Numeric.LinearAlgebra.Util:formatSparse from hmatrix-0.16.1.5

Percentage Accurate: 100.0% → 100.0%

Time: 4.3s

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. Taylor expanded in x around -inf 100.0%

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

   1. fabs-neg 100.0%

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

   2. mul-1-neg 100.0%

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

   3. sub-neg 100.0%

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

   4. fabs-div 100.0%

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

   5. div-sub 100.0%

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

   6. *-inverses 100.0%

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

5. Simplified 100.0%

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

6. Final simplification 100.0%

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

Alternative 2: 72.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.9 \cdot 10^{-115} \lor \neg \left(x \leq 2.4 \cdot 10^{-85} \lor \neg \left(x \leq 1.95 \cdot 10^{+27}\right) \land x \leq 4.6 \cdot 10^{+79}\right):\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.9e-115)
         (not (or (<= x 2.4e-85) (and (not (<= x 1.95e+27)) (<= x 4.6e+79)))))
   (fabs (/ x y))
   1.0))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.9e-115) || !((x <= 2.4e-85) || (!(x <= 1.95e+27) && (x <= 4.6e+79)))) {
		tmp = fabs((x / y));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-1.9d-115)) .or. (.not. (x <= 2.4d-85) .or. (.not. (x <= 1.95d+27)) .and. (x <= 4.6d+79))) then
        tmp = abs((x / y))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -1.9e-115) || !((x <= 2.4e-85) || (!(x <= 1.95e+27) && (x <= 4.6e+79)))) {
		tmp = Math.abs((x / y));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.9e-115) or not ((x <= 2.4e-85) or (not (x <= 1.95e+27) and (x <= 4.6e+79))):
		tmp = math.fabs((x / y))
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.9e-115) || !((x <= 2.4e-85) || (!(x <= 1.95e+27) && (x <= 4.6e+79))))
		tmp = abs(Float64(x / y));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -1.9e-115) || ~(((x <= 2.4e-85) || (~((x <= 1.95e+27)) && (x <= 4.6e+79)))))
		tmp = abs((x / y));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.9e-115], N[Not[Or[LessEqual[x, 2.4e-85], And[N[Not[LessEqual[x, 1.95e+27]], $MachinePrecision], LessEqual[x, 4.6e+79]]]], $MachinePrecision]], N[Abs[N[(x / y), $MachinePrecision]], $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < -1.89999999999999996e-115 or 2.4000000000000001e-85 < x < 1.9499999999999999e27 or 4.6000000000000001e79 < x

   1. Initial program 100.0%

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

      1. add-cbrt-cube 59.3%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{\left|x - y\right|}{\left|y\right|} \cdot \frac{\left|x - y\right|}{\left|y\right|}\right) \cdot \frac{\left|x - y\right|}{\left|y\right|}}} \]

      2. pow1/3 57.7%

         \[\leadsto \color{blue}{{\left(\left(\frac{\left|x - y\right|}{\left|y\right|} \cdot \frac{\left|x - y\right|}{\left|y\right|}\right) \cdot \frac{\left|x - y\right|}{\left|y\right|}\right)}^{0.3333333333333333}} \]

      3. pow3 57.7%

         \[\leadsto {\color{blue}{\left({\left(\frac{\left|x - y\right|}{\left|y\right|}\right)}^{3}\right)}}^{0.3333333333333333} \]

      4. clear-num 57.7%

         \[\leadsto {\left({\color{blue}{\left(\frac{1}{\frac{\left|y\right|}{\left|x - y\right|}}\right)}}^{3}\right)}^{0.3333333333333333} \]

      5. inv-pow 57.7%

         \[\leadsto {\left({\color{blue}{\left({\left(\frac{\left|y\right|}{\left|x - y\right|}\right)}^{-1}\right)}}^{3}\right)}^{0.3333333333333333} \]

      6. pow-pow 57.7%

         \[\leadsto {\color{blue}{\left({\left(\frac{\left|y\right|}{\left|x - y\right|}\right)}^{\left(-1 \cdot 3\right)}\right)}}^{0.3333333333333333} \]

      7. add-sqr-sqrt 22.2%

         \[\leadsto {\left({\left(\frac{\left|\color{blue}{\sqrt{y} \cdot \sqrt{y}}\right|}{\left|x - y\right|}\right)}^{\left(-1 \cdot 3\right)}\right)}^{0.3333333333333333} \]

