Numeric.LinearAlgebra.Util:formatSparse from hmatrix-0.16.1.5

Percentage Accurate: 100.0% → 100.0%

Time: 4.3s

Alternatives: 8

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: 57.2% accurate, 7.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{y}\\ \mathbf{if}\;x \leq -2.65 \cdot 10^{+144}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -9.5 \cdot 10^{+46}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 4.3 \cdot 10^{+36}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.62 \cdot 10^{+59}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{+136}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- x) y)))
   (if (<= x -2.65e+144)
     (/ x y)
     (if (<= x -9.5e+46)
       t_0
       (if (<= x 4.3e+36)
         1.0
         (if (<= x 1.62e+59) t_0 (if (<= x 1.2e+136) 1.0 (/ x y))))))))

C

double code(double x, double y) {
	double t_0 = -x / y;
	double tmp;
	if (x <= -2.65e+144) {
		tmp = x / y;
	} else if (x <= -9.5e+46) {
		tmp = t_0;
	} else if (x <= 4.3e+36) {
		tmp = 1.0;
	} else if (x <= 1.62e+59) {
		tmp = t_0;
	} else if (x <= 1.2e+136) {
		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) :: t_0
    real(8) :: tmp
    t_0 = -x / y
    if (x <= (-2.65d+144)) then
        tmp = x / y
    else if (x <= (-9.5d+46)) then
        tmp = t_0
    else if (x <= 4.3d+36) then
        tmp = 1.0d0
    else if (x <= 1.62d+59) then
        tmp = t_0
    else if (x <= 1.2d+136) then
        tmp = 1.0d0
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = -x / y;
	double tmp;
	if (x <= -2.65e+144) {
		tmp = x / y;
	} else if (x <= -9.5e+46) {
		tmp = t_0;
	} else if (x <= 4.3e+36) {
		tmp = 1.0;
	} else if (x <= 1.62e+59) {
		tmp = t_0;
	} else if (x <= 1.2e+136) {
		tmp = 1.0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = -x / y
	tmp = 0
	if x <= -2.65e+144:
		tmp = x / y
	elif x <= -9.5e+46:
		tmp = t_0
	elif x <= 4.3e+36:
		tmp = 1.0
	elif x <= 1.62e+59:
		tmp = t_0
	elif x <= 1.2e+136:
		tmp = 1.0
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(-x) / y)
	tmp = 0.0
	if (x <= -2.65e+144)
		tmp = Float64(x / y);
	elseif (x <= -9.5e+46)
		tmp = t_0;
	elseif (x <= 4.3e+36)
		tmp = 1.0;
	elseif (x <= 1.62e+59)
		tmp = t_0;
	elseif (x <= 1.2e+136)
		tmp = 1.0;
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = -x / y;
	tmp = 0.0;
	if (x <= -2.65e+144)
		tmp = x / y;
	elseif (x <= -9.5e+46)
		tmp = t_0;
	elseif (x <= 4.3e+36)
		tmp = 1.0;
	elseif (x <= 1.62e+59)
		tmp = t_0;
	elseif (x <= 1.2e+136)
		tmp = 1.0;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[((-x) / y), $MachinePrecision]}, If[LessEqual[x, -2.65e+144], N[(x / y), $MachinePrecision], If[LessEqual[x, -9.5e+46], t$95$0, If[LessEqual[x, 4.3e+36], 1.0, If[LessEqual[x, 1.62e+59], t$95$0, If[LessEqual[x, 1.2e+136], 1.0, N[(x / y), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.6499999999999998e144 or 1.2e136 < x

   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 45.3%

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

      3. fabs-sqr 45.3%

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

      4. add-sqr-sqrt 45.7%

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

      5. add-sqr-sqrt 26.5%

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

      6. fabs-sqr 26.5%

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

      7. add-sqr-sqrt 61.9%

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

      8. associate-/r/ 62.0%

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

   4. Applied egg-rr 62.0%

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

   5. Taylor expanded in y around 0 62.0%

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

   if -2.6499999999999998e144 < x < -9.5000000000000008e46 or 4.30000000000000005e36 < x < 1.6200000000000001e59

