Numeric.LinearAlgebra.Util:formatSparse from hmatrix-0.16.1.5

Percentage Accurate: 100.0% → 100.0%

Time: 5.0s

Alternatives: 8

Speedup: 2.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 2.0× speedup

Math

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

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: 56.8% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{x + y}\\ t_1 := \frac{x \cdot \frac{x}{y}}{y - x}\\ \mathbf{if}\;y \leq -1.15 \cdot 10^{-66}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -1.42 \cdot 10^{-182}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -1.8 \cdot 10^{-189}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -5 \cdot 10^{-262}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-304}:\\ \;\;\;\;\frac{x \cdot x}{y \cdot y}\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{-269}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{-78}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (+ x y))) (t_1 (/ (* x (/ x y)) (- y x))))
   (if (<= y -1.15e-66)
     t_0
     (if (<= y -1.42e-182)
       t_1
       (if (<= y -1.8e-189)
         t_0
         (if (<= y -5e-262)
           (/ x y)
           (if (<= y 2.2e-304)
             (/ (* x x) (* y y))
             (if (<= y 2.3e-269) (/ x y) (if (<= y 1.7e-78) t_1 t_0)))))))))

C

double code(double x, double y) {
	double t_0 = y / (x + y);
	double t_1 = (x * (x / y)) / (y - x);
	double tmp;
	if (y <= -1.15e-66) {
		tmp = t_0;
	} else if (y <= -1.42e-182) {
		tmp = t_1;
	} else if (y <= -1.8e-189) {
		tmp = t_0;
	} else if (y <= -5e-262) {
		tmp = x / y;
	} else if (y <= 2.2e-304) {
		tmp = (x * x) / (y * y);
	} else if (y <= 2.3e-269) {
		tmp = x / y;
	} else if (y <= 1.7e-78) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_0 = y / (x + y)
    t_1 = (x * (x / y)) / (y - x)
    if (y <= (-1.15d-66)) then
        tmp = t_0
    else if (y <= (-1.42d-182)) then
        tmp = t_1
    else if (y <= (-1.8d-189)) then
        tmp = t_0
    else if (y <= (-5d-262)) then
        tmp = x / y
    else if (y <= 2.2d-304) then
        tmp = (x * x) / (y * y)
    else if (y <= 2.3d-269) then
        tmp = x / y
    else if (y <= 1.7d-78) then
        tmp = t_1
    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 t_1 = (x * (x / y)) / (y - x);
	double tmp;
	if (y <= -1.15e-66) {
		tmp = t_0;
	} else if (y <= -1.42e-182) {
		tmp = t_1;
	} else if (y <= -1.8e-189) {
		tmp = t_0;
	} else if (y <= -5e-262) {
		tmp = x / y;
	} else if (y <= 2.2e-304) {
		tmp = (x * x) / (y * y);
	} else if (y <= 2.3e-269) {
		tmp = x / y;
	} else if (y <= 1.7e-78) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y / (x + y)
	t_1 = (x * (x / y)) / (y - x)
	tmp = 0
	if y <= -1.15e-66:
		tmp = t_0
	elif y <= -1.42e-182:
		tmp = t_1
	elif y <= -1.8e-189:
		tmp = t_0
	elif y <= -5e-262:
		tmp = x / y
	elif y <= 2.2e-304:
		tmp = (x * x) / (y * y)
	elif y <= 2.3e-269:
		tmp = x / y
	elif y <= 1.7e-78:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y / Float64(x + y))
	t_1 = Float64(Float64(x * Float64(x / y)) / Float64(y - x))
	tmp = 0.0
	if (y <= -1.15e-66)
		tmp = t_0;
	elseif (y <= -1.42e-182)
		tmp = t_1;
	elseif (y <= -1.8e-189)
		tmp = t_0;
	elseif (y <= -5e-262)
		tmp = Float64(x / y);
	elseif (y <= 2.2e-304)
		tmp = Float64(Float64(x * x) / Float64(y * y));
	elseif (y <= 2.3e-269)
		tmp = Float64(x / y);
	elseif (y <= 1.7e-78)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y / (x + y);
	t_1 = (x * (x / y)) / (y - x);
	tmp = 0.0;
	if (y <= -1.15e-66)
		tmp = t_0;
	elseif (y <= -1.42e-182)
		tmp = t_1;
	elseif (y <= -1.8e-189)
		tmp = t_0;
	elseif (y <= -5e-262)
		tmp = x / y;
	elseif (y <= 2.2e-304)
		tmp = (x * x) / (y * y);
	elseif (y <= 2.3e-269)
		tmp = x / y;
	elseif (y <= 1.7e-78)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision] / N[(y - x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.15e-66], t$95$0, If[LessEqual[y, -1.42e-182], t$95$1, If[LessEqual[y, -1.8e-189], t$95$0, If[LessEqual[y, -5e-262], N[(x / y), $MachinePrecision], If[LessEqual[y, 2.2e-304], N[(N[(x * x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.3e-269], N[(x / y), $MachinePrecision], If[LessEqual[y, 1.7e-78], t$95$1, t$95$0]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -1.14999999999999996e-66 or -1.4199999999999999e-182 < y < -1.80000000000000008e-189 or 1.70000000000000006e-78 < y

   1. Initial program 100.0%

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

      1. div-inv 99.7%

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

      2. add-sqr-sqrt 48.2%

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

      3. fabs-sqr 48.2%

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

      4. add-sqr-sqrt 49.0%

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

      5. *-commutative 49.0%

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

      6. add-sqr-sqrt 6.6%

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

      7. fabs-sqr 6.6%

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

      8. add-sqr-sqrt 13.9%

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

   3. Applied egg-rr 13.9%

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

      1. flip-- 8.3%

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

      2. associate-*r/ 7.1%

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

      3. +-commutative 7.1%

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

   5. Applied egg-rr 7.1%

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

   6. Taylor expanded in x around 0 1.6%

      \[\leadsto \frac{\frac{1}{y} \cdot \color{blue}{\left(-1 \cdot {y}^{2}\right)}}{y + x} \]
   7. Step-by-step derivation