      8. fabs-sqr 22.2%

         \[\leadsto {\left({\left(\frac{\color{blue}{\sqrt{y} \cdot \sqrt{y}}}{\left|x - y\right|}\right)}^{\left(-1 \cdot 3\right)}\right)}^{0.3333333333333333} \]

      9. add-sqr-sqrt 34.4%

         \[\leadsto {\left({\left(\frac{\color{blue}{y}}{\left|x - y\right|}\right)}^{\left(-1 \cdot 3\right)}\right)}^{0.3333333333333333} \]

      10. add-sqr-sqrt 12.7%

          \[\leadsto {\left({\left(\frac{y}{\left|\color{blue}{\sqrt{x - y} \cdot \sqrt{x - y}}\right|}\right)}^{\left(-1 \cdot 3\right)}\right)}^{0.3333333333333333} \]

      11. fabs-sqr 12.7%

          \[\leadsto {\left({\left(\frac{y}{\color{blue}{\sqrt{x - y} \cdot \sqrt{x - y}}}\right)}^{\left(-1 \cdot 3\right)}\right)}^{0.3333333333333333} \]

      12. add-sqr-sqrt 33.5%

          \[\leadsto {\left({\left(\frac{y}{\color{blue}{x - y}}\right)}^{\left(-1 \cdot 3\right)}\right)}^{0.3333333333333333} \]

      13. metadata-eval 33.5%

          \[\leadsto {\left({\left(\frac{y}{x - y}\right)}^{\color{blue}{-3}}\right)}^{0.3333333333333333} \]

   4. Applied egg-rr 33.5%

      \[\leadsto \color{blue}{{\left({\left(\frac{y}{x - y}\right)}^{-3}\right)}^{0.3333333333333333}} \]
   5. Step-by-step derivation

      1. unpow1/3 24.2%

         \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{y}{x - y}\right)}^{-3}}} \]

   6. Simplified 24.2%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{y}{x - y}\right)}^{-3}}} \]

   7. Taylor expanded in y around 0 23.8%

      \[\leadsto \sqrt[3]{{\color{blue}{\left(\frac{y}{x}\right)}}^{-3}} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 23.6%

         \[\leadsto \color{blue}{\sqrt{\sqrt[3]{{\left(\frac{y}{x}\right)}^{-3}}} \cdot \sqrt{\sqrt[3]{{\left(\frac{y}{x}\right)}^{-3}}}} \]

      2. sqrt-unprod 41.9%

         \[\leadsto \color{blue}{\sqrt{\sqrt[3]{{\left(\frac{y}{x}\right)}^{-3}} \cdot \sqrt[3]{{\left(\frac{y}{x}\right)}^{-3}}}} \]

      3. pow1/2 41.9%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{{\left(\frac{y}{x}\right)}^{-3}} \cdot \sqrt[3]{{\left(\frac{y}{x}\right)}^{-3}}\right)}^{0.5}} \]

      4. pow1/3 33.5%

         \[\leadsto {\left(\color{blue}{{\left({\left(\frac{y}{x}\right)}^{-3}\right)}^{0.3333333333333333}} \cdot \sqrt[3]{{\left(\frac{y}{x}\right)}^{-3}}\right)}^{0.5} \]

      5. pow1/3 33.3%

         \[\leadsto {\left({\left({\left(\frac{y}{x}\right)}^{-3}\right)}^{0.3333333333333333} \cdot \color{blue}{{\left({\left(\frac{y}{x}\right)}^{-3}\right)}^{0.3333333333333333}}\right)}^{0.5} \]

      6. pow-prod-up 33.3%

         \[\leadsto {\color{blue}{\left({\left({\left(\frac{y}{x}\right)}^{-3}\right)}^{\left(0.3333333333333333 + 0.3333333333333333\right)}\right)}}^{0.5} \]

      7. pow-pow 51.8%

         \[\leadsto {\color{blue}{\left({\left(\frac{y}{x}\right)}^{\left(-3 \cdot \left(0.3333333333333333 + 0.3333333333333333\right)\right)}\right)}}^{0.5} \]

      8. metadata-eval 51.8%

         \[\leadsto {\left({\left(\frac{y}{x}\right)}^{\left(-3 \cdot \color{blue}{0.6666666666666666}\right)}\right)}^{0.5} \]