   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 55.1%

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

      3. fabs-sqr 55.1%

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

      4. add-sqr-sqrt 55.7%

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

      5. add-sqr-sqrt 3.9%

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

      6. fabs-sqr 3.9%

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

      7. add-sqr-sqrt 23.0%

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

      8. associate-/r/ 22.8%

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

   4. Applied egg-rr 22.8%

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

      1. associate-*l/ 23.0%

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

      2. *-un-lft-identity 23.0%

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

      3. frac-2neg 23.0%

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

      4. sub-neg 23.0%

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

      5. distribute-neg-in 23.0%

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

      6. remove-double-neg 23.0%

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

      7. add-sqr-sqrt 18.7%

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

      8. sqrt-unprod 56.9%

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

      9. sqr-neg 56.9%

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

      10. sqrt-unprod 51.4%

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

      11. add-sqr-sqrt 77.9%

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

   6. Applied egg-rr 77.9%

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

   7. Taylor expanded in x around inf 60.4%

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

      1. associate-*r/ 60.4%

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

      2. mul-1-neg 60.4%

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

   9. Simplified 60.4%

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

   if -9.5000000000000008e46 < x < 4.30000000000000005e36 or 1.6200000000000001e59 < x < 1.2e136

   1. Initial program 100.0%

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

      1. clear-num 99.9%

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

      2. add-sqr-sqrt 46.1%

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

      3. fabs-sqr 46.1%

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

      4. add-sqr-sqrt 47.2%

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

      5. add-sqr-sqrt 9.3%

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

      6. fabs-sqr 9.3%

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

      7. add-sqr-sqrt 14.8%

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

      8. associate-/r/ 14.9%

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

   4. Applied egg-rr 14.9%

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

      1. associate-*l/ 14.9%

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

      2. *-un-lft-identity 14.9%

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

      3. sub-neg 14.9%

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

      4. add-sqr-sqrt 5.6%

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

      5. sqrt-unprod 28.7%

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

      6. sqr-neg 28.7%

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

      7. sqrt-unprod 37.9%

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

      8. add-sqr-sqrt 81.5%

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

   6. Applied egg-rr 81.5%

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

   7. Taylor expanded in x around 0 69.6%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.65 \cdot 10^{+144}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -9.5 \cdot 10^{+46}:\\ \;\;\;\;\frac{-x}{y}\\ \mathbf{elif}\;x \leq 4.3 \cdot 10^{+36}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.62 \cdot 10^{+59}:\\ \;\;\;\;\frac{-x}{y}\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{+136}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.2% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{+104} \lor \neg \left(x \leq -4.5 \cdot 10^{+49} \lor \neg \left(x \leq -3 \cdot 10^{-56}\right) \land x \leq 2.2 \cdot 10^{+49}\right):\\ \;\;\;\;\frac{x + y}{y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.5e+104)
         (not (or (<= x -4.5e+49) (and (not (<= x -3e-56)) (<= x 2.2e+49)))))
   (/ (+ x y) y)
   (- 1.0 (/ x y))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.5e+104) || !((x <= -4.5e+49) || (!(x <= -3e-56) && (x <= 2.2e+49)))) {
		tmp = (x + y) / y;
	} else {
		tmp = 1.0 - (x / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-1.5d+104)) .or. (.not. (x <= (-4.5d+49)) .or. (.not. (x <= (-3d-56))) .and. (x <= 2.2d+49))) then
        tmp = (x + y) / y
    else
        tmp = 1.0d0 - (x / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -1.5e+104) || !((x <= -4.5e+49) || (!(x <= -3e-56) && (x <= 2.2e+49)))) {
		tmp = (x + y) / y;
	} else {
		tmp = 1.0 - (x / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.5e+104) or not ((x <= -4.5e+49) or (not (x <= -3e-56) and (x <= 2.2e+49))):
		tmp = (x + y) / y
	else:
		tmp = 1.0 - (x / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.5e+104) || !((x <= -4.5e+49) || (!(x <= -3e-56) && (x <= 2.2e+49))))
		tmp = Float64(Float64(x + y) / y);
	else
		tmp = Float64(1.0 - Float64(x / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -1.5e+104) || ~(((x <= -4.5e+49) || (~((x <= -3e-56)) && (x <= 2.2e+49)))))
		tmp = (x + y) / y;
	else
		tmp = 1.0 - (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.5e+104], N[Not[Or[LessEqual[x, -4.5e+49], And[N[Not[LessEqual[x, -3e-56]], $MachinePrecision], LessEqual[x, 2.2e+49]]]], $MachinePrecision]], N[(N[(x + y), $MachinePrecision] / y), $MachinePrecision], N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.49999999999999984e104 or -4.49999999999999982e49 < x < -2.99999999999999989e-56 or 2.2000000000000001e49 < x

