Numeric.LinearAlgebra.Util:formatSparse from hmatrix-0.16.1.5

Percentage Accurate: 100.0% → 100.0%

Time: 4.3s

Alternatives: 7

Speedup: 2.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in x around -inf 100.0%

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

   1. fabs-neg 100.0%

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

   2. mul-1-neg 100.0%

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

   3. sub-neg 100.0%

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

   4. fabs-div 100.0%

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

   5. div-sub 100.0%

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

   6. *-inverses 100.0%

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

4. Simplified 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 56.1% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{\frac{x}{y}}{y - x}\\ t_1 := \frac{y}{x + y}\\ \mathbf{if}\;x \leq -6.6 \cdot 10^{+183}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -3.3 \cdot 10^{+115}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -8 \cdot 10^{+96}:\\ \;\;\;\;\frac{x}{y} + -1\\ \mathbf{elif}\;x \leq -1.35 \cdot 10^{-30}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{+69}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.22 \cdot 10^{+92}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{+216}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot x}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (/ (/ x y) (- y x)))) (t_1 (/ y (+ x y))))
   (if (<= x -6.6e+183)
     (/ x y)
     (if (<= x -3.3e+115)
       t_1
       (if (<= x -8e+96)
         (+ (/ x y) -1.0)
         (if (<= x -1.35e-30)
           t_0
           (if (<= x 1.5e+69)
             t_1
             (if (<= x 1.22e+92)
               (/ x y)
               (if (<= x 2.3e+216) t_0 (/ (* x x) (* y y)))))))))))

C

double code(double x, double y) {
	double t_0 = x * ((x / y) / (y - x));
	double t_1 = y / (x + y);
	double tmp;
	if (x <= -6.6e+183) {
		tmp = x / y;
	} else if (x <= -3.3e+115) {
		tmp = t_1;
	} else if (x <= -8e+96) {
		tmp = (x / y) + -1.0;
	} else if (x <= -1.35e-30) {
		tmp = t_0;
	} else if (x <= 1.5e+69) {
		tmp = t_1;
	} else if (x <= 1.22e+92) {
		tmp = x / y;
	} else if (x <= 2.3e+216) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x * ((x / y) / (y - x))
    t_1 = y / (x + y)
    if (x <= (-6.6d+183)) then
        tmp = x / y
    else if (x <= (-3.3d+115)) then
        tmp = t_1
    else if (x <= (-8d+96)) then
        tmp = (x / y) + (-1.0d0)
    else if (x <= (-1.35d-30)) then
        tmp = t_0
    else if (x <= 1.5d+69) then
        tmp = t_1
    else if (x <= 1.22d+92) then
        tmp = x / y
    else if (x <= 2.3d+216) then
        tmp = t_0
    else
        tmp = (x * x) / (y * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x * ((x / y) / (y - x));
	double t_1 = y / (x + y);
	double tmp;
	if (x <= -6.6e+183) {
		tmp = x / y;
	} else if (x <= -3.3e+115) {
		tmp = t_1;
	} else if (x <= -8e+96) {
		tmp = (x / y) + -1.0;
	} else if (x <= -1.35e-30) {
		tmp = t_0;
	} else if (x <= 1.5e+69) {
		tmp = t_1;
	} else if (x <= 1.22e+92) {
		tmp = x / y;
	} else if (x <= 2.3e+216) {
		tmp = t_0;
	} else {
		tmp = (x * x) / (y * y);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x * ((x / y) / (y - x))
	t_1 = y / (x + y)
	tmp = 0
	if x <= -6.6e+183:
		tmp = x / y
	elif x <= -3.3e+115:
		tmp = t_1
	elif x <= -8e+96:
		tmp = (x / y) + -1.0
	elif x <= -1.35e-30:
		tmp = t_0
	elif x <= 1.5e+69:
		tmp = t_1
	elif x <= 1.22e+92:
		tmp = x / y
	elif x <= 2.3e+216:
		tmp = t_0
	else:
		tmp = (x * x) / (y * y)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x * Float64(Float64(x / y) / Float64(y - x)))
	t_1 = Float64(y / Float64(x + y))
	tmp = 0.0
	if (x <= -6.6e+183)
		tmp = Float64(x / y);
	elseif (x <= -3.3e+115)
		tmp = t_1;
	elseif (x <= -8e+96)
		tmp = Float64(Float64(x / y) + -1.0);
	elseif (x <= -1.35e-30)
		tmp = t_0;
	elseif (x <= 1.5e+69)
		tmp = t_1;
	elseif (x <= 1.22e+92)
		tmp = Float64(x / y);
	elseif (x <= 2.3e+216)
		tmp = t_0;
	else
		tmp = Float64(Float64(x * x) / Float64(y * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x * ((x / y) / (y - x));
	t_1 = y / (x + y);
	tmp = 0.0;
	if (x <= -6.6e+183)
		tmp = x / y;
	elseif (x <= -3.3e+115)
		tmp = t_1;
	elseif (x <= -8e+96)
		tmp = (x / y) + -1.0;
	elseif (x <= -1.35e-30)
		tmp = t_0;
	elseif (x <= 1.5e+69)
		tmp = t_1;
	elseif (x <= 1.22e+92)
		tmp = x / y;
	elseif (x <= 2.3e+216)
		tmp = t_0;
	else
		tmp = (x * x) / (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x * N[(N[(x / y), $MachinePrecision] / N[(y - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -6.6e+183], N[(x / y), $MachinePrecision], If[LessEqual[x, -3.3e+115], t$95$1, If[LessEqual[x, -8e+96], N[(N[(x / y), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[x, -1.35e-30], t$95$0, If[LessEqual[x, 1.5e+69], t$95$1, If[LessEqual[x, 1.22e+92], N[(x / y), $MachinePrecision], If[LessEqual[x, 2.3e+216], t$95$0, N[(N[(x * x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -6.60000000000000019e183 or 1.49999999999999992e69 < x < 1.22e92

