Numeric.LinearAlgebra.Util:formatSparse from hmatrix-0.16.1.5

Percentage Accurate: 100.0% → 100.0%

Time: 4.8s

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: 74.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.25 \cdot 10^{+79} \lor \neg \left(y \leq 1.8 \cdot 10^{-37}\right):\\ \;\;\;\;\frac{y}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.25e+79) (not (<= y 1.8e-37))) (/ y (+ x y)) (fabs (/ x y))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.25e+79) || !(y <= 1.8e-37)) {
		tmp = y / (x + y);
	} else {
		tmp = fabs((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 ((y <= (-1.25d+79)) .or. (.not. (y <= 1.8d-37))) then
        tmp = y / (x + y)
    else
        tmp = abs((x / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.25e+79) || !(y <= 1.8e-37)) {
		tmp = y / (x + y);
	} else {
		tmp = Math.abs((x / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.25e+79) or not (y <= 1.8e-37):
		tmp = y / (x + y)
	else:
		tmp = math.fabs((x / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.25e+79) || !(y <= 1.8e-37))
		tmp = Float64(y / Float64(x + y));
	else
		tmp = abs(Float64(x / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.25e+79) || ~((y <= 1.8e-37)))
		tmp = y / (x + y);
	else
		tmp = abs((x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.25e+79], N[Not[LessEqual[y, 1.8e-37]], $MachinePrecision]], N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision], N[Abs[N[(x / y), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.25e79 or 1.80000000000000004e-37 < 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-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 12.5%

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

      7. fabs-sqr 12.5%

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

      8. rem-square-sqrt 13.9%

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

      9. div-sub 13.9%

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

      10. sub-neg 13.9%

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

      11. *-inverses 13.9%

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

      12. metadata-eval 13.9%

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

      13. +-commutative 13.9%

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

   4. Simplified 13.9%

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

      1. +-commutative 13.9%

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

      2. metadata-eval 13.9%

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

      3. sub-neg 13.9%

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

      4. *-inverses 13.9%

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

      5. div-sub 13.9%

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

      6. clear-num 13.8%

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

      7. associate-/r/ 13.9%

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

      8. flip-- 4.2%

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

      9. associate-*r/ 4.2%

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

   6. Applied egg-rr 4.2%

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

   7. Taylor expanded in y around 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 0.6%

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

       8. sqrt-unprod 24.0%

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

       9. sqr-neg 24.0%

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

       10. sqrt-unprod 45.0%

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

       11. add-sqr-sqrt 80.7%

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

   11. Applied egg-rr 80.7%

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

       1. expm1-def 80.9%

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

       2. expm1-log1p 80.9%

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

   13. Simplified 80.9%

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

   if -1.25e79 < y < 1.80000000000000004e-37

   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 77.5%

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

      1. mul-1-neg 77.5%

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

      2. distribute-frac-neg 77.5%

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

   7. Simplified 77.5%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.25 \cdot 10^{+79} \lor \neg \left(y \leq 1.8 \cdot 10^{-37}\right):\\ \;\;\;\;\frac{y}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \end{array} \]

Alternative 3: 58.4% accurate, 13.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{x + y}\\ \mathbf{if}\;y \leq -9.8 \cdot 10^{-132}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 5.1 \cdot 10^{-252}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{-71}:\\ \;\;\;\;x \cdot \frac{\frac{x}{y}}{y - x}\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{-39}:\\ \;\;\;\;\frac{x}{y} + -1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (+ x y))))
   (if (<= y -9.8e-132)
     t_0
     (if (<= y 5.1e-252)
       (/ x y)
       (if (<= y 6.5e-71)
         (* x (/ (/ x y) (- y x)))
         (if (<= y 3.4e-39) (+ (/ x y) -1.0) t_0))))))