      9. metadata-eval 51.8%

         \[\leadsto {\left({\left(\frac{y}{x}\right)}^{\color{blue}{-2}}\right)}^{0.5} \]

   9. Applied egg-rr 51.8%

      \[\leadsto \color{blue}{{\left({\left(\frac{y}{x}\right)}^{-2}\right)}^{0.5}} \]
   10. Step-by-step derivation

       1. unpow1/2 51.8%

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

       2. metadata-eval 51.8%

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

       3. pow-sqr 51.8%

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

       4. unpow-1 51.8%

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

       5. associate-/r/ 51.9%

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

       6. associate-*l/ 51.9%

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

       7. *-lft-identity 51.9%

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

       8. unpow-1 51.9%

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

       9. associate-/r/ 51.9%

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

       10. associate-*l/ 51.9%

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

       11. *-lft-identity 51.9%

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

       12. rem-sqrt-square 82.7%

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

   11. Simplified 82.7%

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

   if -1.89999999999999996e-115 < x < 2.4000000000000001e-85 or 1.9499999999999999e27 < x < 4.6000000000000001e79

   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%

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

      1. fabs-neg 100.0%

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

      2. mul-1-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. fabs-div 100.0%

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

      5. div-sub 100.0%

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

      6. *-inverses 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 83.6%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.9 \cdot 10^{-115} \lor \neg \left(x \leq 2.4 \cdot 10^{-85} \lor \neg \left(x \leq 1.95 \cdot 10^{+27}\right) \land x \leq 4.6 \cdot 10^{+79}\right):\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 59.0% accurate, 12.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{-89}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{-59}:\\ \;\;\;\;\left(1 + \frac{x}{y}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.45e-89) 1.0 (if (<= y 1.1e-59) (+ (+ 1.0 (/ x y)) -1.0) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.45e-89) {
		tmp = 1.0;
	} else if (y <= 1.1e-59) {
		tmp = (1.0 + (x / y)) + -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.45d-89)) then
        tmp = 1.0d0
    else if (y <= 1.1d-59) then
        tmp = (1.0d0 + (x / y)) + (-1.0d0)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.45e-89) {
		tmp = 1.0;
	} else if (y <= 1.1e-59) {
		tmp = (1.0 + (x / y)) + -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.45e-89:
		tmp = 1.0
	elif y <= 1.1e-59:
		tmp = (1.0 + (x / y)) + -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.45e-89)
		tmp = 1.0;
	elseif (y <= 1.1e-59)
		tmp = Float64(Float64(1.0 + Float64(x / y)) + -1.0);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.45e-89)
		tmp = 1.0;
	elseif (y <= 1.1e-59)
		tmp = (1.0 + (x / y)) + -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.45e-89], 1.0, If[LessEqual[y, 1.1e-59], N[(N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -1.44999999999999996e-89 or 1.0999999999999999e-59 < 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%

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

      1. fabs-neg 100.0%

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

      2. mul-1-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. fabs-div 100.0%

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

      5. div-sub 100.0%

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

      6. *-inverses 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 63.4%

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

   if -1.44999999999999996e-89 < y < 1.0999999999999999e-59

   1. Initial program 100.0%

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

      1. add-cbrt-cube 60.4%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{\left|x - y\right|}{\left|y\right|} \cdot \frac{\left|x - y\right|}{\left|y\right|}\right) \cdot \frac{\left|x - y\right|}{\left|y\right|}}} \]

      2. pow1/3 58.9%

         \[\leadsto \color{blue}{{\left(\left(\frac{\left|x - y\right|}{\left|y\right|} \cdot \frac{\left|x - y\right|}{\left|y\right|}\right) \cdot \frac{\left|x - y\right|}{\left|y\right|}\right)}^{0.3333333333333333}} \]

      3. pow3 58.9%

         \[\leadsto {\color{blue}{\left({\left(\frac{\left|x - y\right|}{\left|y\right|}\right)}^{3}\right)}}^{0.3333333333333333} \]

      4. clear-num 58.9%

         \[\leadsto {\left({\color{blue}{\left(\frac{1}{\frac{\left|y\right|}{\left|x - y\right|}}\right)}}^{3}\right)}^{0.3333333333333333} \]

