Numeric.LinearAlgebra.Util:formatSparse from hmatrix-0.16.1.5

Percentage Accurate: 100.0% → 100.0%

Time: 8.3s

Alternatives: 7

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. Taylor expanded in x around -inf 100.0%

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

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

4. Simplified 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 72.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{+50}:\\ \;\;\;\;\frac{y}{x + y}\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{-123}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.1e+50) (/ y (+ x y)) (if (<= y 1.25e-123) (fabs (/ x y)) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.1e+50) {
		tmp = y / (x + y);
	} else if (y <= 1.25e-123) {
		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 (y <= (-1.1d+50)) then
        tmp = y / (x + y)
    else if (y <= 1.25d-123) 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 (y <= -1.1e+50) {
		tmp = y / (x + y);
	} else if (y <= 1.25e-123) {
		tmp = Math.abs((x / y));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.1e+50:
		tmp = y / (x + y)
	elif y <= 1.25e-123:
		tmp = math.fabs((x / y))
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.1e+50)
		tmp = Float64(y / Float64(x + y));
	elseif (y <= 1.25e-123)
		tmp = abs(Float64(x / y));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.1e+50)
		tmp = y / (x + y);
	elseif (y <= 1.25e-123)
		tmp = abs((x / y));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.1e+50], N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.25e-123], N[Abs[N[(x / y), $MachinePrecision]], $MachinePrecision], 1.0]]

Derivation

1. Split input into 3 regimes

2. if y < -1.10000000000000008e50

   1. Initial program 100.0%

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

   2. Taylor expanded in x around -inf 100.0%

      \[\leadsto \color{blue}{\frac{\left|-\left(y + -1 \cdot x\right)\right|}{\left|y\right|}} \]
   3. 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-sub 100.0%

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

      5. fabs-div 100.0%

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

      6. rem-square-sqrt 0.0%

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

      7. fabs-sqr 0.0%

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

      8. rem-square-sqrt 1.5%

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

      9. div-sub 1.5%

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

      10. sub-neg 1.5%

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

      11. *-inverses 1.5%

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

      12. metadata-eval 1.5%

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

      13. +-commutative 1.5%

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

   4. Simplified 1.5%

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

      1. +-commutative 1.5%

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

      2. metadata-eval 1.5%

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

      3. sub-neg 1.5%

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

      4. *-inverses 1.5%

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

      5. div-sub 1.5%

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

      6. clear-num 1.5%

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

      7. associate-/r/ 1.5%

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

      8. flip-- 0.9%

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

      9. associate-*r/ 0.9%

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

   6. Applied egg-rr 0.9%

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

   7. Taylor expanded in y around inf 1.8%

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

      1. neg-mul-1 1.8%

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

   9. Simplified 1.8%

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

       1. expm1-log1p-u 1.8%

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

       2. expm1-udef 1.7%

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

       3. *-un-lft-identity 1.7%

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

       4. *-commutative 1.7%

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

       5. *-commutative 1.7%

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

       6. *-un-lft-identity 1.7%

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

       7. add-sqr-sqrt 1.4%

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

       8. sqrt-unprod 0.7%

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

       9. sqr-neg 0.7%

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

       10. sqrt-unprod 0.0%

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

       11. add-sqr-sqrt 93.5%

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

   11. Applied egg-rr 93.5%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{y}{x + y}\right)} - 1} \]
   12. Step-by-step derivation

       1. expm1-def 93.5%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{y}{x + y}\right)\right)} \]

       2. expm1-log1p 93.5%

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

   13. Simplified 93.5%

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

   if -1.10000000000000008e50 < y < 1.25000000000000007e-123

   1. Initial program 100.0%

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

   2. Taylor expanded in x around -inf 100.0%

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

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

   4. Simplified 100.0%

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

   5. Taylor expanded in x around inf 81.9%

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

      1. mul-1-neg 81.9%

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

      2. distribute-frac-neg 81.9%

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

   7. Simplified 81.9%

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

   if 1.25000000000000007e-123 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in x around -inf 100.0%

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

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

   4. Simplified 100.0%

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

   5. Taylor expanded in x around 0 67.9%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{+50}:\\ \;\;\;\;\frac{y}{x + y}\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{-123}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 3: 58.3% accurate, 13.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\frac{y - x}{\frac{x}{y}}}\\ \mathbf{if}\;x \leq -2.55 \cdot 10^{+268}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -9.2 \cdot 10^{+121}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{+91}:\\ \;\;\;\;\frac{y}{x + y}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (/ (- y x) (/ x y)))))
   (if (<= x -2.55e+268)
     t_0
     (if (<= x -9.2e+121) (/ x y) (if (<= x 1.65e+91) (/ y (+ x y)) t_0)))))