      1. unpow2 1.6%

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

      2. mul-1-neg 1.6%

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

      3. distribute-rgt-neg-out 1.6%

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

   8. Simplified 1.6%

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

      1. associate-*r* 2.0%

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

      2. lft-mult-inverse 2.0%

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

      3. *-un-lft-identity 2.0%

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

      4. add-sqr-sqrt 0.9%

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

      5. sqrt-unprod 16.8%

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

      6. sqr-neg 16.8%

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

      7. sqrt-unprod 32.9%

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

      8. add-sqr-sqrt 69.8%

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

      9. add-log-exp 4.0%

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

      10. *-un-lft-identity 4.0%

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

      11. log-prod 4.0%

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

      12. add-log-exp 69.8%

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

      13. metadata-eval 69.8%

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

   10. Applied egg-rr 69.8%

       \[\leadsto \frac{\color{blue}{0 + y}}{y + x} \]
   11. Step-by-step derivation

       1. +-lft-identity 69.8%

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

   12. Simplified 69.8%

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

   if -1.14999999999999996e-66 < y < -1.4199999999999999e-182 or 2.3e-269 < y < 1.70000000000000006e-78

   1. Initial program 100.0%

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

      1. div-inv 99.8%

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

      2. add-sqr-sqrt 42.3%

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

      3. fabs-sqr 42.3%

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

      4. add-sqr-sqrt 42.8%

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

      5. *-commutative 42.8%

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

      6. add-sqr-sqrt 15.1%

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

      7. fabs-sqr 15.1%

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

      8. add-sqr-sqrt 28.1%

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

   3. Applied egg-rr 28.1%

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

      1. flip-- 28.3%

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

      2. associate-*r/ 28.2%

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

      3. +-commutative 28.2%

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

   5. Applied egg-rr 28.2%

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

   6. Taylor expanded in y around 0 28.3%

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

      1. unpow2 28.3%

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

      2. associate-/l* 28.6%

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

      3. associate-/r/ 28.7%

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

   8. Simplified 28.7%

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

      1. associate-*l/ 28.3%

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

      2. *-un-lft-identity 28.3%

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

      3. associate-*l/ 28.3%

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

      4. frac-2neg 28.3%

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

      5. div-inv 28.4%

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

      6. associate-*l/ 28.4%

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

      7. *-un-lft-identity 28.4%

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

      8. distribute-neg-frac 28.4%

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

      9. add-sqr-sqrt 15.3%

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

      10. sqrt-unprod 28.4%

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

      11. sqr-neg 28.4%

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

      12. sqrt-unprod 21.4%

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

      13. add-sqr-sqrt 49.9%

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

      14. frac-2neg 49.9%

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

      15. associate-*r/ 54.0%

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

      16. distribute-neg-in 54.0%

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

      17. add-sqr-sqrt 21.5%

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

      18. sqrt-unprod 54.1%

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

      19. sqr-neg 54.1%

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

      20. sqrt-unprod 32.8%

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

      21. add-sqr-sqrt 54.2%

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

      22. sub-neg 54.2%

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

   10. Applied egg-rr 54.2%

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

       1. associate-*r/ 54.3%

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

       2. *-rgt-identity 54.3%

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

   12. Simplified 54.3%

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

   if -1.80000000000000008e-189 < y < -4.99999999999999992e-262 or 2.2e-304 < y < 2.3e-269

   1. Initial program 100.0%

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

      1. div-inv 99.7%

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

      2. add-sqr-sqrt 53.2%

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

      3. fabs-sqr 53.2%

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

      4. add-sqr-sqrt 53.3%

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

      5. *-commutative 53.3%

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

      6. add-sqr-sqrt 36.6%

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

      7. fabs-sqr 36.6%

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

      8. add-sqr-sqrt 79.9%

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

   3. Applied egg-rr 79.9%

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

   4. Taylor expanded in y around 0 80.1%

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

   if -4.99999999999999992e-262 < y < 2.2e-304

   1. Initial program 100.0%

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

      1. div-inv 99.9%

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

      2. add-sqr-sqrt 49.7%

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

      3. fabs-sqr 49.7%

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

      4. add-sqr-sqrt 49.9%

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

      5. *-commutative 49.9%

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

      6. add-sqr-sqrt 0.0%

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

      7. fabs-sqr 0.0%

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

      8. add-sqr-sqrt 43.0%

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

   3. Applied egg-rr 43.0%

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

      1. flip-- 43.3%

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

      2. associate-*r/ 43.2%

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

      3. +-commutative 43.2%

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

   5. Applied egg-rr 43.2%

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

   6. Taylor expanded in y around 0 43.3%

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

      1. unpow2 43.3%

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

      2. associate-/l* 44.3%

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

      3. associate-/r/ 44.3%

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

   8. Simplified 44.3%

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

   9. Taylor expanded in x around 0 72.6%

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

       1. unpow2 72.6%

          \[\leadsto \frac{\color{blue}{x \cdot x}}{{y}^{2}} \]

       2. unpow2 72.6%

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

   11. Simplified 72.6%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.15 \cdot 10^{-66}:\\ \;\;\;\;\frac{y}{x + y}\\ \mathbf{elif}\;y \leq -1.42 \cdot 10^{-182}:\\ \;\;\;\;\frac{x \cdot \frac{x}{y}}{y - x}\\ \mathbf{elif}\;y \leq -1.8 \cdot 10^{-189}:\\ \;\;\;\;\frac{y}{x + y}\\ \mathbf{elif}\;y \leq -5 \cdot 10^{-262}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-304}:\\ \;\;\;\;\frac{x \cdot x}{y \cdot y}\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{-269}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{-78}:\\ \;\;\;\;\frac{x \cdot \frac{x}{y}}{y - x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x + y}\\ \end{array} \]