      5. inv-pow 58.9%

         \[\leadsto {\left({\color{blue}{\left({\left(\frac{\left|y\right|}{\left|x - y\right|}\right)}^{-1}\right)}}^{3}\right)}^{0.3333333333333333} \]

      6. pow-pow 58.9%

         \[\leadsto {\color{blue}{\left({\left(\frac{\left|y\right|}{\left|x - y\right|}\right)}^{\left(-1 \cdot 3\right)}\right)}}^{0.3333333333333333} \]

      7. add-sqr-sqrt 23.9%

         \[\leadsto {\left({\left(\frac{\left|\color{blue}{\sqrt{y} \cdot \sqrt{y}}\right|}{\left|x - y\right|}\right)}^{\left(-1 \cdot 3\right)}\right)}^{0.3333333333333333} \]

      8. fabs-sqr 23.9%

         \[\leadsto {\left({\left(\frac{\color{blue}{\sqrt{y} \cdot \sqrt{y}}}{\left|x - y\right|}\right)}^{\left(-1 \cdot 3\right)}\right)}^{0.3333333333333333} \]

      9. add-sqr-sqrt 43.5%

         \[\leadsto {\left({\left(\frac{\color{blue}{y}}{\left|x - y\right|}\right)}^{\left(-1 \cdot 3\right)}\right)}^{0.3333333333333333} \]

      10. add-sqr-sqrt 15.9%

          \[\leadsto {\left({\left(\frac{y}{\left|\color{blue}{\sqrt{x - y} \cdot \sqrt{x - y}}\right|}\right)}^{\left(-1 \cdot 3\right)}\right)}^{0.3333333333333333} \]

      11. fabs-sqr 15.9%

          \[\leadsto {\left({\left(\frac{y}{\color{blue}{\sqrt{x - y} \cdot \sqrt{x - y}}}\right)}^{\left(-1 \cdot 3\right)}\right)}^{0.3333333333333333} \]

      12. add-sqr-sqrt 38.9%

          \[\leadsto {\left({\left(\frac{y}{\color{blue}{x - y}}\right)}^{\left(-1 \cdot 3\right)}\right)}^{0.3333333333333333} \]

      13. metadata-eval 38.9%

          \[\leadsto {\left({\left(\frac{y}{x - y}\right)}^{\color{blue}{-3}}\right)}^{0.3333333333333333} \]

   4. Applied egg-rr 38.9%

      \[\leadsto \color{blue}{{\left({\left(\frac{y}{x - y}\right)}^{-3}\right)}^{0.3333333333333333}} \]
   5. Step-by-step derivation

      1. unpow1/3 23.0%

         \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{y}{x - y}\right)}^{-3}}} \]

   6. Simplified 23.0%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{y}{x - y}\right)}^{-3}}} \]

   7. Taylor expanded in y around 0 23.1%

      \[\leadsto \sqrt[3]{{\color{blue}{\left(\frac{y}{x}\right)}}^{-3}} \]
   8. Step-by-step derivation

      1. expm1-log1p-u 22.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{{\left(\frac{y}{x}\right)}^{-3}}\right)\right)} \]

      2. expm1-undefine 22.4%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt[3]{{\left(\frac{y}{x}\right)}^{-3}}\right)} - 1} \]

      3. log1p-undefine 22.4%

         \[\leadsto e^{\color{blue}{\log \left(1 + \sqrt[3]{{\left(\frac{y}{x}\right)}^{-3}}\right)}} - 1 \]

      4. rem-exp-log 23.1%

         \[\leadsto \color{blue}{\left(1 + \sqrt[3]{{\left(\frac{y}{x}\right)}^{-3}}\right)} - 1 \]

      5. pow1/3 39.0%

         \[\leadsto \left(1 + \color{blue}{{\left({\left(\frac{y}{x}\right)}^{-3}\right)}^{0.3333333333333333}}\right) - 1 \]

      6. pow-pow 50.8%

         \[\leadsto \left(1 + \color{blue}{{\left(\frac{y}{x}\right)}^{\left(-3 \cdot 0.3333333333333333\right)}}\right) - 1 \]

      7. metadata-eval 50.8%

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

      8. inv-pow 50.8%

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

      9. clear-num 51.0%

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

   9. Applied egg-rr 51.0%

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

4. Final simplification 58.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{-89}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{-59}:\\ \;\;\;\;\left(1 + \frac{x}{y}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 25.9% accurate, 68.3× speedup