C

double code(double x, double y) {
	double t_0 = x / ((y - x) / (x / y));
	double tmp;
	if (x <= -2.55e+268) {
		tmp = t_0;
	} else if (x <= -9.2e+121) {
		tmp = x / y;
	} else if (x <= 1.65e+91) {
		tmp = y / (x + y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / ((y - x) / (x / y))
    if (x <= (-2.55d+268)) then
        tmp = t_0
    else if (x <= (-9.2d+121)) then
        tmp = x / y
    else if (x <= 1.65d+91) then
        tmp = y / (x + y)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x / ((y - x) / (x / y));
	double tmp;
	if (x <= -2.55e+268) {
		tmp = t_0;
	} else if (x <= -9.2e+121) {
		tmp = x / y;
	} else if (x <= 1.65e+91) {
		tmp = y / (x + y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / ((y - x) / (x / y))
	tmp = 0
	if x <= -2.55e+268:
		tmp = t_0
	elif x <= -9.2e+121:
		tmp = x / y
	elif x <= 1.65e+91:
		tmp = y / (x + y)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(Float64(y - x) / Float64(x / y)))
	tmp = 0.0
	if (x <= -2.55e+268)
		tmp = t_0;
	elseif (x <= -9.2e+121)
		tmp = Float64(x / y);
	elseif (x <= 1.65e+91)
		tmp = Float64(y / Float64(x + y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / ((y - x) / (x / y));
	tmp = 0.0;
	if (x <= -2.55e+268)
		tmp = t_0;
	elseif (x <= -9.2e+121)
		tmp = x / y;
	elseif (x <= 1.65e+91)
		tmp = y / (x + y);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(N[(y - x), $MachinePrecision] / N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.55e+268], t$95$0, If[LessEqual[x, -9.2e+121], N[(x / y), $MachinePrecision], If[LessEqual[x, 1.65e+91], N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.54999999999999984e268 or 1.65000000000000009e91 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around -inf 100.0%

      \[\leadsto \color{blue}{\frac{\left|-\left(y + -1 \cdot x\right)\right|}{\left|y\right|}} \]
   3. 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-sub 100.0%

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

      5. fabs-div 100.0%

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

      6. rem-square-sqrt 32.6%

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

      7. fabs-sqr 32.6%

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

      8. rem-square-sqrt 33.1%

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

      9. div-sub 33.1%

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

      10. sub-neg 33.1%

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

      11. *-inverses 33.1%

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

      12. metadata-eval 33.1%

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

      13. +-commutative 33.1%

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

   4. Simplified 33.1%

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

      1. +-commutative 33.1%

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

      2. metadata-eval 33.1%

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

      3. sub-neg 33.1%

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

      4. *-inverses 33.1%

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

      5. div-sub 33.1%

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

      6. clear-num 33.1%

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

      7. associate-/r/ 33.1%

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

      8. flip-- 17.2%

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

      9. associate-*r/ 17.2%

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

   6. Applied egg-rr 17.2%

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

   7. Taylor expanded in y around 0 17.8%

      \[\leadsto \frac{\color{blue}{\frac{{x}^{2}}{y}}}{x + y} \]
   8. Step-by-step derivation