Alternative 3: 56.0% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{x + y}\\ t_1 := \frac{\frac{x}{y}}{\frac{y - x}{x}}\\ \mathbf{if}\;y \leq -1.65 \cdot 10^{-66}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -9.6 \cdot 10^{-184}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -6.5 \cdot 10^{-189}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -2.25 \cdot 10^{-262}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-304}:\\ \;\;\;\;\frac{x \cdot x}{y \cdot y}\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{-269}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{+57}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (+ x y))) (t_1 (/ (/ x y) (/ (- y x) x))))
   (if (<= y -1.65e-66)
     t_0
     (if (<= y -9.6e-184)
       t_1
       (if (<= y -6.5e-189)
         t_0
         (if (<= y -2.25e-262)
           (/ x y)
           (if (<= y 2.2e-304)
             (/ (* x x) (* y y))
             (if (<= y 1.5e-269) (/ x y) (if (<= y 5.8e+57) t_1 t_0)))))))))

C

double code(double x, double y) {
	double t_0 = y / (x + y);
	double t_1 = (x / y) / ((y - x) / x);
	double tmp;
	if (y <= -1.65e-66) {
		tmp = t_0;
	} else if (y <= -9.6e-184) {
		tmp = t_1;
	} else if (y <= -6.5e-189) {
		tmp = t_0;
	} else if (y <= -2.25e-262) {
		tmp = x / y;
	} else if (y <= 2.2e-304) {
		tmp = (x * x) / (y * y);
	} else if (y <= 1.5e-269) {
		tmp = x / y;
	} else if (y <= 5.8e+57) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_0 = y / (x + y)
    t_1 = (x / y) / ((y - x) / x)
    if (y <= (-1.65d-66)) then
        tmp = t_0
    else if (y <= (-9.6d-184)) then
        tmp = t_1
    else if (y <= (-6.5d-189)) then
        tmp = t_0
    else if (y <= (-2.25d-262)) then
        tmp = x / y
    else if (y <= 2.2d-304) then
        tmp = (x * x) / (y * y)
    else if (y <= 1.5d-269) then
        tmp = x / y
    else if (y <= 5.8d+57) then
        tmp = t_1
    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 t_1 = (x / y) / ((y - x) / x);
	double tmp;
	if (y <= -1.65e-66) {
		tmp = t_0;
	} else if (y <= -9.6e-184) {
		tmp = t_1;
	} else if (y <= -6.5e-189) {
		tmp = t_0;
	} else if (y <= -2.25e-262) {
		tmp = x / y;
	} else if (y <= 2.2e-304) {
		tmp = (x * x) / (y * y);
	} else if (y <= 1.5e-269) {
		tmp = x / y;
	} else if (y <= 5.8e+57) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y / (x + y)
	t_1 = (x / y) / ((y - x) / x)
	tmp = 0
	if y <= -1.65e-66:
		tmp = t_0
	elif y <= -9.6e-184:
		tmp = t_1
	elif y <= -6.5e-189:
		tmp = t_0
	elif y <= -2.25e-262:
		tmp = x / y
	elif y <= 2.2e-304:
		tmp = (x * x) / (y * y)
	elif y <= 1.5e-269:
		tmp = x / y
	elif y <= 5.8e+57:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y / Float64(x + y))
	t_1 = Float64(Float64(x / y) / Float64(Float64(y - x) / x))
	tmp = 0.0
	if (y <= -1.65e-66)
		tmp = t_0;
	elseif (y <= -9.6e-184)
		tmp = t_1;
	elseif (y <= -6.5e-189)
		tmp = t_0;
	elseif (y <= -2.25e-262)
		tmp = Float64(x / y);
	elseif (y <= 2.2e-304)
		tmp = Float64(Float64(x * x) / Float64(y * y));
	elseif (y <= 1.5e-269)
		tmp = Float64(x / y);
	elseif (y <= 5.8e+57)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y / (x + y);
	t_1 = (x / y) / ((y - x) / x);
	tmp = 0.0;
	if (y <= -1.65e-66)
		tmp = t_0;
	elseif (y <= -9.6e-184)
		tmp = t_1;
	elseif (y <= -6.5e-189)
		tmp = t_0;
	elseif (y <= -2.25e-262)
		tmp = x / y;
	elseif (y <= 2.2e-304)
		tmp = (x * x) / (y * y);
	elseif (y <= 1.5e-269)
		tmp = x / y;
	elseif (y <= 5.8e+57)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x / y), $MachinePrecision] / N[(N[(y - x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.65e-66], t$95$0, If[LessEqual[y, -9.6e-184], t$95$1, If[LessEqual[y, -6.5e-189], t$95$0, If[LessEqual[y, -2.25e-262], N[(x / y), $MachinePrecision], If[LessEqual[y, 2.2e-304], N[(N[(x * x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.5e-269], N[(x / y), $MachinePrecision], If[LessEqual[y, 5.8e+57], t$95$1, t$95$0]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -1.6499999999999999e-66 or -9.60000000000000097e-184 < y < -6.5000000000000001e-189 or 5.8000000000000003e57 < y

   1. Initial program 100.0%

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

      1. div-inv 99.8%

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

      2. add-sqr-sqrt 55.0%

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

      3. fabs-sqr 55.0%

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

      4. add-sqr-sqrt 55.8%

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

      5. *-commutative 55.8%

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

      6. add-sqr-sqrt 2.8%

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

      7. fabs-sqr 2.8%

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

      8. add-sqr-sqrt 11.9%

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

   3. Applied egg-rr 11.9%

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

      1. flip-- 5.6%

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

      2. associate-*r/ 4.9%

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

      3. +-commutative 4.9%

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

   5. Applied egg-rr 4.9%

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

   6. Taylor expanded in x around 0 1.5%

      \[\leadsto \frac{\frac{1}{y} \cdot \color{blue}{\left(-1 \cdot {y}^{2}\right)}}{y + x} \]
   7. Step-by-step derivation

