Numeric.LinearAlgebra.Util:formatSparse from hmatrix-0.16.1.5

Percentage Accurate: 100.0% → 100.0%

Time: 6.4s

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

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

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. neg-fabs N/A

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

   2. div-fabs N/A

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

   3. fabs-lowering-fabs.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{\mathsf{neg}\left(\left(x - y\right)\right)}{y}\right)\right) \]

   4. sub-neg N/A

      \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{\mathsf{neg}\left(\left(x + \left(\mathsf{neg}\left(y\right)\right)\right)\right)}{y}\right)\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(y\right)\right) + x\right)\right)}{y}\right)\right) \]

   6. distribute-neg-in N/A

      \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right) + \left(\mathsf{neg}\left(x\right)\right)}{y}\right)\right) \]

   7. remove-double-neg N/A

      \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{y + \left(\mathsf{neg}\left(x\right)\right)}{y}\right)\right) \]

   8. sub-neg N/A

      \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{y - x}{y}\right)\right) \]

   9. div-sub N/A

      \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{y}{y} - \frac{x}{y}\right)\right) \]

   10. *-inverses N/A

       \[\leadsto \mathsf{fabs.f64}\left(\left(1 - \frac{x}{y}\right)\right) \]

   11. --lowering--.f64 N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{x}{y}\right)\right)\right) \]

   12. /-lowering-/.f64 100.0%

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(x, y\right)\right)\right) \]

4. Applied egg-rr 100.0%

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

Alternative 2: 58.9% accurate, 12.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{+77}:\\ \;\;\;\;0 - \frac{x}{y}\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{+59}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{-1}{\frac{x}{y}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.5e+77)
   (- 0.0 (/ x y))
   (if (<= x 2.1e+59) 1.0 (/ 1.0 (/ -1.0 (/ x y))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.5e+77) {
		tmp = 0.0 - (x / y);
	} else if (x <= 2.1e+59) {
		tmp = 1.0;
	} else {
		tmp = 1.0 / (-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 <= (-1.5d+77)) then
        tmp = 0.0d0 - (x / y)
    else if (x <= 2.1d+59) then
        tmp = 1.0d0
    else
        tmp = 1.0d0 / ((-1.0d0) / (x / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.5e+77) {
		tmp = 0.0 - (x / y);
	} else if (x <= 2.1e+59) {
		tmp = 1.0;
	} else {
		tmp = 1.0 / (-1.0 / (x / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.5e+77:
		tmp = 0.0 - (x / y)
	elif x <= 2.1e+59:
		tmp = 1.0
	else:
		tmp = 1.0 / (-1.0 / (x / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.5e+77)
		tmp = Float64(0.0 - Float64(x / y));
	elseif (x <= 2.1e+59)
		tmp = 1.0;
	else
		tmp = Float64(1.0 / Float64(-1.0 / Float64(x / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.5e+77)
		tmp = 0.0 - (x / y);
	elseif (x <= 2.1e+59)
		tmp = 1.0;
	else
		tmp = 1.0 / (-1.0 / (x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.5e+77], N[(0.0 - N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.1e+59], 1.0, N[(1.0 / N[(-1.0 / N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.4999999999999999e77

   1. Initial program 100.0%

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

      1. div-fabs N/A

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

      2. div-inv N/A

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

      3. flip-- N/A

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

      4. clear-num N/A

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

      5. frac-times N/A

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

      6. metadata-eval N/A

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

      7. fabs-div N/A

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

      8. metadata-eval N/A

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

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\left|\frac{x + y}{x \cdot x - y \cdot y} \cdot y\right|\right)}\right) \]

      10. fabs-lowering-fabs.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fabs.f64}\left(\left(\frac{x + y}{x \cdot x - y \cdot y} \cdot y\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\left(\frac{x + y}{x \cdot x - y \cdot y}\right), y\right)\right)\right) \]

      12. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{\frac{x \cdot x - y \cdot y}{x + y}}\right), y\right)\right)\right) \]

      13. flip-- N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{x - y}\right), y\right)\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(x - y\right)\right), y\right)\right)\right) \]

      15. --lowering--.f64 99.6%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(x, y\right)\right), y\right)\right)\right) \]

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(\frac{1}{x}\right)}, y\right)\right)\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 90.0%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), y\right)\right)\right) \]

