Numeric.LinearAlgebra.Util:formatSparse from hmatrix-0.16.1.5

Percentage Accurate: 100.0% → 100.0%

Time: 5.1s

Alternatives: 6

Speedup: 2.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

public static double code(double x, double y) {
	return Math.abs((x - y)) / Math.abs(y);
}

Python

def code(x, y):
	return math.fabs((x - y)) / math.fabs(y)

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

public static double code(double x, double y) {
	return Math.abs((x - y)) / Math.abs(y);
}

Python

def code(x, y):
	return math.fabs((x - y)) / math.fabs(y)

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 (fabs (+ (/ x y) -1.0)))

C

double code(double x, double y) {
	return fabs(((x / y) + -1.0));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = abs(((x / y) + (-1.0d0)))
end function

Java

public static double code(double x, double y) {
	return Math.abs(((x / y) + -1.0));
}

Python

def code(x, y):
	return math.fabs(((x / y) + -1.0))

Julia

function code(x, y)
	return abs(Float64(Float64(x / y) + -1.0))
end

MATLAB

function tmp = code(x, y)
	tmp = abs(((x / y) + -1.0));
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. div-fabs 100.0%

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

   2. div-sub 100.0%

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

   3. pow1 100.0%

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

   4. pow1 100.0%

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

   5. pow-div 100.0%

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

   6. metadata-eval 100.0%

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

   7. metadata-eval 100.0%

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

4. Applied egg-rr 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 58.6% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{y}\\ \mathbf{if}\;y \leq -2.8 \cdot 10^{-71}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -1.45 \cdot 10^{-99}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -4.5 \cdot 10^{-122}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -9.8 \cdot 10^{-287}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 1.95 \cdot 10^{-172}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- x) y)))
   (if (<= y -2.8e-71)
     1.0
     (if (<= y -1.45e-99)
       t_0
       (if (<= y -4.5e-122)
         1.0
         (if (<= y -9.8e-287) (/ x y) (if (<= y 1.95e-172) t_0 1.0)))))))

C

double code(double x, double y) {
	double t_0 = -x / y;
	double tmp;
	if (y <= -2.8e-71) {
		tmp = 1.0;
	} else if (y <= -1.45e-99) {
		tmp = t_0;
	} else if (y <= -4.5e-122) {
		tmp = 1.0;
	} else if (y <= -9.8e-287) {
		tmp = x / y;
	} else if (y <= 1.95e-172) {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = -x / y
    if (y <= (-2.8d-71)) then
        tmp = 1.0d0
    else if (y <= (-1.45d-99)) then
        tmp = t_0
    else if (y <= (-4.5d-122)) then
        tmp = 1.0d0
    else if (y <= (-9.8d-287)) then
        tmp = x / y
    else if (y <= 1.95d-172) then
        tmp = t_0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = -x / y;
	double tmp;
	if (y <= -2.8e-71) {
		tmp = 1.0;
	} else if (y <= -1.45e-99) {
		tmp = t_0;
	} else if (y <= -4.5e-122) {
		tmp = 1.0;
	} else if (y <= -9.8e-287) {
		tmp = x / y;
	} else if (y <= 1.95e-172) {
		tmp = t_0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = -x / y
	tmp = 0
	if y <= -2.8e-71:
		tmp = 1.0
	elif y <= -1.45e-99:
		tmp = t_0
	elif y <= -4.5e-122:
		tmp = 1.0
	elif y <= -9.8e-287:
		tmp = x / y
	elif y <= 1.95e-172:
		tmp = t_0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(-x) / y)
	tmp = 0.0
	if (y <= -2.8e-71)
		tmp = 1.0;
	elseif (y <= -1.45e-99)
		tmp = t_0;
	elseif (y <= -4.5e-122)
		tmp = 1.0;
	elseif (y <= -9.8e-287)
		tmp = Float64(x / y);
	elseif (y <= 1.95e-172)
		tmp = t_0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = -x / y;
	tmp = 0.0;
	if (y <= -2.8e-71)
		tmp = 1.0;
	elseif (y <= -1.45e-99)
		tmp = t_0;
	elseif (y <= -4.5e-122)
		tmp = 1.0;
	elseif (y <= -9.8e-287)
		tmp = x / y;
	elseif (y <= 1.95e-172)
		tmp = t_0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[((-x) / y), $MachinePrecision]}, If[LessEqual[y, -2.8e-71], 1.0, If[LessEqual[y, -1.45e-99], t$95$0, If[LessEqual[y, -4.5e-122], 1.0, If[LessEqual[y, -9.8e-287], N[(x / y), $MachinePrecision], If[LessEqual[y, 1.95e-172], t$95$0, 1.0]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.8e-71 or -1.44999999999999993e-99 < y < -4.4999999999999998e-122 or 1.94999999999999986e-172 < y