      1. unpow2 1.5%

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

      2. mul-1-neg 1.5%

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

      3. distribute-rgt-neg-out 1.5%

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

   8. Simplified 1.5%

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

      1. associate-*r* 1.9%

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

      2. lft-mult-inverse 1.9%

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

      3. *-un-lft-identity 1.9%

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

      4. add-sqr-sqrt 1.2%

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

      5. sqrt-unprod 12.3%

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

      6. sqr-neg 12.3%

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

      7. sqrt-unprod 32.4%

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

      8. add-sqr-sqrt 78.4%

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

      9. add-log-exp 3.8%

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

      10. *-un-lft-identity 3.8%

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

      11. log-prod 3.8%

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

      12. add-log-exp 78.4%

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

      13. metadata-eval 78.4%

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

   10. Applied egg-rr 78.4%

       \[\leadsto \frac{\color{blue}{0 + y}}{y + x} \]
   11. Step-by-step derivation

       1. +-lft-identity 78.4%

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

   12. Simplified 78.4%

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

   if -1.6499999999999999e-66 < y < -9.60000000000000097e-184 or 1.4999999999999999e-269 < y < 5.8000000000000003e57

   1. Initial program 100.0%

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

      1. div-inv 99.7%

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

      2. add-sqr-sqrt 33.5%

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

      3. fabs-sqr 33.5%

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

      4. add-sqr-sqrt 34.1%

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

      5. *-commutative 34.1%

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

      6. add-sqr-sqrt 17.9%

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

      7. fabs-sqr 17.9%

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

      8. add-sqr-sqrt 25.5%

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

   3. Applied egg-rr 25.5%

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

      1. flip-- 24.4%

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

      2. associate-*r/ 23.2%

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

      3. +-commutative 23.2%

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

   5. Applied egg-rr 23.2%

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

   6. Taylor expanded in y around 0 23.0%

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

      1. unpow2 23.0%

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

      2. associate-/l* 23.1%

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

      3. associate-/r/ 23.2%

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

   8. Simplified 23.2%

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

      1. associate-*l/ 23.0%

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

      2. *-un-lft-identity 23.0%

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

      3. associate-*l/ 23.0%

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

      4. frac-2neg 23.0%

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

      5. div-inv 23.0%

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

      6. associate-*l/ 23.0%

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

      7. *-un-lft-identity 23.0%

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

      8. distribute-neg-frac 23.0%

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

      9. add-sqr-sqrt 15.3%

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

      10. sqrt-unprod 23.0%

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

      11. sqr-neg 23.0%

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

      12. sqrt-unprod 12.6%

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

      13. add-sqr-sqrt 34.0%

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

      14. frac-2neg 34.0%

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

      15. associate-*r/ 36.4%

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

      16. distribute-neg-in 36.4%

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

      17. add-sqr-sqrt 12.6%

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

      18. sqrt-unprod 36.2%

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

      19. sqr-neg 36.2%

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

      20. sqrt-unprod 23.7%

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

      21. add-sqr-sqrt 36.2%

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

      22. sub-neg 36.2%

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

   10. Applied egg-rr 36.2%

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

       1. associate-*r/ 36.3%

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

       2. *-rgt-identity 36.3%

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

       3. *-commutative 36.3%

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

       4. associate-/l* 52.6%

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

   12. Simplified 52.6%

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

   if -6.5000000000000001e-189 < y < -2.24999999999999999e-262 or 2.2e-304 < y < 1.4999999999999999e-269

   1. Initial program 100.0%

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

      1. div-inv 99.7%

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

      2. add-sqr-sqrt 53.2%

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

      3. fabs-sqr 53.2%

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

      4. add-sqr-sqrt 53.3%

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

      5. *-commutative 53.3%

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

      6. add-sqr-sqrt 36.6%

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

      7. fabs-sqr 36.6%

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

      8. add-sqr-sqrt 79.9%

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

   3. Applied egg-rr 79.9%

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

   4. Taylor expanded in y around 0 80.1%

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

   if -2.24999999999999999e-262 < y < 2.2e-304

   1. Initial program 100.0%

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

      1. div-inv 99.9%

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

      2. add-sqr-sqrt 49.7%

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

      3. fabs-sqr 49.7%

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

      4. add-sqr-sqrt 49.9%

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

      5. *-commutative 49.9%

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

      6. add-sqr-sqrt 0.0%

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

      7. fabs-sqr 0.0%

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

      8. add-sqr-sqrt 43.0%

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

   3. Applied egg-rr 43.0%

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

      1. flip-- 43.3%

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

      2. associate-*r/ 43.2%

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

      3. +-commutative 43.2%

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

   5. Applied egg-rr 43.2%

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

   6. Taylor expanded in y around 0 43.3%

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

      1. unpow2 43.3%

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

      2. associate-/l* 44.3%

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

      3. associate-/r/ 44.3%

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

   8. Simplified 44.3%

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

   9. Taylor expanded in x around 0 72.6%

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

       1. unpow2 72.6%

          \[\leadsto \frac{\color{blue}{x \cdot x}}{{y}^{2}} \]

       2. unpow2 72.6%

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

   11. Simplified 72.6%

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

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.65 \cdot 10^{-66}:\\ \;\;\;\;\frac{y}{x + y}\\ \mathbf{elif}\;y \leq -9.6 \cdot 10^{-184}:\\ \;\;\;\;\frac{\frac{x}{y}}{\frac{y - x}{x}}\\ \mathbf{elif}\;y \leq -6.5 \cdot 10^{-189}:\\ \;\;\;\;\frac{y}{x + y}\\ \mathbf{elif}\;y \leq -2.25 \cdot 10^{-262}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-304}:\\ \;\;\;\;\frac{x \cdot x}{y \cdot y}\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{-269}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{+57}:\\ \;\;\;\;\frac{\frac{x}{y}}{\frac{y - x}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x + y}\\ \end{array} \]