   7. Simplified 90.0%

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

      1. inv-pow N/A

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

      2. sqr-pow N/A

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

      3. pow-prod-down N/A

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

      4. remove-double-neg N/A

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

      5. remove-double-neg N/A

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

      6. sqr-neg N/A

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

      7. pow-prod-down N/A

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

      8. sqr-pow N/A

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

      9. inv-pow N/A

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

      10. distribute-frac-neg2 N/A

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

      11. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\frac{1}{\left|\frac{1}{x} \cdot y\right|}\right)\right) \]

      12. inv-pow N/A

          \[\leadsto \mathsf{neg.f64}\left(\left({\left(\left|\frac{1}{x} \cdot y\right|\right)}^{-1}\right)\right) \]

      13. sqr-pow N/A

          \[\leadsto \mathsf{neg.f64}\left(\left({\left(\left|\frac{1}{x} \cdot y\right|\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(\left|\frac{1}{x} \cdot y\right|\right)}^{\left(\frac{-1}{2}\right)}\right)\right) \]

      14. pow-prod-down N/A

          \[\leadsto \mathsf{neg.f64}\left(\left({\left(\left|\frac{1}{x} \cdot y\right| \cdot \left|\frac{1}{x} \cdot y\right|\right)}^{\left(\frac{-1}{2}\right)}\right)\right) \]

      15. sqr-abs N/A

          \[\leadsto \mathsf{neg.f64}\left(\left({\left(\left(\frac{1}{x} \cdot y\right) \cdot \left(\frac{1}{x} \cdot y\right)\right)}^{\left(\frac{-1}{2}\right)}\right)\right) \]

      16. pow2 N/A

          \[\leadsto \mathsf{neg.f64}\left(\left({\left({\left(\frac{1}{x} \cdot y\right)}^{2}\right)}^{\left(\frac{-1}{2}\right)}\right)\right) \]

      17. pow-pow N/A

          \[\leadsto \mathsf{neg.f64}\left(\left({\left(\frac{1}{x} \cdot y\right)}^{\left(2 \cdot \frac{-1}{2}\right)}\right)\right) \]

      18. metadata-eval N/A

          \[\leadsto \mathsf{neg.f64}\left(\left({\left(\frac{1}{x} \cdot y\right)}^{\left(2 \cdot \frac{-1}{2}\right)}\right)\right) \]

      19. metadata-eval N/A

          \[\leadsto \mathsf{neg.f64}\left(\left({\left(\frac{1}{x} \cdot y\right)}^{-1}\right)\right) \]

      20. inv-pow N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\frac{1}{\frac{1}{x} \cdot y}\right)\right) \]

   9. Applied egg-rr 56.9%

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

   if -1.4999999999999999e77 < x < 2.09999999999999984e59

   1. Initial program 100.0%

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

      1. neg-fabs N/A

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

      2. div-fabs N/A

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

      3. fabs-lowering-fabs.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{\mathsf{neg}\left(\left(x - y\right)\right)}{y}\right)\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{\mathsf{neg}\left(\left(x + \left(\mathsf{neg}\left(y\right)\right)\right)\right)}{y}\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(y\right)\right) + x\right)\right)}{y}\right)\right) \]

      6. distribute-neg-in N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right) + \left(\mathsf{neg}\left(x\right)\right)}{y}\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{y + \left(\mathsf{neg}\left(x\right)\right)}{y}\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{y - x}{y}\right)\right) \]

      9. div-sub N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{y}{y} - \frac{x}{y}\right)\right) \]

      10. *-inverses N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(1 - \frac{x}{y}\right)\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{x}{y}\right)\right)\right) \]

      12. /-lowering-/.f64 100.0%

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(x, y\right)\right)\right) \]

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{1}\right) \]
   6. Step-by-step derivation

   7. Simplified 73.4%

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

      1. metadata-eval 73.4%

         \[\leadsto 1 \]

   9. Applied egg-rr 73.4%

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

   if 2.09999999999999984e59 < x

   1. Initial program 100.0%

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

      1. div-fabs N/A

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

      2. div-inv N/A

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

      3. flip-- N/A

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

      4. clear-num N/A

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

      5. frac-times N/A

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

      6. metadata-eval N/A

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

      7. fabs-div N/A

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

      8. metadata-eval N/A

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

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\left|\frac{x + y}{x \cdot x - y \cdot y} \cdot y\right|\right)}\right) \]

