower-/.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 99.7

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

   7. Simplified 99.7%

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

      1. lift-neg.f64 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. sqr-pow N/A

         \[\leadsto \left|\color{blue}{{\left(\frac{\mathsf{neg}\left(y\right)}{x}\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(\frac{\mathsf{neg}\left(y\right)}{x}\right)}^{\left(\frac{-1}{2}\right)}}\right| \]

      5. fabs-sqr N/A

         \[\leadsto \color{blue}{{\left(\frac{\mathsf{neg}\left(y\right)}{x}\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(\frac{\mathsf{neg}\left(y\right)}{x}\right)}^{\left(\frac{-1}{2}\right)}} \]

      6. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{\mathsf{neg}\left(y\right)}{x} \cdot \frac{\mathsf{neg}\left(y\right)}{x}\right)}^{\left(\frac{-1}{2}\right)}} \]

      7. sqr-neg N/A

         \[\leadsto {\color{blue}{\left(\left(\mathsf{neg}\left(\frac{\mathsf{neg}\left(y\right)}{x}\right)\right) \cdot \left(\mathsf{neg}\left(\frac{\mathsf{neg}\left(y\right)}{x}\right)\right)\right)}}^{\left(\frac{-1}{2}\right)} \]

      8. lift-neg.f64 N/A

         \[\leadsto {\left(\left(\mathsf{neg}\left(\frac{\color{blue}{\mathsf{neg}\left(y\right)}}{x}\right)\right) \cdot \left(\mathsf{neg}\left(\frac{\mathsf{neg}\left(y\right)}{x}\right)\right)\right)}^{\left(\frac{-1}{2}\right)} \]

      9. distribute-frac-neg N/A

         \[\leadsto {\left(\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\frac{y}{x}\right)\right)}\right)\right) \cdot \left(\mathsf{neg}\left(\frac{\mathsf{neg}\left(y\right)}{x}\right)\right)\right)}^{\left(\frac{-1}{2}\right)} \]

      10. remove-double-neg N/A

          \[\leadsto {\left(\color{blue}{\frac{y}{x}} \cdot \left(\mathsf{neg}\left(\frac{\mathsf{neg}\left(y\right)}{x}\right)\right)\right)}^{\left(\frac{-1}{2}\right)} \]

      11. lift-neg.f64 N/A

          \[\leadsto {\left(\frac{y}{x} \cdot \left(\mathsf{neg}\left(\frac{\color{blue}{\mathsf{neg}\left(y\right)}}{x}\right)\right)\right)}^{\left(\frac{-1}{2}\right)} \]

      12. distribute-frac-neg N/A

          \[\leadsto {\left(\frac{y}{x} \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\frac{y}{x}\right)\right)}\right)\right)\right)}^{\left(\frac{-1}{2}\right)} \]

      13. remove-double-neg N/A

          \[\leadsto {\left(\frac{y}{x} \cdot \color{blue}{\frac{y}{x}}\right)}^{\left(\frac{-1}{2}\right)} \]

      14. pow-prod-down N/A

          \[\leadsto \color{blue}{{\left(\frac{y}{x}\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(\frac{y}{x}\right)}^{\left(\frac{-1}{2}\right)}} \]

      15. sqr-pow N/A

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

      16. inv-pow N/A

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

      17. clear-num N/A

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

      18. lower-/.f64 45.4

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

   9. Applied egg-rr 45.4%

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

Alternative 5: 51.4% accurate, 19.0× 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. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. neg-fabs N/A

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

   3. div-fabs N/A

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

   4. lower-fabs.f64 N/A

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

   5. lift--.f64 N/A

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

   6. sub-neg N/A

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

   7. +-commutative N/A

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

   8. distribute-neg-in N/A

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

   9. remove-double-neg N/A

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

   10. sub-neg N/A

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

   11. div-sub N/A

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

   12. *-inverses N/A

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

   13. lower--.f64 N/A

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

   14. lower-/.f64 100.0

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

4. Applied egg-rr 100.0%

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

5. Taylor expanded in x around 0

   \[\leadsto \left|\color{blue}{1}\right| \]
6. Step-by-step derivation

7. Simplified 47.7%

   \[\leadsto \left|\color{blue}{1}\right| \]
8. Step-by-step derivation

   1. metadata-eval 47.7

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

9. Applied egg-rr 47.7%

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

Reproduce

herbie shell --seed 2024207 
(FPCore (x y)
  :name "Numeric.LinearAlgebra.Util:formatSparse from hmatrix-0.16.1.5"
  :precision binary64
  (/ (fabs (- x y)) (fabs y)))