Alternative 4: 55.3% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{x + y}\\ \mathbf{if}\;y \leq -1.25 \cdot 10^{-66}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -2 \cdot 10^{-183}:\\ \;\;\;\;\frac{\frac{x}{y} \cdot \left(-x\right)}{x + y}\\ \mathbf{elif}\;y \leq -1 \cdot 10^{-188}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -6.5 \cdot 10^{-262}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 2.35 \cdot 10^{-304}:\\ \;\;\;\;\frac{x \cdot x}{y \cdot y}\\ \mathbf{elif}\;y \leq 9.6 \cdot 10^{-270}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{+57}:\\ \;\;\;\;\frac{\frac{x}{y}}{\frac{y - x}{x}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (+ x y))))
   (if (<= y -1.25e-66)
     t_0
     (if (<= y -2e-183)
       (/ (* (/ x y) (- x)) (+ x y))
       (if (<= y -1e-188)
         t_0
         (if (<= y -6.5e-262)
           (/ x y)
           (if (<= y 2.35e-304)
             (/ (* x x) (* y y))
             (if (<= y 9.6e-270)
               (/ x y)
               (if (<= y 3.3e+57) (/ (/ x y) (/ (- y x) x)) t_0)))))))))

C

double code(double x, double y) {
	double t_0 = y / (x + y);
	double tmp;
	if (y <= -1.25e-66) {
		tmp = t_0;
	} else if (y <= -2e-183) {
		tmp = ((x / y) * -x) / (x + y);
	} else if (y <= -1e-188) {
		tmp = t_0;
	} else if (y <= -6.5e-262) {
		tmp = x / y;
	} else if (y <= 2.35e-304) {
		tmp = (x * x) / (y * y);
	} else if (y <= 9.6e-270) {
		tmp = x / y;
	} else if (y <= 3.3e+57) {
		tmp = (x / y) / ((y - x) / x);
	} 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 <= (-1.25d-66)) then
        tmp = t_0
    else if (y <= (-2d-183)) then
        tmp = ((x / y) * -x) / (x + y)
    else if (y <= (-1d-188)) then
        tmp = t_0
    else if (y <= (-6.5d-262)) then
        tmp = x / y
    else if (y <= 2.35d-304) then
        tmp = (x * x) / (y * y)
    else if (y <= 9.6d-270) then
        tmp = x / y
    else if (y <= 3.3d+57) then
        tmp = (x / y) / ((y - x) / x)
    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 <= -1.25e-66) {
		tmp = t_0;
	} else if (y <= -2e-183) {
		tmp = ((x / y) * -x) / (x + y);
	} else if (y <= -1e-188) {
		tmp = t_0;
	} else if (y <= -6.5e-262) {
		tmp = x / y;
	} else if (y <= 2.35e-304) {
		tmp = (x * x) / (y * y);
	} else if (y <= 9.6e-270) {
		tmp = x / y;
	} else if (y <= 3.3e+57) {
		tmp = (x / y) / ((y - x) / x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y / (x + y)
	tmp = 0
	if y <= -1.25e-66:
		tmp = t_0
	elif y <= -2e-183:
		tmp = ((x / y) * -x) / (x + y)
	elif y <= -1e-188:
		tmp = t_0
	elif y <= -6.5e-262:
		tmp = x / y
	elif y <= 2.35e-304:
		tmp = (x * x) / (y * y)
	elif y <= 9.6e-270:
		tmp = x / y
	elif y <= 3.3e+57:
		tmp = (x / y) / ((y - x) / x)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y / Float64(x + y))
	tmp = 0.0
	if (y <= -1.25e-66)
		tmp = t_0;
	elseif (y <= -2e-183)
		tmp = Float64(Float64(Float64(x / y) * Float64(-x)) / Float64(x + y));
	elseif (y <= -1e-188)
		tmp = t_0;
	elseif (y <= -6.5e-262)
		tmp = Float64(x / y);
	elseif (y <= 2.35e-304)
		tmp = Float64(Float64(x * x) / Float64(y * y));
	elseif (y <= 9.6e-270)
		tmp = Float64(x / y);
	elseif (y <= 3.3e+57)
		tmp = Float64(Float64(x / y) / Float64(Float64(y - x) / x));
	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 <= -1.25e-66)
		tmp = t_0;
	elseif (y <= -2e-183)
		tmp = ((x / y) * -x) / (x + y);
	elseif (y <= -1e-188)
		tmp = t_0;
	elseif (y <= -6.5e-262)
		tmp = x / y;
	elseif (y <= 2.35e-304)
		tmp = (x * x) / (y * y);
	elseif (y <= 9.6e-270)
		tmp = x / y;
	elseif (y <= 3.3e+57)
		tmp = (x / y) / ((y - x) / x);
	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, -1.25e-66], t$95$0, If[LessEqual[y, -2e-183], N[(N[(N[(x / y), $MachinePrecision] * (-x)), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -1e-188], t$95$0, If[LessEqual[y, -6.5e-262], N[(x / y), $MachinePrecision], If[LessEqual[y, 2.35e-304], N[(N[(x * x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 9.6e-270], N[(x / y), $MachinePrecision], If[LessEqual[y, 3.3e+57], N[(N[(x / y), $MachinePrecision] / N[(N[(y - x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if y < -1.2499999999999999e-66 or -2.00000000000000001e-183 < y < -9.9999999999999995e-189 or 3.3000000000000001e57 < y

   1. Initial program 100.0%

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

      1. div-inv 99.8%

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

      2. add-sqr-sqrt 55.0%

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

      3. fabs-sqr 55.0%

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

      4. add-sqr-sqrt 55.8%

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

      5. *-commutative 55.8%

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

      6. add-sqr-sqrt 2.8%

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

      7. fabs-sqr 2.8%

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

      8. add-sqr-sqrt 11.9%

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

   3. Applied egg-rr 11.9%

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

      1. flip-- 5.6%

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

      2. associate-*r/ 4.9%

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

      3. +-commutative 4.9%

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

   5. Applied egg-rr 4.9%

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

   6. Taylor expanded in x around 0 1.5%

      \[\leadsto \frac{\frac{1}{y} \cdot \color{blue}{\left(-1 \cdot {y}^{2}\right)}}{y + x} \]
   7. Step-by-step derivation