      1. unpow2 17.8%

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

      2. associate-/l* 22.8%

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

      3. associate-/r/ 22.8%

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

   9. Simplified 22.8%

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

       1. associate-*l/ 17.8%

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

       2. *-un-lft-identity 17.8%

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

       3. associate-*l/ 17.8%

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

       4. frac-2neg 17.8%

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

       5. div-inv 17.8%

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

       6. associate-*l/ 17.8%

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

       7. *-un-lft-identity 17.8%

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

       8. distribute-neg-frac 17.8%

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

       9. add-sqr-sqrt 15.6%

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

       10. sqrt-unprod 40.2%

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

       11. sqr-neg 40.2%

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

       12. sqrt-unprod 25.0%

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

       13. add-sqr-sqrt 31.1%

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

       14. frac-2neg 31.1%

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

       15. associate-*r/ 34.6%

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

       16. distribute-neg-in 34.6%

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

       17. neg-mul-1 34.6%

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

       18. add-sqr-sqrt 26.8%

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

       19. sqrt-unprod 31.1%

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

       20. sqr-neg 31.1%

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

       21. sqrt-unprod 7.4%

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

       22. add-sqr-sqrt 34.3%

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

       23. fma-def 34.3%

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

   11. Applied egg-rr 34.3%

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

       1. associate-*r/ 34.3%

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

       2. *-rgt-identity 34.3%

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

       3. associate-/l* 56.4%

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

       4. fma-udef 56.4%

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

       5. neg-mul-1 56.4%

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

       6. +-commutative 56.4%

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

       7. sub-neg 56.4%

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

   13. Simplified 56.4%

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

   if -2.54999999999999984e268 < x < -9.1999999999999995e121

   1. Initial program 100.0%

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

   2. Taylor expanded in x around -inf 100.0%

      \[\leadsto \color{blue}{\frac{\left|-\left(y + -1 \cdot x\right)\right|}{\left|y\right|}} \]
   3. 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-sub 100.0%

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

      5. fabs-div 100.0%

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

      6. rem-square-sqrt 51.6%

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

      7. fabs-sqr 51.6%

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

      8. rem-square-sqrt 52.1%

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

      9. div-sub 52.1%

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

      10. sub-neg 52.1%

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

      11. *-inverses 52.1%

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

      12. metadata-eval 52.1%

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

      13. +-commutative 52.1%

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

   4. Simplified 52.1%

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

   5. Taylor expanded in x around inf 52.9%

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

   if -9.1999999999999995e121 < x < 1.65000000000000009e91

   1. Initial program 100.0%

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

   2. Taylor expanded in x around -inf 100.0%

      \[\leadsto \color{blue}{\frac{\left|-\left(y + -1 \cdot x\right)\right|}{\left|y\right|}} \]
   3. 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-sub 100.0%

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

      5. fabs-div 100.0%

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

      6. rem-square-sqrt 12.8%

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

      7. fabs-sqr 12.8%

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

      8. rem-square-sqrt 14.2%

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

      9. div-sub 14.2%

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

      10. sub-neg 14.2%

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

      11. *-inverses 14.2%

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

      12. metadata-eval 14.2%

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

      13. +-commutative 14.2%

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

   4. Simplified 14.2%

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

      1. +-commutative 14.2%

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

      2. metadata-eval 14.2%

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

      3. sub-neg 14.2%

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

      4. *-inverses 14.2%

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

      5. div-sub 14.2%

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

      6. clear-num 14.1%

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

      7. associate-/r/ 14.1%

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

      8. flip-- 11.7%

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

      9. associate-*r/ 11.1%

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

   6. Applied egg-rr 11.1%

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

   7. Taylor expanded in y around inf 1.9%

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

      1. neg-mul-1 1.9%

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

   9. Simplified 1.9%

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

       1. expm1-log1p-u 1.9%

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

       2. expm1-udef 1.7%

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

       3. *-un-lft-identity 1.7%

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

       4. *-commutative 1.7%

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

       5. *-commutative 1.7%

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

       6. *-un-lft-identity 1.7%

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

       7. add-sqr-sqrt 0.7%

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

       8. sqrt-unprod 26.1%

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

       9. sqr-neg 26.1%

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

       10. sqrt-unprod 39.1%

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

       11. add-sqr-sqrt 72.3%

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

   11. Applied egg-rr 72.3%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{y}{x + y}\right)} - 1} \]
   12. Step-by-step derivation

       1. expm1-def 72.5%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{y}{x + y}\right)\right)} \]

       2. expm1-log1p 72.5%

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

   13. Simplified 72.5%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.55 \cdot 10^{+268}:\\ \;\;\;\;\frac{x}{\frac{y - x}{\frac{x}{y}}}\\ \mathbf{elif}\;x \leq -9.2 \cdot 10^{+121}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{+91}:\\ \;\;\;\;\frac{y}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{y - x}{\frac{x}{y}}}\\ \end{array} \]