C

double code(double x, double y) {
	double t_0 = y / (x + y);
	double tmp;
	if (y <= -9.8e-132) {
		tmp = t_0;
	} else if (y <= 5.1e-252) {
		tmp = x / y;
	} else if (y <= 6.5e-71) {
		tmp = x * ((x / y) / (y - x));
	} else if (y <= 3.4e-39) {
		tmp = (x / y) + -1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y / (x + y)
    if (y <= (-9.8d-132)) then
        tmp = t_0
    else if (y <= 5.1d-252) then
        tmp = x / y
    else if (y <= 6.5d-71) then
        tmp = x * ((x / y) / (y - x))
    else if (y <= 3.4d-39) then
        tmp = (x / y) + (-1.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y / (x + y);
	double tmp;
	if (y <= -9.8e-132) {
		tmp = t_0;
	} else if (y <= 5.1e-252) {
		tmp = x / y;
	} else if (y <= 6.5e-71) {
		tmp = x * ((x / y) / (y - x));
	} else if (y <= 3.4e-39) {
		tmp = (x / y) + -1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y / (x + y)
	tmp = 0
	if y <= -9.8e-132:
		tmp = t_0
	elif y <= 5.1e-252:
		tmp = x / y
	elif y <= 6.5e-71:
		tmp = x * ((x / y) / (y - x))
	elif y <= 3.4e-39:
		tmp = (x / y) + -1.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y / Float64(x + y))
	tmp = 0.0
	if (y <= -9.8e-132)
		tmp = t_0;
	elseif (y <= 5.1e-252)
		tmp = Float64(x / y);
	elseif (y <= 6.5e-71)
		tmp = Float64(x * Float64(Float64(x / y) / Float64(y - x)));
	elseif (y <= 3.4e-39)
		tmp = Float64(Float64(x / y) + -1.0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y / (x + y);
	tmp = 0.0;
	if (y <= -9.8e-132)
		tmp = t_0;
	elseif (y <= 5.1e-252)
		tmp = x / y;
	elseif (y <= 6.5e-71)
		tmp = x * ((x / y) / (y - x));
	elseif (y <= 3.4e-39)
		tmp = (x / y) + -1.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -9.8e-132], t$95$0, If[LessEqual[y, 5.1e-252], N[(x / y), $MachinePrecision], If[LessEqual[y, 6.5e-71], N[(x * N[(N[(x / y), $MachinePrecision] / N[(y - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.4e-39], N[(N[(x / y), $MachinePrecision] + -1.0), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -9.79999999999999961e-132 or 3.3999999999999999e-39 < 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-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 15.5%