      1. unpow2 1.5%

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

      2. mul-1-neg 1.5%

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

      3. distribute-rgt-neg-out 1.5%

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

   8. Simplified 1.5%

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

      1. associate-*r* 1.9%

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

      2. lft-mult-inverse 1.9%

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

      3. *-un-lft-identity 1.9%

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

      4. add-sqr-sqrt 1.2%

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

      5. sqrt-unprod 12.3%

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

      6. sqr-neg 12.3%

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

      7. sqrt-unprod 32.4%

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

      8. add-sqr-sqrt 78.4%

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

      9. add-log-exp 3.8%

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

      10. *-un-lft-identity 3.8%

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

      11. log-prod 3.8%

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

      12. add-log-exp 78.4%

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

      13. metadata-eval 78.4%

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

   10. Applied egg-rr 78.4%

       \[\leadsto \frac{\color{blue}{0 + y}}{y + x} \]
   11. Step-by-step derivation

       1. +-lft-identity 78.4%

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

   12. Simplified 78.4%

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

   if -1.2499999999999999e-66 < y < -2.00000000000000001e-183

   1. Initial program 100.0%

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

      1. div-inv 99.8%

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

      2. add-sqr-sqrt 67.9%

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

      3. fabs-sqr 67.9%

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

      4. add-sqr-sqrt 68.4%

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

      5. *-commutative 68.4%

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

      6. add-sqr-sqrt 0.0%

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

      7. fabs-sqr 0.0%

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

      8. add-sqr-sqrt 32.1%

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

   3. Applied egg-rr 32.1%

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

      1. flip-- 32.1%

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

      2. associate-*r/ 32.0%

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

      3. +-commutative 32.0%

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

   5. Applied egg-rr 32.0%

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

   6. Taylor expanded in y around 0 32.2%

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

      1. unpow2 32.2%

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

      2. associate-/l* 32.2%

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

      3. associate-/r/ 32.3%

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

   8. Simplified 32.3%

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

      1. associate-*l/ 32.2%

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

      2. frac-2neg 32.2%

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

      3. add-sqr-sqrt 32.1%

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

      4. sqrt-unprod 32.0%

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

      5. sqr-neg 32.0%

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

      6. sqrt-unprod 0.0%

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

      7. add-sqr-sqrt 53.3%

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

      8. distribute-neg-frac 53.3%

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

      9. *-un-lft-identity 53.3%

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

      10. associate-*l/ 53.2%

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

      11. neg-sub0 53.2%

          \[\leadsto \frac{\color{blue}{0 - \frac{1}{y} \cdot \left(x \cdot x\right)}}{y + x} \]

      12. *-commutative 53.2%

          \[\leadsto \frac{0 - \color{blue}{\left(x \cdot x\right) \cdot \frac{1}{y}}}{y + x} \]

      13. associate-*r* 53.1%

          \[\leadsto \frac{0 - \color{blue}{x \cdot \left(x \cdot \frac{1}{y}\right)}}{y + x} \]

      14. div-inv 53.3%

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

   10. Applied egg-rr 53.3%

       \[\leadsto \frac{\color{blue}{0 - x \cdot \frac{x}{y}}}{y + x} \]
   11. Step-by-step derivation

       1. neg-sub0 53.3%

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

       2. distribute-lft-neg-in 53.3%

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

       3. *-commutative 53.3%

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

   12. Simplified 53.3%

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

   if -9.9999999999999995e-189 < y < -6.5000000000000003e-262 or 2.35e-304 < y < 9.60000000000000007e-270

   1. Initial program 100.0%

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

      1. div-inv 99.7%

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

      2. add-sqr-sqrt 53.2%

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

      3. fabs-sqr 53.2%

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

      4. add-sqr-sqrt 53.3%

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

      5. *-commutative 53.3%

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

      6. add-sqr-sqrt 36.6%

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

      7. fabs-sqr 36.6%

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

      8. add-sqr-sqrt 79.9%

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

   3. Applied egg-rr 79.9%

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

   4. Taylor expanded in y around 0 80.1%

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

   if -6.5000000000000003e-262 < y < 2.35e-304

   1. Initial program 100.0%

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

      1. div-inv 99.9%

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

      2. add-sqr-sqrt 49.7%

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

      3. fabs-sqr 49.7%

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

      4. add-sqr-sqrt 49.9%

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

      5. *-commutative 49.9%

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

      6. add-sqr-sqrt 0.0%

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

      7. fabs-sqr 0.0%

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

      8. add-sqr-sqrt 43.0%

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

   3. Applied egg-rr 43.0%

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

      1. flip-- 43.3%

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

      2. associate-*r/ 43.2%

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

      3. +-commutative 43.2%

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

   5. Applied egg-rr 43.2%

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

   6. Taylor expanded in y around 0 43.3%

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

      1. unpow2 43.3%

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

      2. associate-/l* 44.3%

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

      3. associate-/r/ 44.3%

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

   8. Simplified 44.3%

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

   9. Taylor expanded in x around 0 72.6%

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

       1. unpow2 72.6%

          \[\leadsto \frac{\color{blue}{x \cdot x}}{{y}^{2}} \]