Alternative 4: 58.4% accurate, 13.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{+269}:\\ \;\;\;\;x \cdot \frac{\frac{-x}{y}}{x + y}\\ \mathbf{elif}\;x \leq -2.9 \cdot 10^{+121}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{+95}:\\ \;\;\;\;\frac{y}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{y - x}{\frac{x}{y}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -5.8e+269)
   (* x (/ (/ (- x) y) (+ x y)))
   (if (<= x -2.9e+121)
     (/ x y)
     (if (<= x 5.2e+95) (/ y (+ x y)) (/ x (/ (- y x) (/ x y)))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -5.8e+269) {
		tmp = x * ((-x / y) / (x + y));
	} else if (x <= -2.9e+121) {
		tmp = x / y;
	} else if (x <= 5.2e+95) {
		tmp = y / (x + y);
	} else {
		tmp = x / ((y - x) / (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 <= (-5.8d+269)) then
        tmp = x * ((-x / y) / (x + y))
    else if (x <= (-2.9d+121)) then
        tmp = x / y
    else if (x <= 5.2d+95) then
        tmp = y / (x + y)
    else
        tmp = x / ((y - x) / (x / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -5.8e+269) {
		tmp = x * ((-x / y) / (x + y));
	} else if (x <= -2.9e+121) {
		tmp = x / y;
	} else if (x <= 5.2e+95) {
		tmp = y / (x + y);
	} else {
		tmp = x / ((y - x) / (x / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -5.8e+269:
		tmp = x * ((-x / y) / (x + y))
	elif x <= -2.9e+121:
		tmp = x / y
	elif x <= 5.2e+95:
		tmp = y / (x + y)
	else:
		tmp = x / ((y - x) / (x / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -5.8e+269)
		tmp = Float64(x * Float64(Float64(Float64(-x) / y) / Float64(x + y)));
	elseif (x <= -2.9e+121)
		tmp = Float64(x / y);
	elseif (x <= 5.2e+95)
		tmp = Float64(y / Float64(x + y));
	else
		tmp = Float64(x / Float64(Float64(y - x) / Float64(x / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -5.8e+269)
		tmp = x * ((-x / y) / (x + y));
	elseif (x <= -2.9e+121)
		tmp = x / y;
	elseif (x <= 5.2e+95)
		tmp = y / (x + y);
	else
		tmp = x / ((y - x) / (x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -5.8e+269], N[(x * N[(N[((-x) / y), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2.9e+121], N[(x / y), $MachinePrecision], If[LessEqual[x, 5.2e+95], N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[(y - x), $MachinePrecision] / N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -5.80000000000000051e269

   1. Initial program 100.0%

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

   2. Taylor expanded in x around -inf 100.0%

      \[\leadsto \color{blue}{\frac{\left|-\left(y + -1 \cdot x\right)\right|}{\left|y\right|}} \]
   3. 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-sub 100.0%

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

      5. fabs-div 100.0%

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

      6. rem-square-sqrt 11.1%

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

      7. fabs-sqr 11.1%

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

      8. rem-square-sqrt 11.6%

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

      9. div-sub 11.6%

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

      10. sub-neg 11.6%

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

      11. *-inverses 11.6%

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

      12. metadata-eval 11.6%

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

      13. +-commutative 11.6%

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

   4. Simplified 11.6%

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

      1. +-commutative 11.6%

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

      2. metadata-eval 11.6%

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

      3. sub-neg 11.6%

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

      4. *-inverses 11.6%

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

      5. div-sub 11.6%

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

      6. clear-num 11.6%

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

      7. associate-/r/ 11.6%

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

      8. flip-- 11.2%

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

      9. associate-*r/ 11.2%

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

   6. Applied egg-rr 11.2%

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

   7. Taylor expanded in y around 0 11.3%

      \[\leadsto \frac{\color{blue}{\frac{{x}^{2}}{y}}}{x + y} \]
   8. Step-by-step derivation

      1. unpow2 11.3%

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

      2. associate-/l* 11.4%

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

      3. associate-/r/ 11.4%

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

   9. Simplified 11.4%

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

       1. associate-*l/ 11.3%

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

       2. *-un-lft-identity 11.3%

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

       3. associate-*l/ 11.3%

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

       4. frac-2neg 11.3%

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

       5. distribute-frac-neg 11.3%

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

       6. *-commutative 11.3%

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

       7. div-inv 11.3%

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

       8. frac-2neg 11.3%

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

       9. add-sqr-sqrt 11.1%

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

       10. sqrt-unprod 46.0%

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

       11. sqr-neg 46.0%

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

       12. sqrt-unprod 35.7%

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

       13. add-sqr-sqrt 35.7%

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

       14. distribute-neg-frac 35.7%

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

       15. associate-*l/ 46.3%

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

       16. frac-2neg 46.3%

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

       17. associate-/l* 88.9%

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

       18. associate-/r/ 88.5%

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

   11. Applied egg-rr 88.5%

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

       1. distribute-rgt-neg-in 88.5%

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

   13. Simplified 88.5%

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

   if -5.80000000000000051e269 < x < -2.8999999999999999e121

   1. Initial program 100.0%

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

   2. Taylor expanded in x around -inf 100.0%

      \[\leadsto \color{blue}{\frac{\left|-\left(y + -1 \cdot x\right)\right|}{\left|y\right|}} \]
   3. 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-sub 100.0%