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

      7. fabs-sqr 15.5%

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

      8. rem-square-sqrt 16.8%

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

      9. div-sub 16.8%

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

      10. sub-neg 16.8%

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

      11. *-inverses 16.8%

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

      12. metadata-eval 16.8%

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

      13. +-commutative 16.8%

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

   4. Simplified 16.8%

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

      1. +-commutative 16.8%

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

      2. metadata-eval 16.8%

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

      3. sub-neg 16.8%

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

      4. *-inverses 16.8%

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

      5. div-sub 16.8%

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

      6. clear-num 16.7%

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

      7. associate-/r/ 16.7%

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

      8. flip-- 6.4%

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

      9. associate-*r/ 6.3%

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

   6. Applied egg-rr 6.3%

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

   7. Taylor expanded in y around inf 2.0%

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

      1. neg-mul-1 2.0%

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

   9. Simplified 2.0%

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

       1. expm1-log1p-u 2.0%

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

       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 1.0%

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

       8. sqrt-unprod 18.2%

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

       9. sqr-neg 18.2%

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

       10. sqrt-unprod 33.0%

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

       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 70.2%

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

       2. expm1-log1p 70.2%

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

   13. Simplified 70.2%

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

   if -9.79999999999999961e-132 < y < 5.1000000000000004e-252

   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 64.8%

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

      7. fabs-sqr 64.8%

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

      8. rem-square-sqrt 65.2%

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

      9. div-sub 65.2%

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

      10. sub-neg 65.2%

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

      11. *-inverses 65.2%

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

      12. metadata-eval 65.2%

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

      13. +-commutative 65.2%

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

   4. Simplified 65.2%

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

   5. Taylor expanded in x around inf 65.4%

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

   if 5.1000000000000004e-252 < y < 6.50000000000000005e-71

   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 31.2%

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

      7. fabs-sqr 31.2%

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

      8. rem-square-sqrt 31.9%

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

      9. div-sub 31.9%

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

      10. sub-neg 31.9%

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

      11. *-inverses 31.9%

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

      12. metadata-eval 31.9%

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

      13. +-commutative 31.9%

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

   4. Simplified 31.9%

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

      1. +-commutative 31.9%

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

      2. metadata-eval 31.9%

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

      3. sub-neg 31.9%

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

      4. *-inverses 31.9%

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

      5. div-sub 31.9%

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

      6. clear-num 31.9%

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

      7. associate-/r/ 31.8%

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

      8. flip-- 28.9%

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

      9. associate-*r/ 23.2%

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

   6. Applied egg-rr 23.2%

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

   7. Taylor expanded in y around 0 24.0%

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

      1. unpow2 24.0%

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

      2. associate-/l* 27.5%

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

      3. associate-/r/ 27.5%

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

   9. Simplified 27.5%

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

       1. associate-*l/ 24.0%

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

       2. *-un-lft-identity 24.0%

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

       3. associate-*l/ 24.0%

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

       4. frac-2neg 24.0%

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

       5. div-inv 24.0%

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

       6. associate-*l/ 24.0%

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

       7. *-un-lft-identity 24.0%

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

       8. distribute-neg-frac 24.0%

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

       9. add-sqr-sqrt 24.0%

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

       10. sqrt-unprod 17.5%

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

       11. sqr-neg 17.5%

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

       12. sqrt-unprod 0.0%

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

       13. add-sqr-sqrt 38.8%

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

       14. frac-2neg 38.8%

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

       15. associate-*r/ 38.7%

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

       16. distribute-neg-in 38.7%

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

       17. neg-mul-1 38.7%

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

       18. add-sqr-sqrt 0.0%

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

       19. sqrt-unprod 39.5%

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

       20. sqr-neg 39.5%

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

       21. sqrt-unprod 39.4%

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

       22. add-sqr-sqrt 39.4%

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

       23. fma-def 39.4%

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

   11. Applied egg-rr 39.4%

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

       1. associate-*l* 45.3%

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

       2. associate-*r/ 45.3%

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

       3. *-rgt-identity 45.3%

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

       4. fma-udef 45.3%

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

       5. neg-mul-1 45.3%

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

       6. +-commutative 45.3%

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

       7. sub-neg 45.3%

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

   13. Simplified 45.3%

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

   if 6.50000000000000005e-71 < y < 3.3999999999999999e-39

   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 63.2%

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

      7. fabs-sqr 63.2%

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

      8. rem-square-sqrt 64.0%

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

      9. div-sub 64.0%

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

      10. sub-neg 64.0%

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

      11. *-inverses 64.0%

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

      12. metadata-eval 64.0%

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

      13. +-commutative 64.0%

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

   4. Simplified 64.0%

      \[\leadsto \color{blue}{-1 + \frac{x}{y}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.8 \cdot 10^{-132}:\\ \;\;\;\;\frac{y}{x + y}\\ \mathbf{elif}\;y \leq 5.1 \cdot 10^{-252}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{-71}:\\ \;\;\;\;x \cdot \frac{\frac{x}{y}}{y - x}\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{-39}:\\ \;\;\;\;\frac{x}{y} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x + y}\\ \end{array} \]