       2. unpow2 72.6%

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

   11. Simplified 72.6%

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

   if 9.60000000000000007e-270 < y < 3.3000000000000001e57

   1. Initial program 99.9%

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

      1. div-inv 99.6%

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

      2. add-sqr-sqrt 22.8%

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

      3. fabs-sqr 22.8%

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

      4. add-sqr-sqrt 23.5%

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

      5. *-commutative 23.5%

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

      6. add-sqr-sqrt 23.5%

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

      7. fabs-sqr 23.5%

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

      8. add-sqr-sqrt 23.5%

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

   3. Applied egg-rr 23.5%

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

      1. flip-- 22.0%

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

      2. associate-*r/ 20.5%

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

      3. +-commutative 20.5%

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

   5. Applied egg-rr 20.5%

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

   6. Taylor expanded in y around 0 20.2%

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

      1. unpow2 20.2%

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

      2. associate-/l* 20.3%

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

      3. associate-/r/ 20.4%

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

   8. Simplified 20.4%

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

      1. associate-*l/ 20.2%

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

      2. *-un-lft-identity 20.2%

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

      3. associate-*l/ 20.1%

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

      4. frac-2neg 20.1%

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

      5. div-inv 20.2%

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

      6. associate-*l/ 20.1%

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

      7. *-un-lft-identity 20.1%

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

      8. distribute-neg-frac 20.1%

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

      9. add-sqr-sqrt 20.1%

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

      10. sqrt-unprod 15.2%

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

      11. sqr-neg 15.2%

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

      12. sqrt-unprod 0.0%

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

      13. add-sqr-sqrt 28.0%

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

      14. frac-2neg 28.0%

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

      15. associate-*r/ 31.2%

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

      16. distribute-neg-in 31.2%

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

      17. add-sqr-sqrt 0.0%

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

      18. sqrt-unprod 30.9%

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

      19. sqr-neg 30.9%

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

      20. sqrt-unprod 31.0%

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

      21. add-sqr-sqrt 31.0%

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

      22. sub-neg 31.0%

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

   10. Applied egg-rr 31.0%

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

       1. associate-*r/ 31.1%

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

       2. *-rgt-identity 31.1%

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

       3. *-commutative 31.1%

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

       4. associate-/l* 52.5%

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

   12. Simplified 52.5%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.25 \cdot 10^{-66}:\\ \;\;\;\;\frac{y}{x + y}\\ \mathbf{elif}\;y \leq -2 \cdot 10^{-183}:\\ \;\;\;\;\frac{\frac{x}{y} \cdot \left(-x\right)}{x + y}\\ \mathbf{elif}\;y \leq -1 \cdot 10^{-188}:\\ \;\;\;\;\frac{y}{x + y}\\ \mathbf{elif}\;y \leq -6.5 \cdot 10^{-262}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 2.35 \cdot 10^{-304}:\\ \;\;\;\;\frac{x \cdot x}{y \cdot y}\\ \mathbf{elif}\;y \leq 9.6 \cdot 10^{-270}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{+57}:\\ \;\;\;\;\frac{\frac{x}{y}}{\frac{y - x}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x + y}\\ \end{array} \]

Alternative 5: 56.7% accurate, 18.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{+40}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{+156}:\\ \;\;\;\;\frac{y}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot x}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -7e+40)
   (/ x y)
   (if (<= x 1.65e+156) (/ y (+ x y)) (/ (* x x) (* y y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -7e+40) {
		tmp = x / y;
	} else if (x <= 1.65e+156) {
		tmp = y / (x + y);
	} else {
		tmp = (x * x) / (y * 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 <= (-7d+40)) then
        tmp = x / y
    else if (x <= 1.65d+156) then
        tmp = y / (x + y)
    else
        tmp = (x * x) / (y * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -7e+40) {
		tmp = x / y;
	} else if (x <= 1.65e+156) {
		tmp = y / (x + y);
	} else {
		tmp = (x * x) / (y * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -7e+40:
		tmp = x / y
	elif x <= 1.65e+156:
		tmp = y / (x + y)
	else:
		tmp = (x * x) / (y * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -7e+40)
		tmp = Float64(x / y);
	elseif (x <= 1.65e+156)
		tmp = Float64(y / Float64(x + y));
	else
		tmp = Float64(Float64(x * x) / Float64(y * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -7e+40)
		tmp = x / y;
	elseif (x <= 1.65e+156)
		tmp = y / (x + y);
	else
		tmp = (x * x) / (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -7e+40], N[(x / y), $MachinePrecision], If[LessEqual[x, 1.65e+156], N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.9999999999999998e40

   1. Initial program 100.0%

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

      1. div-inv 99.7%

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

      2. add-sqr-sqrt 12.2%

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

      3. fabs-sqr 12.2%

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

      4. add-sqr-sqrt 12.6%

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

      5. *-commutative 12.6%

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

      6. add-sqr-sqrt 0.2%

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

      7. fabs-sqr 0.2%

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

      8. add-sqr-sqrt 41.9%

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

   3. Applied egg-rr 41.9%

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

   4. Taylor expanded in y around 0 42.5%

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

   if -6.9999999999999998e40 < x < 1.6499999999999999e156

   1. Initial program 100.0%

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

      1. div-inv 99.7%

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

      2. add-sqr-sqrt 53.7%

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

      3. fabs-sqr 53.7%

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

      4. add-sqr-sqrt 54.6%

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

      5. *-commutative 54.6%

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

      6. add-sqr-sqrt 9.3%

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

      7. fabs-sqr 9.3%

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

      8. add-sqr-sqrt 15.6%

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

   3. Applied egg-rr 15.6%

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

      1. flip-- 12.4%

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

      2. associate-*r/ 11.8%

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

      3. +-commutative 11.8%

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

   5. Applied egg-rr 11.8%

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

   6. Taylor expanded in x around 0 1.9%

      \[\leadsto \frac{\frac{1}{y} \cdot \color{blue}{\left(-1 \cdot {y}^{2}\right)}}{y + x} \]
   7. Step-by-step derivation

      1. unpow2 1.9%

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

      2. mul-1-neg 1.9%

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

      3. distribute-rgt-neg-out 1.9%

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

   8. Simplified 1.9%

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

      1. associate-*r* 2.0%

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

      2. lft-mult-inverse 2.0%

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

      3. *-un-lft-identity 2.0%

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

      4. add-sqr-sqrt 1.0%

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

      5. sqrt-unprod 18.2%

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

      6. sqr-neg 18.2%

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

      7. sqrt-unprod 33.3%

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

      8. add-sqr-sqrt 68.1%

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

      9. add-log-exp 4.1%

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

      10. *-un-lft-identity 4.1%

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

      11. log-prod 4.1%

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

      12. add-log-exp 68.1%

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

      13. metadata-eval 68.1%

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

   10. Applied egg-rr 68.1%

       \[\leadsto \frac{\color{blue}{0 + y}}{y + x} \]
   11. Step-by-step derivation

       1. +-lft-identity 68.1%

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

   12. Simplified 68.1%

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

   if 1.6499999999999999e156 < x

   1. Initial program 100.0%

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

      1. div-inv 100.0%

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

      2. add-sqr-sqrt 90.4%

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

      3. fabs-sqr 90.4%

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

      4. add-sqr-sqrt 90.7%

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

      5. *-commutative 90.7%

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

      6. add-sqr-sqrt 43.8%

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

      7. fabs-sqr 43.8%

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

      8. add-sqr-sqrt 44.1%

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

   3. Applied egg-rr 44.1%

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

      1. flip-- 31.6%

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

      2. associate-*r/ 31.6%

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

      3. +-commutative 31.6%

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

   5. Applied egg-rr 31.6%

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

   6. Taylor expanded in y around 0 32.3%

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

      1. unpow2 32.3%

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

      2. associate-/l* 37.3%

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

      3. associate-/r/ 37.3%

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

   8. Simplified 37.3%

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

   9. Taylor expanded in x around 0 57.3%

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

       1. unpow2 57.3%

          \[\leadsto \frac{\color{blue}{x \cdot x}}{{y}^{2}} \]