      10. fabs-lowering-fabs.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fabs.f64}\left(\left(\frac{x + y}{x \cdot x - y \cdot y} \cdot y\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\left(\frac{x + y}{x \cdot x - y \cdot y}\right), y\right)\right)\right) \]

      12. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{\frac{x \cdot x - y \cdot y}{x + y}}\right), y\right)\right)\right) \]

      13. flip-- N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{x - y}\right), y\right)\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(x - y\right)\right), y\right)\right)\right) \]

      15. --lowering--.f64 99.7%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(x, y\right)\right), y\right)\right)\right) \]

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(\frac{1}{x}\right)}, y\right)\right)\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 88.5%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), y\right)\right)\right) \]

   7. Simplified 88.5%

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

      1. remove-double-div N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{1}{\color{blue}{\frac{1}{\left|\frac{1}{x} \cdot y\right|}}}\right)\right) \]

      2. frac-2neg N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\mathsf{neg}\left(1\right)}{\color{blue}{\mathsf{neg}\left(\frac{1}{\left|\frac{1}{x} \cdot y\right|}\right)}}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{-1}{\mathsf{neg}\left(\color{blue}{\frac{1}{\left|\frac{1}{x} \cdot y\right|}}\right)}\right)\right) \]

      4. distribute-frac-neg2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{-1}{\frac{1}{\color{blue}{\mathsf{neg}\left(\left|\frac{1}{x} \cdot y\right|\right)}}}\right)\right) \]

      5. inv-pow N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{-1}{{\left(\mathsf{neg}\left(\left|\frac{1}{x} \cdot y\right|\right)\right)}^{\color{blue}{-1}}}\right)\right) \]

      6. sqr-pow N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{-1}{{\left(\mathsf{neg}\left(\left|\frac{1}{x} \cdot y\right|\right)\right)}^{\left(\frac{-1}{2}\right)} \cdot \color{blue}{{\left(\mathsf{neg}\left(\left|\frac{1}{x} \cdot y\right|\right)\right)}^{\left(\frac{-1}{2}\right)}}}\right)\right) \]

      7. pow-prod-down N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{-1}{{\left(\left(\mathsf{neg}\left(\left|\frac{1}{x} \cdot y\right|\right)\right) \cdot \left(\mathsf{neg}\left(\left|\frac{1}{x} \cdot y\right|\right)\right)\right)}^{\color{blue}{\left(\frac{-1}{2}\right)}}}\right)\right) \]

      8. sqr-neg N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{-1}{{\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left|\frac{1}{x} \cdot y\right|\right)\right)\right)\right) \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left|\frac{1}{x} \cdot y\right|\right)\right)\right)\right)\right)}^{\left(\frac{\color{blue}{-1}}{2}\right)}}\right)\right) \]

      9. remove-double-neg N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{-1}{{\left(\left|\frac{1}{x} \cdot y\right| \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left|\frac{1}{x} \cdot y\right|\right)\right)\right)\right)\right)}^{\left(\frac{-1}{2}\right)}}\right)\right) \]

      10. remove-double-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{-1}{{\left(\left|\frac{1}{x} \cdot y\right| \cdot \left|\frac{1}{x} \cdot y\right|\right)}^{\left(\frac{-1}{2}\right)}}\right)\right) \]

      11. pow-prod-down N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{-1}{{\left(\left|\frac{1}{x} \cdot y\right|\right)}^{\left(\frac{-1}{2}\right)} \cdot \color{blue}{{\left(\left|\frac{1}{x} \cdot y\right|\right)}^{\left(\frac{-1}{2}\right)}}}\right)\right) \]

      12. sqr-pow N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{-1}{{\left(\left|\frac{1}{x} \cdot y\right|\right)}^{\color{blue}{-1}}}\right)\right) \]

      13. inv-pow N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{-1}{\frac{1}{\color{blue}{\left|\frac{1}{x} \cdot y\right|}}}\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(-1, \color{blue}{\left(\frac{1}{\left|\frac{1}{x} \cdot y\right|}\right)}\right)\right) \]

      15. inv-pow N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(-1, \left({\left(\left|\frac{1}{x} \cdot y\right|\right)}^{\color{blue}{-1}}\right)\right)\right) \]

      16. sqr-pow N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(-1, \left({\left(\left|\frac{1}{x} \cdot y\right|\right)}^{\left(\frac{-1}{2}\right)} \cdot \color{blue}{{\left(\left|\frac{1}{x} \cdot y\right|\right)}^{\left(\frac{-1}{2}\right)}}\right)\right)\right) \]