Alternative 4: 58.6% accurate, 13.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{-132}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{-255}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{-73}:\\ \;\;\;\;x \cdot \frac{\frac{x}{y}}{y - x}\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{-39}:\\ \;\;\;\;\frac{x}{y} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x + y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.45e-132)
   1.0
   (if (<= y 4.5e-255)
     (/ x y)
     (if (<= y 2.7e-73)
       (* x (/ (/ x y) (- y x)))
       (if (<= y 3.8e-39) (+ (/ x y) -1.0) (/ y (+ x y)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.45e-132) {
		tmp = 1.0;
	} else if (y <= 4.5e-255) {
		tmp = x / y;
	} else if (y <= 2.7e-73) {
		tmp = x * ((x / y) / (y - x));
	} else if (y <= 3.8e-39) {
		tmp = (x / y) + -1.0;
	} else {
		tmp = y / (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 (y <= (-1.45d-132)) then
        tmp = 1.0d0
    else if (y <= 4.5d-255) then
        tmp = x / y
    else if (y <= 2.7d-73) then
        tmp = x * ((x / y) / (y - x))
    else if (y <= 3.8d-39) then
        tmp = (x / y) + (-1.0d0)
    else
        tmp = y / (x + y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.45e-132) {
		tmp = 1.0;
	} else if (y <= 4.5e-255) {
		tmp = x / y;
	} else if (y <= 2.7e-73) {
		tmp = x * ((x / y) / (y - x));
	} else if (y <= 3.8e-39) {
		tmp = (x / y) + -1.0;
	} else {
		tmp = y / (x + y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.45e-132:
		tmp = 1.0
	elif y <= 4.5e-255:
		tmp = x / y
	elif y <= 2.7e-73:
		tmp = x * ((x / y) / (y - x))
	elif y <= 3.8e-39:
		tmp = (x / y) + -1.0
	else:
		tmp = y / (x + y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.45e-132)
		tmp = 1.0;
	elseif (y <= 4.5e-255)
		tmp = Float64(x / y);
	elseif (y <= 2.7e-73)
		tmp = Float64(x * Float64(Float64(x / y) / Float64(y - x)));
	elseif (y <= 3.8e-39)
		tmp = Float64(Float64(x / y) + -1.0);
	else
		tmp = Float64(y / Float64(x + y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.45e-132)
		tmp = 1.0;
	elseif (y <= 4.5e-255)
		tmp = x / y;
	elseif (y <= 2.7e-73)
		tmp = x * ((x / y) / (y - x));
	elseif (y <= 3.8e-39)
		tmp = (x / y) + -1.0;
	else
		tmp = y / (x + y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.45e-132], 1.0, If[LessEqual[y, 4.5e-255], N[(x / y), $MachinePrecision], If[LessEqual[y, 2.7e-73], N[(x * N[(N[(x / y), $MachinePrecision] / N[(y - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.8e-39], N[(N[(x / y), $MachinePrecision] + -1.0), $MachinePrecision], N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if y < -1.44999999999999992e-132