   1. Initial program 100.0%

      \[\frac{\left|x - y\right|}{\left|y\right|} \]
   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 22.4%

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

      3. fabs-sqr 22.4%

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

      4. add-sqr-sqrt 22.9%

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

      5. *-commutative 22.9%

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

      6. add-sqr-sqrt 19.5%

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

      7. fabs-sqr 19.5%

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

      8. add-sqr-sqrt 64.6%

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

   3. Applied egg-rr 64.6%

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

   4. Taylor expanded in y around 0 65.0%

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

   if -6.60000000000000019e183 < x < -3.30000000000000005e115 or -1.34999999999999994e-30 < x < 1.49999999999999992e69

   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 56.8%

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

      3. fabs-sqr 56.8%

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

      4. add-sqr-sqrt 57.6%

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

      5. *-commutative 57.6%

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

      6. add-sqr-sqrt 8.8%

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

      7. fabs-sqr 8.8%

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

      8. add-sqr-sqrt 13.6%

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

   3. Applied egg-rr 13.6%

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

      1. flip-- 9.5%

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

      2. associate-*r/ 9.5%

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

      3. +-commutative 9.5%

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

   5. Applied egg-rr 9.5%

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

   6. Taylor expanded in x around 0 1.8%

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

      1. unpow2 1.8%

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

      2. mul-1-neg 1.8%

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

      3. distribute-rgt-neg-out 1.8%

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

   8. Simplified 1.8%

      \[\leadsto \frac{\frac{1}{y} \cdot \color{blue}{\left(y \cdot \left(-y\right)\right)}}{y + x} \]
   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. neg-sub0 2.0%

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

      5. metadata-eval 2.0%

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

      6. sub-neg 2.0%

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

      7. metadata-eval 2.0%

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

      8. add-sqr-sqrt 1.1%

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

      9. sqrt-unprod 18.6%

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

      10. sqr-neg 18.6%

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

      11. sqrt-unprod 31.6%

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

      12. add-sqr-sqrt 73.4%

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

   10. Applied egg-rr 73.4%

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

       1. +-lft-identity 73.4%

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

   12. Simplified 73.4%

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

   if -3.30000000000000005e115 < x < -8.0000000000000004e96

   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 66.4%

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

      7. fabs-sqr 66.4%

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

      8. rem-square-sqrt 67.0%

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

      9. div-sub 67.0%

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

      10. sub-neg 67.0%

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

      11. *-inverses 67.0%

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

      12. metadata-eval 67.0%

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

      13. +-commutative 67.0%

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

   4. Simplified 67.0%

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

   if -8.0000000000000004e96 < x < -1.34999999999999994e-30 or 1.22e92 < x < 2.29999999999999996e216