   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 42.4%

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

      3. fabs-sqr 42.4%

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

      4. add-sqr-sqrt 43.0%

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

      5. add-sqr-sqrt 20.7%

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

      6. fabs-sqr 20.7%

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

      7. add-sqr-sqrt 49.2%

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

      8. associate-/r/ 49.2%

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

   4. Applied egg-rr 49.2%

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

      1. associate-*l/ 49.3%

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

      2. *-un-lft-identity 49.3%

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

      3. sub-neg 49.3%

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

      4. add-sqr-sqrt 28.6%

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

      5. sqrt-unprod 48.3%

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

      6. sqr-neg 48.3%

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

      7. sqrt-unprod 31.3%

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

      8. add-sqr-sqrt 79.2%

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

   6. Applied egg-rr 79.2%

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

   if -1.49999999999999984e104 < x < -4.49999999999999982e49 or -2.99999999999999989e-56 < x < 2.2000000000000001e49

   1. Initial program 100.0%

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

      1. clear-num 99.9%

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

      2. add-sqr-sqrt 50.6%

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

      3. fabs-sqr 50.6%

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

      4. add-sqr-sqrt 51.7%

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

      5. add-sqr-sqrt 7.0%

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

      6. fabs-sqr 7.0%

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

      7. add-sqr-sqrt 10.6%

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

      8. associate-/r/ 10.6%

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

   4. Applied egg-rr 10.6%

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

      1. associate-*l/ 10.6%

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

      2. *-un-lft-identity 10.6%

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

      3. frac-2neg 10.6%

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

      4. sub-neg 10.6%

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

      5. distribute-neg-in 10.6%

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

      6. remove-double-neg 10.6%

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

      7. add-sqr-sqrt 3.5%

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

      8. sqrt-unprod 24.7%

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

      9. sqr-neg 24.7%

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

      10. sqrt-unprod 44.2%

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

      11. add-sqr-sqrt 90.7%

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

   6. Applied egg-rr 90.7%

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

   7. Taylor expanded in x around 0 90.7%

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

      1. mul-1-neg 90.7%

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

      2. unsub-neg 90.7%

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

   9. Simplified 90.7%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{+104} \lor \neg \left(x \leq -4.5 \cdot 10^{+49} \lor \neg \left(x \leq -3 \cdot 10^{-56}\right) \land x \leq 2.2 \cdot 10^{+49}\right):\\ \;\;\;\;\frac{x + y}{y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 70.6% accurate, 13.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{+144} \lor \neg \left(x \leq 1.15 \cdot 10^{+137}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.1e+144) (not (<= x 1.15e+137))) (/ x y) (- 1.0 (/ x y))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.1e+144) || !(x <= 1.15e+137)) {
		tmp = x / y;
	} else {
		tmp = 1.0 - (x / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-1.1d+144)) .or. (.not. (x <= 1.15d+137))) then
        tmp = x / y
    else
        tmp = 1.0d0 - (x / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -1.1e+144) || !(x <= 1.15e+137)) {
		tmp = x / y;
	} else {
		tmp = 1.0 - (x / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.1e+144) or not (x <= 1.15e+137):
		tmp = x / y
	else:
		tmp = 1.0 - (x / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.1e+144) || !(x <= 1.15e+137))
		tmp = Float64(x / y);
	else
		tmp = Float64(1.0 - Float64(x / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -1.1e+144) || ~((x <= 1.15e+137)))
		tmp = x / y;
	else
		tmp = 1.0 - (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.1e+144], N[Not[LessEqual[x, 1.15e+137]], $MachinePrecision]], N[(x / y), $MachinePrecision], N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.09999999999999994e144 or 1.15e137 < x