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

      5. fabs-div 100.0%

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

      6. rem-square-sqrt 51.6%

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

      7. fabs-sqr 51.6%

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

      8. rem-square-sqrt 52.1%

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

      9. div-sub 52.1%

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

      10. sub-neg 52.1%

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

      11. *-inverses 52.1%

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

      12. metadata-eval 52.1%

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

      13. +-commutative 52.1%

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

   4. Simplified 52.1%

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

   5. Taylor expanded in x around inf 52.9%

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

   if -2.8999999999999999e121 < x < 5.19999999999999981e95

   1. Initial program 100.0%

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

   2. Taylor expanded in x around -inf 100.0%

      \[\leadsto \color{blue}{\frac{\left|-\left(y + -1 \cdot x\right)\right|}{\left|y\right|}} \]
   3. 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-sub 100.0%

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

      5. fabs-div 100.0%

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

      6. rem-square-sqrt 12.8%

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

      7. fabs-sqr 12.8%

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

      8. rem-square-sqrt 14.2%

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

      9. div-sub 14.2%

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

      10. sub-neg 14.2%

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

      11. *-inverses 14.2%

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

      12. metadata-eval 14.2%

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

      13. +-commutative 14.2%

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

   4. Simplified 14.2%

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

      1. +-commutative 14.2%

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

      2. metadata-eval 14.2%

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

      3. sub-neg 14.2%

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

      4. *-inverses 14.2%

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

      5. div-sub 14.2%

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

      6. clear-num 14.1%

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

      7. associate-/r/ 14.1%

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

      8. flip-- 11.7%

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

      9. associate-*r/ 11.1%

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

   6. Applied egg-rr 11.1%

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

   7. Taylor expanded in y around inf 1.9%

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

      1. neg-mul-1 1.9%

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

   9. Simplified 1.9%

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

       1. expm1-log1p-u 1.9%

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

       2. expm1-udef 1.7%

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

       3. *-un-lft-identity 1.7%

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

       4. *-commutative 1.7%

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

       5. *-commutative 1.7%

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

       6. *-un-lft-identity 1.7%

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

       7. add-sqr-sqrt 0.7%

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

       8. sqrt-unprod 26.1%

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

       9. sqr-neg 26.1%

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

       10. sqrt-unprod 39.1%

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

       11. add-sqr-sqrt 72.3%

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

   11. Applied egg-rr 72.3%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{y}{x + y}\right)} - 1} \]
   12. Step-by-step derivation

       1. expm1-def 72.5%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{y}{x + y}\right)\right)} \]

       2. expm1-log1p 72.5%

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

   13. Simplified 72.5%

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

   if 5.19999999999999981e95 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around -inf 100.0%

      \[\leadsto \color{blue}{\frac{\left|-\left(y + -1 \cdot x\right)\right|}{\left|y\right|}} \]
   3. 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-sub 100.0%

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

      5. fabs-div 100.0%

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

      6. rem-square-sqrt 36.8%

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

      7. fabs-sqr 36.8%

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

      8. rem-square-sqrt 37.3%

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

      9. div-sub 37.3%

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

      10. sub-neg 37.3%

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

      11. *-inverses 37.3%

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

      12. metadata-eval 37.3%

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

      13. +-commutative 37.3%

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

   4. Simplified 37.3%

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

      1. +-commutative 37.3%

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

      2. metadata-eval 37.3%

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

      3. sub-neg 37.3%

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

      4. *-inverses 37.3%

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

      5. div-sub 37.3%

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

      6. clear-num 37.3%

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

      7. associate-/r/ 37.3%

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

      8. flip-- 18.4%

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

      9. associate-*r/ 18.3%

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

   6. Applied egg-rr 18.3%

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

   7. Taylor expanded in y around 0 19.1%

      \[\leadsto \frac{\color{blue}{\frac{{x}^{2}}{y}}}{x + y} \]
   8. Step-by-step derivation