   1. Initial program 100.0%

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

      1. clear-num 99.8%

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

      2. add-sqr-sqrt 52.3%

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

      3. fabs-sqr 52.3%

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

      4. add-sqr-sqrt 53.3%

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

      5. add-sqr-sqrt 10.3%

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

      6. fabs-sqr 10.3%

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

      7. add-sqr-sqrt 17.4%

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

      8. associate-/r/ 17.4%

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

   4. Applied egg-rr 17.4%

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

      1. associate-*l/ 17.5%

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

      2. *-un-lft-identity 17.5%

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

      3. sub-neg 17.5%

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

      4. add-sqr-sqrt 7.1%

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

      5. sqrt-unprod 31.4%

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

      6. sqr-neg 31.4%

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

      7. sqrt-unprod 41.4%

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

      8. add-sqr-sqrt 82.5%

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

   6. Applied egg-rr 82.5%

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

   7. Taylor expanded in x around 0 68.6%

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

   if -2.8e-71 < y < -1.44999999999999993e-99 or -9.8000000000000002e-287 < y < 1.94999999999999986e-172

   1. Initial program 100.0%

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

      1. clear-num 99.7%

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

      2. add-sqr-sqrt 59.1%

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

      3. fabs-sqr 59.1%

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

      4. add-sqr-sqrt 59.5%

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

      5. add-sqr-sqrt 16.6%

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

      6. fabs-sqr 16.6%

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

      7. add-sqr-sqrt 28.8%

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

      8. associate-/r/ 28.9%

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

   4. Applied egg-rr 28.9%

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

      1. associate-*l/ 28.9%

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

      2. *-un-lft-identity 28.9%

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

      3. frac-2neg 28.9%

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

      4. sub-neg 28.9%

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

      5. distribute-neg-in 28.9%

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

      6. remove-double-neg 28.9%

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

      7. add-sqr-sqrt 12.0%

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

      8. sqrt-unprod 20.7%

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

      9. sqr-neg 20.7%

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

      10. sqrt-unprod 42.8%

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

      11. add-sqr-sqrt 71.6%

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

   6. Applied egg-rr 71.6%

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

   7. Taylor expanded in x around inf 63.7%

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

      1. associate-*r/ 63.7%

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

      2. mul-1-neg 63.7%

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

   9. Simplified 63.7%

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

   if -4.4999999999999998e-122 < y < -9.8000000000000002e-287

   1. Initial program 99.9%

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

      1. clear-num 99.7%

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

      2. add-sqr-sqrt 0.0%

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

      3. fabs-sqr 0.0%

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

      4. add-sqr-sqrt 0.7%

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

      5. add-sqr-sqrt 0.4%

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

      6. fabs-sqr 0.4%

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

      7. add-sqr-sqrt 53.5%

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

      8. associate-/r/ 53.6%

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

   4. Applied egg-rr 53.6%

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

   5. Taylor expanded in y around 0 54.1%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{-71}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -1.45 \cdot 10^{-99}:\\ \;\;\;\;\frac{-x}{y}\\ \mathbf{elif}\;y \leq -4.5 \cdot 10^{-122}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -9.8 \cdot 10^{-287}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 1.95 \cdot 10^{-172}:\\ \;\;\;\;\frac{-x}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 58.4% accurate, 15.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.15 \cdot 10^{-120}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{-219}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.15e-120) 1.0 (if (<= y 8.2e-219) (/ x y) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.15e-120) {
		tmp = 1.0;
	} else if (y <= 8.2e-219) {
		tmp = x / y;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-2.15d-120)) then
        tmp = 1.0d0
    else if (y <= 8.2d-219) then
        tmp = x / y
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2.15e-120) {
		tmp = 1.0;
	} else if (y <= 8.2e-219) {
		tmp = x / y;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.15e-120:
		tmp = 1.0
	elif y <= 8.2e-219:
		tmp = x / y
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.15e-120)
		tmp = 1.0;
	elseif (y <= 8.2e-219)
		tmp = Float64(x / y);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.15e-120)
		tmp = 1.0;
	elseif (y <= 8.2e-219)
		tmp = x / y;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.15e-120], 1.0, If[LessEqual[y, 8.2e-219], N[(x / y), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -2.14999999999999991e-120 or 8.2e-219 < y