       2. unpow2 57.3%

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

   11. Simplified 57.3%

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

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{+40}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{+156}:\\ \;\;\;\;\frac{y}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot x}{y \cdot y}\\ \end{array} \]

Alternative 6: 58.6% accurate, 22.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.52 \cdot 10^{-58} \lor \neg \left(y \leq 1.75 \cdot 10^{-69}\right):\\ \;\;\;\;\frac{y}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.52e-58) (not (<= y 1.75e-69))) (/ y (+ x y)) (/ x y)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.52e-58) || !(y <= 1.75e-69)) {
		tmp = y / (x + y);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-1.52d-58)) .or. (.not. (y <= 1.75d-69))) then
        tmp = y / (x + y)
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.52e-58) || !(y <= 1.75e-69)) {
		tmp = y / (x + y);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.52e-58) or not (y <= 1.75e-69):
		tmp = y / (x + y)
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.52e-58) || !(y <= 1.75e-69))
		tmp = Float64(y / Float64(x + y));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.52e-58) || ~((y <= 1.75e-69)))
		tmp = y / (x + y);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.52e-58], N[Not[LessEqual[y, 1.75e-69]], $MachinePrecision]], N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.51999999999999993e-58 or 1.7500000000000001e-69 < y

   1. Initial program 100.0%

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

      1. div-inv 99.7%

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

      2. add-sqr-sqrt 47.1%

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

      3. fabs-sqr 47.1%

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

      4. add-sqr-sqrt 48.0%

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

      5. *-commutative 48.0%

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

      6. add-sqr-sqrt 5.1%

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

      7. fabs-sqr 5.1%

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

      8. add-sqr-sqrt 11.5%

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

   3. Applied egg-rr 11.5%

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

      1. flip-- 5.5%

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

      2. associate-*r/ 4.9%

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

      3. +-commutative 4.9%

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

   5. Applied egg-rr 4.9%

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

   6. Taylor expanded in x around 0 1.6%

      \[\leadsto \frac{\frac{1}{y} \cdot \color{blue}{\left(-1 \cdot {y}^{2}\right)}}{y + x} \]
   7. Step-by-step derivation

      1. unpow2 1.6%

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

      2. mul-1-neg 1.6%

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

      3. distribute-rgt-neg-out 1.6%

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

   8. Simplified 1.6%

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

      1. associate-*r* 2.0%

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

      2. lft-mult-inverse 2.0%

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

      3. *-un-lft-identity 2.0%

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

      4. add-sqr-sqrt 0.9%

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

      5. sqrt-unprod 17.0%

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

      6. sqr-neg 17.0%

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

      7. sqrt-unprod 34.1%

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

      8. add-sqr-sqrt 71.1%

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

      9. add-log-exp 4.0%

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

      10. *-un-lft-identity 4.0%

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

      11. log-prod 4.0%

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

      12. add-log-exp 71.1%

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

      13. metadata-eval 71.1%

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

   10. Applied egg-rr 71.1%

       \[\leadsto \frac{\color{blue}{0 + y}}{y + x} \]
   11. Step-by-step derivation

       1. +-lft-identity 71.1%

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

   12. Simplified 71.1%

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

   if -1.51999999999999993e-58 < y < 1.7500000000000001e-69

   1. Initial program 100.0%

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

      1. div-inv 99.8%

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

      2. add-sqr-sqrt 48.8%

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

      3. fabs-sqr 48.8%

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

      4. add-sqr-sqrt 49.1%

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

      5. *-commutative 49.1%

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

      6. add-sqr-sqrt 21.1%

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

      7. fabs-sqr 21.1%

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

      8. add-sqr-sqrt 48.3%

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

   3. Applied egg-rr 48.3%

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

   4. Taylor expanded in y around 0 48.6%

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

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.52 \cdot 10^{-58} \lor \neg \left(y \leq 1.75 \cdot 10^{-69}\right):\\ \;\;\;\;\frac{y}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 7: 26.4% 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. Step-by-step derivation

   1. div-inv 99.8%

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

   2. add-sqr-sqrt 47.8%

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

   3. fabs-sqr 47.8%

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

   4. add-sqr-sqrt 48.4%

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

   5. *-commutative 48.4%

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

   6. add-sqr-sqrt 11.3%

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

   7. fabs-sqr 11.3%

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

   8. add-sqr-sqrt 25.8%

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

3. Applied egg-rr 25.8%

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

4. Taylor expanded in y around 0 26.1%

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

5. Final simplification 26.1%

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

Alternative 8: 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. Step-by-step derivation

   1. div-inv 99.8%

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

   2. add-sqr-sqrt 47.8%

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

   3. fabs-sqr 47.8%

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

   4. add-sqr-sqrt 48.4%

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

   5. *-commutative 48.4%

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

   6. add-sqr-sqrt 11.3%

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

   7. fabs-sqr 11.3%

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

   8. add-sqr-sqrt 25.8%

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

3. Applied egg-rr 25.8%

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

4. Taylor expanded in y around inf 1.3%

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

5. Final simplification 1.3%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023224 
(FPCore (x y)
  :name "Numeric.LinearAlgebra.Util:formatSparse from hmatrix-0.16.1.5"
  :precision binary64
  (/ (fabs (- x y)) (fabs y)))