      17. pow-prod-down N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(-1, \left({\left(\left|\frac{1}{x} \cdot y\right| \cdot \left|\frac{1}{x} \cdot y\right|\right)}^{\color{blue}{\left(\frac{-1}{2}\right)}}\right)\right)\right) \]

   9. Applied egg-rr 48.3%

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

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{+77}:\\ \;\;\;\;0 - \frac{x}{y}\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{+59}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{-1}{\frac{x}{y}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 59.0% accurate, 13.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0 - \frac{x}{y}\\ \mathbf{if}\;x \leq -7.5 \cdot 10^{+78}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+62}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 0.0 (/ x y))))
   (if (<= x -7.5e+78) t_0 (if (<= x 4e+62) 1.0 t_0))))

C

double code(double x, double y) {
	double t_0 = 0.0 - (x / y);
	double tmp;
	if (x <= -7.5e+78) {
		tmp = t_0;
	} else if (x <= 4e+62) {
		tmp = 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.0d0 - (x / y)
    if (x <= (-7.5d+78)) then
        tmp = t_0
    else if (x <= 4d+62) then
        tmp = 1.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 0.0 - (x / y);
	double tmp;
	if (x <= -7.5e+78) {
		tmp = t_0;
	} else if (x <= 4e+62) {
		tmp = 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 0.0 - (x / y)
	tmp = 0
	if x <= -7.5e+78:
		tmp = t_0
	elif x <= 4e+62:
		tmp = 1.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(0.0 - Float64(x / y))
	tmp = 0.0
	if (x <= -7.5e+78)
		tmp = t_0;
	elseif (x <= 4e+62)
		tmp = 1.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 0.0 - (x / y);
	tmp = 0.0;
	if (x <= -7.5e+78)
		tmp = t_0;
	elseif (x <= 4e+62)
		tmp = 1.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(0.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -7.5e+78], t$95$0, If[LessEqual[x, 4e+62], 1.0, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -7.49999999999999934e78 or 4.00000000000000014e62 < x

   1. Initial program 100.0%

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

      1. div-fabs N/A

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

      2. div-inv N/A

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

      3. flip-- N/A

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

      4. clear-num N/A

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

      5. frac-times N/A

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

      6. metadata-eval N/A

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

      7. fabs-div N/A

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

      8. metadata-eval N/A

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

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\left|\frac{x + y}{x \cdot x - y \cdot y} \cdot y\right|\right)}\right) \]

      10. fabs-lowering-fabs.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fabs.f64}\left(\left(\frac{x + y}{x \cdot x - y \cdot y} \cdot y\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\left(\frac{x + y}{x \cdot x - y \cdot y}\right), y\right)\right)\right) \]

      12. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{\frac{x \cdot x - y \cdot y}{x + y}}\right), y\right)\right)\right) \]

      13. flip-- N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{x - y}\right), y\right)\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(x - y\right)\right), y\right)\right)\right) \]

      15. --lowering--.f64 99.7%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(x, y\right)\right), y\right)\right)\right) \]

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(\frac{1}{x}\right)}, y\right)\right)\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 89.1%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), y\right)\right)\right) \]

   7. Simplified 89.1%

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

      1. inv-pow N/A

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

      2. sqr-pow N/A

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

      3. pow-prod-down N/A

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

      4. remove-double-neg N/A

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

      5. remove-double-neg N/A

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

      6. sqr-neg N/A

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

      7. pow-prod-down N/A

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

      8. sqr-pow N/A

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

      9. inv-pow N/A

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

      10. distribute-frac-neg2 N/A

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

      11. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\frac{1}{\left|\frac{1}{x} \cdot y\right|}\right)\right) \]

      12. inv-pow N/A

          \[\leadsto \mathsf{neg.f64}\left(\left({\left(\left|\frac{1}{x} \cdot y\right|\right)}^{-1}\right)\right) \]

      13. sqr-pow N/A

          \[\leadsto \mathsf{neg.f64}\left(\left({\left(\left|\frac{1}{x} \cdot y\right|\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(\left|\frac{1}{x} \cdot y\right|\right)}^{\left(\frac{-1}{2}\right)}\right)\right) \]