   1. Initial program 100.0%

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

      1. clear-num 99.8%

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

      2. add-sqr-sqrt 53.0%

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

      3. fabs-sqr 53.0%

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

      4. add-sqr-sqrt 54.0%

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

      5. add-sqr-sqrt 10.5%

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

      6. fabs-sqr 10.5%

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

      7. add-sqr-sqrt 17.6%

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

      8. associate-/r/ 17.6%

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

   4. Applied egg-rr 17.6%

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

      1. associate-*l/ 17.7%

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

      2. *-un-lft-identity 17.7%

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

      3. sub-neg 17.7%

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

      4. add-sqr-sqrt 7.1%

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

      5. sqrt-unprod 30.5%

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

      6. sqr-neg 30.5%

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

      7. sqrt-unprod 40.6%

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

      8. add-sqr-sqrt 78.9%

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

   6. Applied egg-rr 78.9%

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

   7. Taylor expanded in x around 0 64.9%

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

   if -2.14999999999999991e-120 < y < 8.2e-219

   1. Initial program 100.0%

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

      1. clear-num 99.8%

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

      2. add-sqr-sqrt 26.6%

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

      3. fabs-sqr 26.6%

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

      4. add-sqr-sqrt 27.2%

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

      5. add-sqr-sqrt 9.1%

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

      6. fabs-sqr 9.1%

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

      7. add-sqr-sqrt 44.8%

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

      8. associate-/r/ 44.8%

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

   4. Applied egg-rr 44.8%

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

   5. Taylor expanded in y around 0 44.3%

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

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.15 \cdot 10^{-120}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{-219}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.5% accurate, 41.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 (- 1.0 (/ x y)))

C

double code(double x, double y) {
	return 1.0 - (x / y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 - (x / y)
end function

Java

public static double code(double x, double y) {
	return 1.0 - (x / y);
}

Python

def code(x, y):
	return 1.0 - (x / y)

Julia

function code(x, y)
	return Float64(1.0 - Float64(x / y))
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 - (x / y);
end

Wolfram

code[x_, y_] := N[(1.0 - N[(x / y), $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. clear-num 99.8%

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

   2. add-sqr-sqrt 47.3%

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

   3. fabs-sqr 47.3%

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

   4. add-sqr-sqrt 48.1%

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

   5. add-sqr-sqrt 10.2%

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

   6. fabs-sqr 10.2%

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

   7. add-sqr-sqrt 23.5%

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

   8. associate-/r/ 23.5%

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

4. Applied egg-rr 23.5%

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

   1. associate-*l/ 23.6%

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

   2. *-un-lft-identity 23.6%

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

   3. frac-2neg 23.6%

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

   4. sub-neg 23.6%

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

   5. distribute-neg-in 23.6%

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

   6. remove-double-neg 23.6%

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

   7. add-sqr-sqrt 13.3%

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

   8. sqrt-unprod 31.4%

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

   9. sqr-neg 31.4%

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

   10. sqrt-unprod 37.7%

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

   11. add-sqr-sqrt 77.5%

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

6. Applied egg-rr 77.5%

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

7. Taylor expanded in x around 0 77.5%

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

   1. mul-1-neg 77.5%

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

   2. unsub-neg 77.5%

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

9. Simplified 77.5%

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

10. Final simplification 77.5%

    \[\leadsto 1 - \frac{x}{y} \]
11. 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. clear-num 99.8%

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

   2. add-sqr-sqrt 47.3%

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

   3. fabs-sqr 47.3%

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

   4. add-sqr-sqrt 48.1%

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

   5. add-sqr-sqrt 10.2%

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

   6. fabs-sqr 10.2%

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

   7. add-sqr-sqrt 23.5%

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

   8. associate-/r/ 23.5%

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

4. Applied egg-rr 23.5%

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

5. Taylor expanded in y around inf 1.3%

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

6. Final simplification 1.3%

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

Alternative 6: 51.7% 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. clear-num 99.8%

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

   2. add-sqr-sqrt 47.3%

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

   3. fabs-sqr 47.3%

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

   4. add-sqr-sqrt 48.1%

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

   5. add-sqr-sqrt 10.2%

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

   6. fabs-sqr 10.2%

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

   7. add-sqr-sqrt 23.5%

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

   8. associate-/r/ 23.5%

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

4. Applied egg-rr 23.5%

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

   1. associate-*l/ 23.6%

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

   2. *-un-lft-identity 23.6%

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

   3. sub-neg 23.6%

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

   4. add-sqr-sqrt 13.4%

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

   5. sqrt-unprod 33.5%

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

   6. sqr-neg 33.5%

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

   7. sqrt-unprod 33.5%

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

   8. add-sqr-sqrt 72.5%

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

6. Applied egg-rr 72.5%

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

7. Taylor expanded in x around 0 52.9%

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

8. Final simplification 52.9%

   \[\leadsto 1 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024031 
(FPCore (x y)
  :name "Numeric.LinearAlgebra.Util:formatSparse from hmatrix-0.16.1.5"
  :precision binary64
  (/ (fabs (- x y)) (fabs y)))