   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 45.3%

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

      3. fabs-sqr 45.3%

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

      4. add-sqr-sqrt 45.7%

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

      5. add-sqr-sqrt 26.5%

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

      6. fabs-sqr 26.5%

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

      7. add-sqr-sqrt 61.9%

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

      8. associate-/r/ 62.0%

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

   4. Applied egg-rr 62.0%

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

   5. Taylor expanded in y around 0 62.0%

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

   if -1.09999999999999994e144 < x < 1.15e137

   1. Initial program 100.0%

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

      1. clear-num 99.9%

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

      2. add-sqr-sqrt 47.4%

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

      3. fabs-sqr 47.4%

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

      4. add-sqr-sqrt 48.4%

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

      5. add-sqr-sqrt 8.5%

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

      6. fabs-sqr 8.5%

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

      7. add-sqr-sqrt 16.0%

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

      8. associate-/r/ 16.0%

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

   4. Applied egg-rr 16.0%

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

      1. associate-*l/ 16.1%

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

      2. *-un-lft-identity 16.1%

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

      3. frac-2neg 16.1%

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

      4. sub-neg 16.1%

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

      5. distribute-neg-in 16.1%

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

      6. remove-double-neg 16.1%

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

      7. add-sqr-sqrt 7.5%

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

      8. sqrt-unprod 23.7%

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

      9. sqr-neg 23.7%

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

      10. sqrt-unprod 39.6%

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

      11. add-sqr-sqrt 85.2%

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

   6. Applied egg-rr 85.2%

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

   7. Taylor expanded in x around 0 85.2%

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

      1. mul-1-neg 85.2%

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

      2. unsub-neg 85.2%

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

   9. Simplified 85.2%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{+144} \lor \neg \left(x \leq 1.15 \cdot 10^{+137}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 70.5% accurate, 13.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{+144}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{+136}:\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.3e+144)
   (/ x y)
   (if (<= x 3.3e+136) (- 1.0 (/ x y)) (+ (/ x y) -1.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.3e+144) {
		tmp = x / y;
	} else if (x <= 3.3e+136) {
		tmp = 1.0 - (x / y);
	} else {
		tmp = (x / y) + -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 <= (-2.3d+144)) then
        tmp = x / y
    else if (x <= 3.3d+136) then
        tmp = 1.0d0 - (x / y)
    else
        tmp = (x / y) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -2.3e+144) {
		tmp = x / y;
	} else if (x <= 3.3e+136) {
		tmp = 1.0 - (x / y);
	} else {
		tmp = (x / y) + -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.3e+144:
		tmp = x / y
	elif x <= 3.3e+136:
		tmp = 1.0 - (x / y)
	else:
		tmp = (x / y) + -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.3e+144)
		tmp = Float64(x / y);
	elseif (x <= 3.3e+136)
		tmp = Float64(1.0 - Float64(x / y));
	else
		tmp = Float64(Float64(x / y) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.3e+144)
		tmp = x / y;
	elseif (x <= 3.3e+136)
		tmp = 1.0 - (x / y);
	else
		tmp = (x / y) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.3e+144], N[(x / y), $MachinePrecision], If[LessEqual[x, 3.3e+136], N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.3000000000000001e144