      14. pow-prod-down N/A

          \[\leadsto \mathsf{neg.f64}\left(\left({\left(\left|\frac{1}{x} \cdot y\right| \cdot \left|\frac{1}{x} \cdot y\right|\right)}^{\left(\frac{-1}{2}\right)}\right)\right) \]

      15. sqr-abs N/A

          \[\leadsto \mathsf{neg.f64}\left(\left({\left(\left(\frac{1}{x} \cdot y\right) \cdot \left(\frac{1}{x} \cdot y\right)\right)}^{\left(\frac{-1}{2}\right)}\right)\right) \]

      16. pow2 N/A

          \[\leadsto \mathsf{neg.f64}\left(\left({\left({\left(\frac{1}{x} \cdot y\right)}^{2}\right)}^{\left(\frac{-1}{2}\right)}\right)\right) \]

      17. pow-pow N/A

          \[\leadsto \mathsf{neg.f64}\left(\left({\left(\frac{1}{x} \cdot y\right)}^{\left(2 \cdot \frac{-1}{2}\right)}\right)\right) \]

      18. metadata-eval N/A

          \[\leadsto \mathsf{neg.f64}\left(\left({\left(\frac{1}{x} \cdot y\right)}^{\left(2 \cdot \frac{-1}{2}\right)}\right)\right) \]

      19. metadata-eval N/A

          \[\leadsto \mathsf{neg.f64}\left(\left({\left(\frac{1}{x} \cdot y\right)}^{-1}\right)\right) \]

      20. inv-pow N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\frac{1}{\frac{1}{x} \cdot y}\right)\right) \]

   9. Applied egg-rr 52.0%

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

   if -7.49999999999999934e78 < x < 4.00000000000000014e62

   1. Initial program 100.0%

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

      1. neg-fabs N/A

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

      2. div-fabs N/A

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

      3. fabs-lowering-fabs.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{\mathsf{neg}\left(\left(x - y\right)\right)}{y}\right)\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{\mathsf{neg}\left(\left(x + \left(\mathsf{neg}\left(y\right)\right)\right)\right)}{y}\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(y\right)\right) + x\right)\right)}{y}\right)\right) \]

      6. distribute-neg-in N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right) + \left(\mathsf{neg}\left(x\right)\right)}{y}\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{y + \left(\mathsf{neg}\left(x\right)\right)}{y}\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{y - x}{y}\right)\right) \]

      9. div-sub N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{y}{y} - \frac{x}{y}\right)\right) \]

      10. *-inverses N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(1 - \frac{x}{y}\right)\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{x}{y}\right)\right)\right) \]

      12. /-lowering-/.f64 100.0%

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(x, y\right)\right)\right) \]

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{1}\right) \]
   6. Step-by-step derivation

   7. Simplified 73.4%

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

      1. metadata-eval 73.4%

         \[\leadsto 1 \]

   9. Applied egg-rr 73.4%

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

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{+78}:\\ \;\;\;\;0 - \frac{x}{y}\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+62}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 59.2% accurate, 15.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{+131}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 4.1 \cdot 10^{+106}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -7.5e+131) (/ x y) (if (<= x 4.1e+106) 1.0 (/ x y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -7.5e+131) {
		tmp = x / y;
	} else if (x <= 4.1e+106) {
		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 (x <= (-7.5d+131)) then
        tmp = x / y
    else if (x <= 4.1d+106) 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 (x <= -7.5e+131) {
		tmp = x / y;
	} else if (x <= 4.1e+106) {
		tmp = 1.0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -7.5e+131:
		tmp = x / y
	elif x <= 4.1e+106:
		tmp = 1.0
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -7.5e+131)
		tmp = Float64(x / y);
	elseif (x <= 4.1e+106)
		tmp = 1.0;
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -7.5e+131)
		tmp = x / y;
	elseif (x <= 4.1e+106)
		tmp = 1.0;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -7.5e+131], N[(x / y), $MachinePrecision], If[LessEqual[x, 4.1e+106], 1.0, N[(x / y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -7.4999999999999995e131 or 4.1000000000000002e106 < x

   1. Initial program 100.0%

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

      1. div-fabs N/A

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

      2. div-inv N/A

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

      3. flip-- N/A

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

      4. clear-num N/A

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

      5. frac-times N/A

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

      6. metadata-eval N/A

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

      7. fabs-div N/A

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

      8. metadata-eval N/A

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

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\left|\frac{x + y}{x \cdot x - y \cdot y} \cdot y\right|\right)}\right) \]