   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 50.6%

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

      3. fabs-sqr 50.6%

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

      4. add-sqr-sqrt 51.3%

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

      5. *-commutative 51.3%

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

      6. add-sqr-sqrt 13.1%

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

      7. fabs-sqr 13.1%

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

      8. add-sqr-sqrt 18.9%

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

   3. Applied egg-rr 18.9%

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

      1. flip-- 13.5%

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

      2. associate-*r/ 13.4%

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

      3. +-commutative 13.4%

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

   5. Applied egg-rr 13.4%

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

   6. Taylor expanded in y around 0 14.5%

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

      1. unpow2 14.5%

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

      2. associate-/l* 14.7%

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

      3. associate-/r/ 14.7%

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

   8. Simplified 14.7%

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

      1. associate-*l/ 14.5%

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

      2. *-un-lft-identity 14.5%

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

      3. associate-*l/ 14.5%

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

      4. frac-2neg 14.5%

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

      5. div-inv 14.5%

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

      6. associate-*l/ 14.5%

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

      7. *-un-lft-identity 14.5%

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

      8. distribute-neg-frac 14.5%

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

      9. add-sqr-sqrt 8.4%

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

      10. sqrt-unprod 20.5%

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

      11. sqr-neg 20.5%

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

      12. sqrt-unprod 12.1%

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

      13. add-sqr-sqrt 36.9%

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

      14. frac-2neg 36.9%

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

      15. associate-*r/ 38.8%

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

      16. distribute-neg-in 38.8%

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

      17. add-sqr-sqrt 14.0%

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

      18. sqrt-unprod 39.0%

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

      19. sqr-neg 39.0%

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

      20. sqrt-unprod 25.1%

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

      21. add-sqr-sqrt 39.3%

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

      22. sub-neg 39.3%

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

   10. Applied egg-rr 39.3%

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

       1. associate-*l* 52.6%

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

       2. associate-*r/ 52.8%

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

       3. *-rgt-identity 52.8%

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

   12. Simplified 52.8%

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

   if 2.29999999999999996e216 < x

   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 94.5%

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

      3. fabs-sqr 94.5%

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

      4. add-sqr-sqrt 94.7%

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

      5. *-commutative 94.7%

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

      6. add-sqr-sqrt 42.2%

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

      7. fabs-sqr 42.2%

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

      8. add-sqr-sqrt 42.3%

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

   3. Applied egg-rr 42.3%

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

      1. flip-- 32.8%

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

      2. associate-*r/ 32.8%

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

      3. +-commutative 32.8%

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

   5. Applied egg-rr 32.8%

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

   6. Taylor expanded in y around 0 33.0%

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

      1. unpow2 33.0%

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

      2. associate-/l* 33.2%

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

      3. associate-/r/ 33.2%

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

   8. Simplified 33.2%

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

   9. Taylor expanded in x around 0 60.3%

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

       1. unpow2 60.3%

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

       2. unpow2 60.3%

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

   11. Simplified 60.3%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.6 \cdot 10^{+183}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -3.3 \cdot 10^{+115}:\\ \;\;\;\;\frac{y}{x + y}\\ \mathbf{elif}\;x \leq -8 \cdot 10^{+96}:\\ \;\;\;\;\frac{x}{y} + -1\\ \mathbf{elif}\;x \leq -1.35 \cdot 10^{-30}:\\ \;\;\;\;x \cdot \frac{\frac{x}{y}}{y - x}\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{+69}:\\ \;\;\;\;\frac{y}{x + y}\\ \mathbf{elif}\;x \leq 1.22 \cdot 10^{+92}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{+216}:\\ \;\;\;\;x \cdot \frac{\frac{x}{y}}{y - x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot x}{y \cdot y}\\ \end{array} \]

Alternative 3: 56.2% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{\frac{x}{y}}{y - x}\\ t_1 := \frac{y}{x + y}\\ \mathbf{if}\;x \leq -6.5 \cdot 10^{+183}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -2.4 \cdot 10^{+123}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.8 \cdot 10^{+97}:\\ \;\;\;\;\frac{x}{y} + -1\\ \mathbf{elif}\;x \leq -3.3 \cdot 10^{-31}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 6.9 \cdot 10^{+71}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+93}:\\ \;\;\;\;\frac{x \cdot \frac{x}{y}}{x + y}\\ \mathbf{elif}\;x \leq 1.06 \cdot 10^{+216}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot x}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (/ (/ x y) (- y x)))) (t_1 (/ y (+ x y))))
   (if (<= x -6.5e+183)
     (/ x y)
     (if (<= x -2.4e+123)
       t_1
       (if (<= x -1.8e+97)
         (+ (/ x y) -1.0)
         (if (<= x -3.3e-31)
           t_0
           (if (<= x 6.9e+71)
             t_1
             (if (<= x 1.35e+93)
               (/ (* x (/ x y)) (+ x y))
               (if (<= x 1.06e+216) t_0 (/ (* x x) (* y y)))))))))))