   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 27.4%

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

      3. fabs-sqr 27.4%

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

      4. add-sqr-sqrt 27.9%

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

      5. add-sqr-sqrt 0.2%

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

      6. fabs-sqr 0.2%

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

      7. add-sqr-sqrt 60.3%

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

      8. associate-/r/ 60.4%

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

   4. Applied egg-rr 60.4%

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

   5. Taylor expanded in y around 0 61.2%

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

   if -2.3000000000000001e144 < x < 3.29999999999999992e136

   1. Initial program 100.0%

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

      1. clear-num 99.9%

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

      2. add-sqr-sqrt 47.4%

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

      3. fabs-sqr 47.4%

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

      4. add-sqr-sqrt 48.4%

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

      5. add-sqr-sqrt 8.5%

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

      6. fabs-sqr 8.5%

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

      7. add-sqr-sqrt 16.0%

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

      8. associate-/r/ 16.0%

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

   4. Applied egg-rr 16.0%

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

      1. associate-*l/ 16.1%

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

      2. *-un-lft-identity 16.1%

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

      3. frac-2neg 16.1%

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

      4. sub-neg 16.1%

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

      5. distribute-neg-in 16.1%

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

      6. remove-double-neg 16.1%

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

      7. add-sqr-sqrt 7.5%

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

      8. sqrt-unprod 23.7%

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

      9. sqr-neg 23.7%

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

      10. sqrt-unprod 39.6%

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

      11. add-sqr-sqrt 85.2%

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

   6. Applied egg-rr 85.2%

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

   7. Taylor expanded in x around 0 85.2%

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

      1. mul-1-neg 85.2%

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

      2. unsub-neg 85.2%

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

   9. Simplified 85.2%

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

   if 3.29999999999999992e136 < x

   1. Initial program 100.0%

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

      1. add-sqr-sqrt 92.3%

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

      2. fabs-sqr 92.3%

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

      3. add-sqr-sqrt 63.9%

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

      4. fabs-sqr 63.9%

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

      5. add-sqr-sqrt 64.2%

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

      6. add-sqr-sqrt 64.6%

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

      7. div-sub 64.6%

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

      8. pow1 64.6%

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

      9. pow1 64.6%

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

      10. pow-div 64.6%

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

      11. metadata-eval 64.6%

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

      12. metadata-eval 64.6%

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

   4. Applied egg-rr 64.6%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{+144}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{+136}:\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 58.2% accurate, 15.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{+69} \lor \neg \left(x \leq 9.6 \cdot 10^{+135}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.8e+69) (not (<= x 9.6e+135))) (/ x y) 1.0))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.8e+69) || !(x <= 9.6e+135)) {
		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 ((x <= (-1.8d+69)) .or. (.not. (x <= 9.6d+135))) 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 ((x <= -1.8e+69) || !(x <= 9.6e+135)) {
		tmp = x / y;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.8e+69) or not (x <= 9.6e+135):
		tmp = x / y
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.8e+69) || !(x <= 9.6e+135))
		tmp = Float64(x / y);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -1.8e+69) || ~((x <= 9.6e+135)))
		tmp = x / y;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.8e+69], N[Not[LessEqual[x, 9.6e+135]], $MachinePrecision]], N[(x / y), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < -1.8000000000000001e69 or 9.59999999999999989e135 < x

   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 48.4%

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

      3. fabs-sqr 48.4%

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

      4. add-sqr-sqrt 48.9%

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

      5. add-sqr-sqrt 22.5%

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

      6. fabs-sqr 22.5%

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

      7. add-sqr-sqrt 57.7%

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

      8. associate-/r/ 57.7%

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

   4. Applied egg-rr 57.7%

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

   5. Taylor expanded in y around 0 57.3%

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

   if -1.8000000000000001e69 < x < 9.59999999999999989e135

   1. Initial program 100.0%

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

      1. clear-num 99.9%

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

      2. add-sqr-sqrt 46.1%

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

      3. fabs-sqr 46.1%

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

      4. add-sqr-sqrt 47.2%

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

      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 14.8%

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

      8. associate-/r/ 14.8%

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

   4. Applied egg-rr 14.8%

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

      1. associate-*l/ 14.9%

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

      2. *-un-lft-identity 14.9%

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

      3. sub-neg 14.9%

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

      4. add-sqr-sqrt 5.8%

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

      5. sqrt-unprod 28.6%

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

      6. sqr-neg 28.6%

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

      7. sqrt-unprod 36.9%

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

      8. add-sqr-sqrt 77.7%

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

   6. Applied egg-rr 77.7%

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

   7. Taylor expanded in x around 0 66.0%

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

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{+69} \lor \neg \left(x \leq 9.6 \cdot 10^{+135}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 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 46.8%

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

   3. fabs-sqr 46.8%

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

   4. add-sqr-sqrt 47.7%

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

   5. add-sqr-sqrt 13.3%

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

   6. fabs-sqr 13.3%

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

   7. add-sqr-sqrt 28.2%

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

   8. associate-/r/ 28.2%

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

4. Applied egg-rr 28.2%

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

5. Taylor expanded in y around inf 1.3%

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

6. Final simplification 1.3%

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

Alternative 8: 51.5% 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 46.8%

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

   3. fabs-sqr 46.8%

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

   4. add-sqr-sqrt 47.7%

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

   5. add-sqr-sqrt 13.3%

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

   6. fabs-sqr 13.3%

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

   7. add-sqr-sqrt 28.2%

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

   8. associate-/r/ 28.2%

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

4. Applied egg-rr 28.2%

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

   1. associate-*l/ 28.3%

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

   2. *-un-lft-identity 28.3%

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

   3. sub-neg 28.3%

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

   4. add-sqr-sqrt 15.0%

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

   5. sqrt-unprod 36.6%

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

   6. sqr-neg 36.6%

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

   7. sqrt-unprod 34.3%

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

   8. add-sqr-sqrt 75.9%

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

6. Applied egg-rr 75.9%

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

7. Taylor expanded in x around 0 51.4%

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

8. Final simplification 51.4%

   \[\leadsto 1 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024019 
(FPCore (x y)
  :name "Numeric.LinearAlgebra.Util:formatSparse from hmatrix-0.16.1.5"
  :precision binary64
  (/ (fabs (- x y)) (fabs y)))