   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 64.8%

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

   if -1.44999999999999992e-132 < y < 4.49999999999999979e-255

   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 64.8%

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

      7. fabs-sqr 64.8%

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

      8. rem-square-sqrt 65.2%

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

      9. div-sub 65.2%

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

      10. sub-neg 65.2%

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

      11. *-inverses 65.2%

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

      12. metadata-eval 65.2%

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

      13. +-commutative 65.2%

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

   4. Simplified 65.2%

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

   5. Taylor expanded in x around inf 65.4%

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

   if 4.49999999999999979e-255 < y < 2.69999999999999994e-73

   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 31.2%

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

      7. fabs-sqr 31.2%

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

      8. rem-square-sqrt 31.9%

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

      9. div-sub 31.9%

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

      10. sub-neg 31.9%

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

      11. *-inverses 31.9%

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

      12. metadata-eval 31.9%

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

      13. +-commutative 31.9%

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

   4. Simplified 31.9%

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

      1. +-commutative 31.9%

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

      2. metadata-eval 31.9%

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

      3. sub-neg 31.9%

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

      4. *-inverses 31.9%

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

      5. div-sub 31.9%

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

      6. clear-num 31.9%

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

      7. associate-/r/ 31.8%

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

      8. flip-- 28.9%

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

      9. associate-*r/ 23.2%

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

   6. Applied egg-rr 23.2%

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

   7. Taylor expanded in y around 0 24.0%

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

      1. unpow2 24.0%

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

      2. associate-/l* 27.5%

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

      3. associate-/r/ 27.5%

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

   9. Simplified 27.5%

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

       1. associate-*l/ 24.0%

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

       2. *-un-lft-identity 24.0%

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

       3. associate-*l/ 24.0%

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

       4. frac-2neg 24.0%

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

       5. div-inv 24.0%

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

       6. associate-*l/ 24.0%

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

       7. *-un-lft-identity 24.0%

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

       8. distribute-neg-frac 24.0%

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

       9. add-sqr-sqrt 24.0%

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

       10. sqrt-unprod 17.5%

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

       11. sqr-neg 17.5%

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

       12. sqrt-unprod 0.0%

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

       13. add-sqr-sqrt 38.8%

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

       14. frac-2neg 38.8%

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

       15. associate-*r/ 38.7%

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

       16. distribute-neg-in 38.7%

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

       17. neg-mul-1 38.7%

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

       18. add-sqr-sqrt 0.0%

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

       19. sqrt-unprod 39.5%

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

       20. sqr-neg 39.5%

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

       21. sqrt-unprod 39.4%

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

       22. add-sqr-sqrt 39.4%

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

       23. fma-def 39.4%

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

   11. Applied egg-rr 39.4%

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

       1. associate-*l* 45.3%

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

       2. associate-*r/ 45.3%

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

       3. *-rgt-identity 45.3%

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

       4. fma-udef 45.3%

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

       5. neg-mul-1 45.3%

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

       6. +-commutative 45.3%

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

       7. sub-neg 45.3%

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

   13. Simplified 45.3%

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

   if 2.69999999999999994e-73 < y < 3.8000000000000002e-39

   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 63.2%

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

      7. fabs-sqr 63.2%

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

      8. rem-square-sqrt 64.0%

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

      9. div-sub 64.0%

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

      10. sub-neg 64.0%

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

      11. *-inverses 64.0%

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

      12. metadata-eval 64.0%

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

      13. +-commutative 64.0%

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

   4. Simplified 64.0%

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

   if 3.8000000000000002e-39 < 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-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 15.0%

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

      9. div-sub 15.0%

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

      10. sub-neg 15.0%

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

      11. *-inverses 15.0%

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

      12. metadata-eval 15.0%

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

      13. +-commutative 15.0%

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

   4. Simplified 15.0%

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

      1. +-commutative 15.0%

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

      2. metadata-eval 15.0%

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

      3. sub-neg 15.0%

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

      4. *-inverses 15.0%

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

      5. div-sub 15.0%

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

      6. clear-num 14.9%

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

      7. associate-/r/ 14.9%

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

      8. flip-- 6.7%

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

      9. associate-*r/ 6.6%

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

   6. Applied egg-rr 6.6%

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

   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 0.0%

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

       8. sqrt-unprod 41.3%

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

       9. sqr-neg 41.3%

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

       10. sqrt-unprod 78.3%

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

       11. add-sqr-sqrt 78.4%

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

   11. Applied egg-rr 78.4%

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

       1. expm1-def 78.6%

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

       2. expm1-log1p 78.6%

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

   13. Simplified 78.6%

       \[\leadsto \color{blue}{\frac{y}{x + y}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{-132}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{-255}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{-73}:\\ \;\;\;\;x \cdot \frac{\frac{x}{y}}{y - x}\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{-39}:\\ \;\;\;\;\frac{x}{y} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x + y}\\ \end{array} \]