      10. fabs-lowering-fabs.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fabs.f64}\left(\left(\frac{x + y}{x \cdot x - y \cdot y} \cdot y\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\left(\frac{x + y}{x \cdot x - y \cdot y}\right), y\right)\right)\right) \]

      12. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{\frac{x \cdot x - y \cdot y}{x + y}}\right), y\right)\right)\right) \]

      13. flip-- N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{x - y}\right), y\right)\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(x - y\right)\right), y\right)\right)\right) \]

      15. --lowering--.f64 99.6%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(x, y\right)\right), y\right)\right)\right) \]

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(\frac{1}{x}\right)}, y\right)\right)\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 92.9%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), y\right)\right)\right) \]

   7. Simplified 92.9%

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

      1. inv-pow N/A

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

      2. sqr-pow N/A

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

      3. pow-prod-down N/A

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

      4. sqr-abs N/A

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

      5. pow2 N/A

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

      6. pow-pow N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. inv-pow N/A

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

      10. *-commutative N/A

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

      11. un-div-inv N/A

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

      12. clear-num N/A

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

      13. /-lowering-/.f64 44.2%

          \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{y}\right) \]

   9. Applied egg-rr 44.2%

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

   if -7.4999999999999995e131 < x < 4.1000000000000002e106

   1. Initial program 100.0%

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

      1. neg-fabs N/A

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

      2. div-fabs N/A

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

      3. fabs-lowering-fabs.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{\mathsf{neg}\left(\left(x - y\right)\right)}{y}\right)\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{\mathsf{neg}\left(\left(x + \left(\mathsf{neg}\left(y\right)\right)\right)\right)}{y}\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(y\right)\right) + x\right)\right)}{y}\right)\right) \]

      6. distribute-neg-in N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right) + \left(\mathsf{neg}\left(x\right)\right)}{y}\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{y + \left(\mathsf{neg}\left(x\right)\right)}{y}\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{y - x}{y}\right)\right) \]

      9. div-sub N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{y}{y} - \frac{x}{y}\right)\right) \]

      10. *-inverses N/A

          \[\leadsto \mathsf{fabs.f64}\left(\left(1 - \frac{x}{y}\right)\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{x}{y}\right)\right)\right) \]

      12. /-lowering-/.f64 100.0%

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(x, y\right)\right)\right) \]

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{1}\right) \]
   6. Step-by-step derivation

   7. Simplified 68.5%

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

      1. metadata-eval 68.5%

         \[\leadsto 1 \]

   9. Applied egg-rr 68.5%

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

Alternative 5: 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. Step-by-step derivation

   1. neg-fabs N/A

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

   2. div-fabs N/A

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

   3. fabs-lowering-fabs.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{\mathsf{neg}\left(\left(x - y\right)\right)}{y}\right)\right) \]

   4. sub-neg N/A

      \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{\mathsf{neg}\left(\left(x + \left(\mathsf{neg}\left(y\right)\right)\right)\right)}{y}\right)\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(y\right)\right) + x\right)\right)}{y}\right)\right) \]

   6. distribute-neg-in N/A

      \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right) + \left(\mathsf{neg}\left(x\right)\right)}{y}\right)\right) \]

   7. remove-double-neg N/A

      \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{y + \left(\mathsf{neg}\left(x\right)\right)}{y}\right)\right) \]

   8. sub-neg N/A

      \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{y - x}{y}\right)\right) \]

   9. div-sub N/A

      \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{y}{y} - \frac{x}{y}\right)\right) \]

   10. *-inverses N/A

       \[\leadsto \mathsf{fabs.f64}\left(\left(1 - \frac{x}{y}\right)\right) \]

   11. --lowering--.f64 N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{x}{y}\right)\right)\right) \]

   12. /-lowering-/.f64 100.0%

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(x, y\right)\right)\right) \]

4. Applied egg-rr 100.0%

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

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{1}\right) \]
6. Step-by-step derivation

7. Simplified 52.3%

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

   1. metadata-eval 52.3%

      \[\leadsto 1 \]

9. Applied egg-rr 52.3%

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

Reproduce

herbie shell --seed 2024161 
(FPCore (x y)
  :name "Numeric.LinearAlgebra.Util:formatSparse from hmatrix-0.16.1.5"
  :precision binary64
  (/ (fabs (- x y)) (fabs y)))