Math

\[\begin{array}{l} \\ \frac{x}{y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ x y))

C

double code(double x, double y) {
	return x / y;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x / y
end function

Java

public static double code(double x, double y) {
	return x / y;
}

Python

def code(x, y):
	return x / y

Julia

function code(x, y)
	return Float64(x / y)
end

MATLAB

function tmp = code(x, y)
	tmp = x / y;
end

Wolfram

code[x_, y_] := N[(x / y), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. add-cbrt-cube 71.9%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{\left|x - y\right|}{\left|y\right|} \cdot \frac{\left|x - y\right|}{\left|y\right|}\right) \cdot \frac{\left|x - y\right|}{\left|y\right|}}} \]

   2. pow1/3 70.6%

      \[\leadsto \color{blue}{{\left(\left(\frac{\left|x - y\right|}{\left|y\right|} \cdot \frac{\left|x - y\right|}{\left|y\right|}\right) \cdot \frac{\left|x - y\right|}{\left|y\right|}\right)}^{0.3333333333333333}} \]

   3. pow3 70.6%

      \[\leadsto {\color{blue}{\left({\left(\frac{\left|x - y\right|}{\left|y\right|}\right)}^{3}\right)}}^{0.3333333333333333} \]

   4. clear-num 70.6%

      \[\leadsto {\left({\color{blue}{\left(\frac{1}{\frac{\left|y\right|}{\left|x - y\right|}}\right)}}^{3}\right)}^{0.3333333333333333} \]

   5. inv-pow 70.6%

      \[\leadsto {\left({\color{blue}{\left({\left(\frac{\left|y\right|}{\left|x - y\right|}\right)}^{-1}\right)}}^{3}\right)}^{0.3333333333333333} \]

   6. pow-pow 70.6%

      \[\leadsto {\color{blue}{\left({\left(\frac{\left|y\right|}{\left|x - y\right|}\right)}^{\left(-1 \cdot 3\right)}\right)}}^{0.3333333333333333} \]

   7. add-sqr-sqrt 30.0%

      \[\leadsto {\left({\left(\frac{\left|\color{blue}{\sqrt{y} \cdot \sqrt{y}}\right|}{\left|x - y\right|}\right)}^{\left(-1 \cdot 3\right)}\right)}^{0.3333333333333333} \]

   8. fabs-sqr 30.0%

      \[\leadsto {\left({\left(\frac{\color{blue}{\sqrt{y} \cdot \sqrt{y}}}{\left|x - y\right|}\right)}^{\left(-1 \cdot 3\right)}\right)}^{0.3333333333333333} \]

   9. add-sqr-sqrt 38.5%

      \[\leadsto {\left({\left(\frac{\color{blue}{y}}{\left|x - y\right|}\right)}^{\left(-1 \cdot 3\right)}\right)}^{0.3333333333333333} \]

   10. add-sqr-sqrt 9.0%

       \[\leadsto {\left({\left(\frac{y}{\left|\color{blue}{\sqrt{x - y} \cdot \sqrt{x - y}}\right|}\right)}^{\left(-1 \cdot 3\right)}\right)}^{0.3333333333333333} \]

   11. fabs-sqr 9.0%

       \[\leadsto {\left({\left(\frac{y}{\color{blue}{\sqrt{x - y} \cdot \sqrt{x - y}}}\right)}^{\left(-1 \cdot 3\right)}\right)}^{0.3333333333333333} \]

   12. add-sqr-sqrt 22.8%

       \[\leadsto {\left({\left(\frac{y}{\color{blue}{x - y}}\right)}^{\left(-1 \cdot 3\right)}\right)}^{0.3333333333333333} \]

   13. metadata-eval 22.8%

       \[\leadsto {\left({\left(\frac{y}{x - y}\right)}^{\color{blue}{-3}}\right)}^{0.3333333333333333} \]

4. Applied egg-rr 22.8%

   \[\leadsto \color{blue}{{\left({\left(\frac{y}{x - y}\right)}^{-3}\right)}^{0.3333333333333333}} \]
5. Step-by-step derivation

   1. unpow1/3 16.9%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{y}{x - y}\right)}^{-3}}} \]

6. Simplified 16.9%

   \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{y}{x - y}\right)}^{-3}}} \]