Alternative 5: 58.2% accurate, 22.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.08 \cdot 10^{-131} \lor \neg \left(y \leq 3.4 \cdot 10^{-39}\right):\\ \;\;\;\;\frac{y}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.08e-131) (not (<= y 3.4e-39)))
   (/ y (+ x y))
   (+ (/ x y) -1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.08e-131) || !(y <= 3.4e-39)) {
		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 ((y <= (-1.08d-131)) .or. (.not. (y <= 3.4d-39))) 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 ((y <= -1.08e-131) || !(y <= 3.4e-39)) {
		tmp = y / (x + y);
	} else {
		tmp = (x / y) + -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.08e-131) or not (y <= 3.4e-39):
		tmp = y / (x + y)
	else:
		tmp = (x / y) + -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.08e-131) || !(y <= 3.4e-39))
		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 ((y <= -1.08e-131) || ~((y <= 3.4e-39)))
		tmp = y / (x + y);
	else
		tmp = (x / y) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.08e-131], N[Not[LessEqual[y, 3.4e-39]], $MachinePrecision]], N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.07999999999999996e-131 or 3.3999999999999999e-39 < 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-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 15.5%

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

      7. fabs-sqr 15.5%

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

      8. rem-square-sqrt 16.8%

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

      9. div-sub 16.8%

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

      10. sub-neg 16.8%

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

      11. *-inverses 16.8%

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

      12. metadata-eval 16.8%

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

      13. +-commutative 16.8%

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

   4. Simplified 16.8%

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

      1. +-commutative 16.8%

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

      2. metadata-eval 16.8%

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

      3. sub-neg 16.8%

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

      4. *-inverses 16.8%

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

      5. div-sub 16.8%

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

      6. clear-num 16.7%

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

      7. associate-/r/ 16.7%

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

      8. flip-- 6.4%

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

      9. associate-*r/ 6.3%

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

   6. Applied egg-rr 6.3%

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

   7. Taylor expanded in y around inf 2.0%

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

      1. neg-mul-1 2.0%

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

   9. Simplified 2.0%

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

       1. expm1-log1p-u 2.0%

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

       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 1.0%

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

       8. sqrt-unprod 18.2%

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

       9. sqr-neg 18.2%

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

       10. sqrt-unprod 33.0%

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

       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 70.2%

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

       2. expm1-log1p 70.2%

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

   13. Simplified 70.2%

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

   if -1.07999999999999996e-131 < y < 3.3999999999999999e-39

   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 52.2%

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

      9. div-sub 52.2%

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

      10. sub-neg 52.2%

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

      11. *-inverses 52.2%

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

      12. metadata-eval 52.2%

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

      13. +-commutative 52.2%

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

   4. Simplified 52.2%

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

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.08 \cdot 10^{-131} \lor \neg \left(y \leq 3.4 \cdot 10^{-39}\right):\\ \;\;\;\;\frac{y}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + -1\\ \end{array} \]

Alternative 6: 26.0% 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 27.2%

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

   7. fabs-sqr 27.2%

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

   8. rem-square-sqrt 28.3%

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

   9. div-sub 28.3%

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

   10. sub-neg 28.3%

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

   11. *-inverses 28.3%

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

   12. metadata-eval 28.3%

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

   13. +-commutative 28.3%

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

4. Simplified 28.3%

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

5. Taylor expanded in x around inf 29.0%

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

6. Final simplification 29.0%

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

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 27.2%

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

   7. fabs-sqr 27.2%

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

   8. rem-square-sqrt 28.3%

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

   9. div-sub 28.3%

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

   10. sub-neg 28.3%

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

   11. *-inverses 28.3%

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

   12. metadata-eval 28.3%

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

   13. +-commutative 28.3%

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

4. Simplified 28.3%

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

5. Taylor expanded in x around 0 1.3%

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

6. Final simplification 1.3%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023293 
(FPCore (x y)
  :name "Numeric.LinearAlgebra.Util:formatSparse from hmatrix-0.16.1.5"
  :precision binary64
  (/ (fabs (- x y)) (fabs y)))