      1. unpow2 19.1%

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

      2. associate-/l* 25.0%

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

      3. associate-/r/ 25.1%

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

   9. Simplified 25.1%

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

       1. associate-*l/ 19.1%

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

       2. *-un-lft-identity 19.1%

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

       3. associate-*l/ 19.1%

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

       4. frac-2neg 19.1%

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

       5. div-inv 19.1%

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

       6. associate-*l/ 19.1%

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

       7. *-un-lft-identity 19.1%

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

       8. distribute-neg-frac 19.1%

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

       9. add-sqr-sqrt 18.6%

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

       10. sqrt-unprod 48.1%

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

       11. sqr-neg 48.1%

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

       12. sqrt-unprod 29.9%

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

       13. add-sqr-sqrt 30.1%

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

       14. frac-2neg 30.1%

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

       15. associate-*r/ 32.4%

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

       16. distribute-neg-in 32.4%

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

       17. neg-mul-1 32.4%

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

       18. add-sqr-sqrt 32.0%

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

       19. sqrt-unprod 30.3%

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

       20. sqr-neg 30.3%

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

       21. sqrt-unprod 0.5%

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

       22. add-sqr-sqrt 32.7%

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

       23. fma-def 32.7%

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

   11. Applied egg-rr 32.7%

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

       1. associate-*r/ 32.7%

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

       2. *-rgt-identity 32.7%

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

       3. associate-/l* 50.7%

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

       4. fma-udef 50.7%

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

       5. neg-mul-1 50.7%

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

       6. +-commutative 50.7%

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

       7. sub-neg 50.7%

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

   13. Simplified 50.7%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{+269}:\\ \;\;\;\;x \cdot \frac{\frac{-x}{y}}{x + y}\\ \mathbf{elif}\;x \leq -2.9 \cdot 10^{+121}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{+95}:\\ \;\;\;\;\frac{y}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{y - x}{\frac{x}{y}}}\\ \end{array} \]

Alternative 5: 58.1% accurate, 22.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{+121}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{+141}:\\ \;\;\;\;\frac{y}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -3.6e+121)
   (/ x y)
   (if (<= x 2.7e+141) (/ y (+ x y)) (+ (/ x y) -1.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -3.6e+121) {
		tmp = x / y;
	} else if (x <= 2.7e+141) {
		tmp = y / (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 <= (-3.6d+121)) then
        tmp = x / y
    else if (x <= 2.7d+141) then
        tmp = y / (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 <= -3.6e+121) {
		tmp = x / y;
	} else if (x <= 2.7e+141) {
		tmp = y / (x + y);
	} else {
		tmp = (x / y) + -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -3.6e+121:
		tmp = x / y
	elif x <= 2.7e+141:
		tmp = y / (x + y)
	else:
		tmp = (x / y) + -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -3.6e+121)
		tmp = Float64(x / y);
	elseif (x <= 2.7e+141)
		tmp = Float64(y / 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 <= -3.6e+121)
		tmp = x / y;
	elseif (x <= 2.7e+141)
		tmp = y / (x + y);
	else
		tmp = (x / y) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -3.6e+121], N[(x / y), $MachinePrecision], If[LessEqual[x, 2.7e+141], N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.59999999999999981e121

   1. Initial program 100.0%

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

   2. Taylor expanded in x around -inf 100.0%

      \[\leadsto \color{blue}{\frac{\left|-\left(y + -1 \cdot x\right)\right|}{\left|y\right|}} \]
   3. 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-sub 100.0%

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

      5. fabs-div 100.0%

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

      6. rem-square-sqrt 42.5%

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

      7. fabs-sqr 42.5%

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

      8. rem-square-sqrt 43.0%

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

      9. div-sub 43.0%

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

      10. sub-neg 43.0%

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

      11. *-inverses 43.0%

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

      12. metadata-eval 43.0%

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

      13. +-commutative 43.0%

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

   4. Simplified 43.0%

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

   5. Taylor expanded in x around inf 43.6%

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

   if -3.59999999999999981e121 < x < 2.7000000000000001e141

   1. Initial program 100.0%

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

   2. Taylor expanded in x around -inf 100.0%

      \[\leadsto \color{blue}{\frac{\left|-\left(y + -1 \cdot x\right)\right|}{\left|y\right|}} \]
   3. 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-sub 100.0%