7. Taylor expanded in y around 0 33.1%

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

8. Final simplification 33.1%

   \[\leadsto \frac{x}{y} \]
9. Add Preprocessing

Alternative 5: 1.3% 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. add-cbrt-cube 71.9%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{\left|x - y\right|}{\left|y\right|} \cdot \frac{\left|x - y\right|}{\left|y\right|}\right) \cdot \frac{\left|x - y\right|}{\left|y\right|}}} \]

   2. pow1/3 70.6%

      \[\leadsto \color{blue}{{\left(\left(\frac{\left|x - y\right|}{\left|y\right|} \cdot \frac{\left|x - y\right|}{\left|y\right|}\right) \cdot \frac{\left|x - y\right|}{\left|y\right|}\right)}^{0.3333333333333333}} \]

   3. pow3 70.6%

      \[\leadsto {\color{blue}{\left({\left(\frac{\left|x - y\right|}{\left|y\right|}\right)}^{3}\right)}}^{0.3333333333333333} \]

   4. clear-num 70.6%

      \[\leadsto {\left({\color{blue}{\left(\frac{1}{\frac{\left|y\right|}{\left|x - y\right|}}\right)}}^{3}\right)}^{0.3333333333333333} \]

   5. inv-pow 70.6%

      \[\leadsto {\left({\color{blue}{\left({\left(\frac{\left|y\right|}{\left|x - y\right|}\right)}^{-1}\right)}}^{3}\right)}^{0.3333333333333333} \]

   6. pow-pow 70.6%

      \[\leadsto {\color{blue}{\left({\left(\frac{\left|y\right|}{\left|x - y\right|}\right)}^{\left(-1 \cdot 3\right)}\right)}}^{0.3333333333333333} \]

   7. add-sqr-sqrt 30.0%

      \[\leadsto {\left({\left(\frac{\left|\color{blue}{\sqrt{y} \cdot \sqrt{y}}\right|}{\left|x - y\right|}\right)}^{\left(-1 \cdot 3\right)}\right)}^{0.3333333333333333} \]

   8. fabs-sqr 30.0%

      \[\leadsto {\left({\left(\frac{\color{blue}{\sqrt{y} \cdot \sqrt{y}}}{\left|x - y\right|}\right)}^{\left(-1 \cdot 3\right)}\right)}^{0.3333333333333333} \]

   9. add-sqr-sqrt 38.5%

      \[\leadsto {\left({\left(\frac{\color{blue}{y}}{\left|x - y\right|}\right)}^{\left(-1 \cdot 3\right)}\right)}^{0.3333333333333333} \]

   10. add-sqr-sqrt 9.0%

       \[\leadsto {\left({\left(\frac{y}{\left|\color{blue}{\sqrt{x - y} \cdot \sqrt{x - y}}\right|}\right)}^{\left(-1 \cdot 3\right)}\right)}^{0.3333333333333333} \]

   11. fabs-sqr 9.0%

       \[\leadsto {\left({\left(\frac{y}{\color{blue}{\sqrt{x - y} \cdot \sqrt{x - y}}}\right)}^{\left(-1 \cdot 3\right)}\right)}^{0.3333333333333333} \]

   12. add-sqr-sqrt 22.8%

       \[\leadsto {\left({\left(\frac{y}{\color{blue}{x - y}}\right)}^{\left(-1 \cdot 3\right)}\right)}^{0.3333333333333333} \]

   13. metadata-eval 22.8%

       \[\leadsto {\left({\left(\frac{y}{x - y}\right)}^{\color{blue}{-3}}\right)}^{0.3333333333333333} \]

4. Applied egg-rr 22.8%

   \[\leadsto \color{blue}{{\left({\left(\frac{y}{x - y}\right)}^{-3}\right)}^{0.3333333333333333}} \]
5. Step-by-step derivation

   1. unpow1/3 16.9%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{y}{x - y}\right)}^{-3}}} \]

6. Simplified 16.9%

   \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{y}{x - y}\right)}^{-3}}} \]

7. Taylor expanded in y around inf 1.2%

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

8. Final simplification 1.2%

   \[\leadsto -1 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024055 
(FPCore (x y)
  :name "Numeric.LinearAlgebra.Util:formatSparse from hmatrix-0.16.1.5"
  :precision binary64
  (/ (fabs (- x y)) (fabs y)))