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

      5. fabs-div 100.0%

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

      6. rem-square-sqrt 13.6%

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

      7. fabs-sqr 13.6%

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

      8. rem-square-sqrt 14.9%

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

      9. div-sub 14.9%

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

      10. sub-neg 14.9%

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

      11. *-inverses 14.9%

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

      12. metadata-eval 14.9%

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

      13. +-commutative 14.9%

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

   4. Simplified 14.9%

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

      1. +-commutative 14.9%

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

      2. metadata-eval 14.9%

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

      3. sub-neg 14.9%

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

      4. *-inverses 14.9%

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

      5. div-sub 14.9%

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

      6. clear-num 14.8%

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

      7. associate-/r/ 14.8%

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

      8. flip-- 12.5%

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

      9. associate-*r/ 12.0%

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

   6. Applied egg-rr 12.0%

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

   7. Taylor expanded in y around inf 1.9%

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

      1. neg-mul-1 1.9%

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

   9. Simplified 1.9%

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

       1. expm1-log1p-u 1.9%

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

       2. expm1-udef 1.8%

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

       3. *-un-lft-identity 1.8%

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

       4. *-commutative 1.8%

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

       5. *-commutative 1.8%

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

       6. *-un-lft-identity 1.8%

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

       7. add-sqr-sqrt 0.7%

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

       8. sqrt-unprod 24.4%

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

       9. sqr-neg 24.4%

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

       10. sqrt-unprod 37.5%

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

       11. add-sqr-sqrt 70.0%

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

   11. Applied egg-rr 70.0%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{y}{x + y}\right)} - 1} \]
   12. Step-by-step derivation

       1. expm1-def 70.1%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{y}{x + y}\right)\right)} \]

       2. expm1-log1p 70.1%

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

   13. Simplified 70.1%

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

   if 2.7000000000000001e141 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around -inf 100.0%

      \[\leadsto \color{blue}{\frac{\left|-\left(y + -1 \cdot x\right)\right|}{\left|y\right|}} \]
   3. 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-sub 100.0%

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

      5. fabs-div 100.0%

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

      6. rem-square-sqrt 42.1%

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

      7. fabs-sqr 42.1%

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

      8. rem-square-sqrt 42.6%

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

      9. div-sub 42.6%

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

      10. sub-neg 42.6%

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

      11. *-inverses 42.6%

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

      12. metadata-eval 42.6%

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

      13. +-commutative 42.6%

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

   4. Simplified 42.6%

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

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{+121}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{+141}:\\ \;\;\;\;\frac{y}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + -1\\ \end{array} \]

Alternative 6: 26.7% 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. Taylor expanded in x around -inf 100.0%

   \[\leadsto \color{blue}{\frac{\left|-\left(y + -1 \cdot x\right)\right|}{\left|y\right|}} \]
3. 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-sub 100.0%

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

   5. fabs-div 100.0%

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

   6. rem-square-sqrt 21.8%

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

   7. fabs-sqr 21.8%

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

   8. rem-square-sqrt 22.8%

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

   9. div-sub 22.8%

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

   10. sub-neg 22.8%

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

   11. *-inverses 22.8%

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

   12. metadata-eval 22.8%

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

   13. +-commutative 22.8%

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

4. Simplified 22.8%

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

5. Taylor expanded in x around inf 23.4%

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

6. Final simplification 23.4%

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

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. Taylor expanded in x around -inf 100.0%

   \[\leadsto \color{blue}{\frac{\left|-\left(y + -1 \cdot x\right)\right|}{\left|y\right|}} \]
3. 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-sub 100.0%

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

   5. fabs-div 100.0%

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

   6. rem-square-sqrt 21.8%

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

   7. fabs-sqr 21.8%

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

   8. rem-square-sqrt 22.8%

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

   9. div-sub 22.8%

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

   10. sub-neg 22.8%

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

   11. *-inverses 22.8%

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

   12. metadata-eval 22.8%

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

   13. +-commutative 22.8%

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

4. Simplified 22.8%

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

5. Taylor expanded in x around 0 1.3%

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

6. Final simplification 1.3%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023274 
(FPCore (x y)
  :name "Numeric.LinearAlgebra.Util:formatSparse from hmatrix-0.16.1.5"
  :precision binary64
  (/ (fabs (- x y)) (fabs y)))