C

double code(double x, double y) {
	double t_0 = x * ((x / y) / (y - x));
	double t_1 = y / (x + y);
	double tmp;
	if (x <= -6.5e+183) {
		tmp = x / y;
	} else if (x <= -2.4e+123) {
		tmp = t_1;
	} else if (x <= -1.8e+97) {
		tmp = (x / y) + -1.0;
	} else if (x <= -3.3e-31) {
		tmp = t_0;
	} else if (x <= 6.9e+71) {
		tmp = t_1;
	} else if (x <= 1.35e+93) {
		tmp = (x * (x / y)) / (x + y);
	} else if (x <= 1.06e+216) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x * ((x / y) / (y - x))
    t_1 = y / (x + y)
    if (x <= (-6.5d+183)) then
        tmp = x / y
    else if (x <= (-2.4d+123)) then
        tmp = t_1
    else if (x <= (-1.8d+97)) then
        tmp = (x / y) + (-1.0d0)
    else if (x <= (-3.3d-31)) then
        tmp = t_0
    else if (x <= 6.9d+71) then
        tmp = t_1
    else if (x <= 1.35d+93) then
        tmp = (x * (x / y)) / (x + y)
    else if (x <= 1.06d+216) then
        tmp = t_0
    else
        tmp = (x * x) / (y * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x * ((x / y) / (y - x));
	double t_1 = y / (x + y);
	double tmp;
	if (x <= -6.5e+183) {
		tmp = x / y;
	} else if (x <= -2.4e+123) {
		tmp = t_1;
	} else if (x <= -1.8e+97) {
		tmp = (x / y) + -1.0;
	} else if (x <= -3.3e-31) {
		tmp = t_0;
	} else if (x <= 6.9e+71) {
		tmp = t_1;
	} else if (x <= 1.35e+93) {
		tmp = (x * (x / y)) / (x + y);
	} else if (x <= 1.06e+216) {
		tmp = t_0;
	} else {
		tmp = (x * x) / (y * y);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x * ((x / y) / (y - x))
	t_1 = y / (x + y)
	tmp = 0
	if x <= -6.5e+183:
		tmp = x / y
	elif x <= -2.4e+123:
		tmp = t_1
	elif x <= -1.8e+97:
		tmp = (x / y) + -1.0
	elif x <= -3.3e-31:
		tmp = t_0
	elif x <= 6.9e+71:
		tmp = t_1
	elif x <= 1.35e+93:
		tmp = (x * (x / y)) / (x + y)
	elif x <= 1.06e+216:
		tmp = t_0
	else:
		tmp = (x * x) / (y * y)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x * Float64(Float64(x / y) / Float64(y - x)))
	t_1 = Float64(y / Float64(x + y))
	tmp = 0.0
	if (x <= -6.5e+183)
		tmp = Float64(x / y);
	elseif (x <= -2.4e+123)
		tmp = t_1;
	elseif (x <= -1.8e+97)
		tmp = Float64(Float64(x / y) + -1.0);
	elseif (x <= -3.3e-31)
		tmp = t_0;
	elseif (x <= 6.9e+71)
		tmp = t_1;
	elseif (x <= 1.35e+93)
		tmp = Float64(Float64(x * Float64(x / y)) / Float64(x + y));
	elseif (x <= 1.06e+216)
		tmp = t_0;
	else
		tmp = Float64(Float64(x * x) / Float64(y * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x * ((x / y) / (y - x));
	t_1 = y / (x + y);
	tmp = 0.0;
	if (x <= -6.5e+183)
		tmp = x / y;
	elseif (x <= -2.4e+123)
		tmp = t_1;
	elseif (x <= -1.8e+97)
		tmp = (x / y) + -1.0;
	elseif (x <= -3.3e-31)
		tmp = t_0;
	elseif (x <= 6.9e+71)
		tmp = t_1;
	elseif (x <= 1.35e+93)
		tmp = (x * (x / y)) / (x + y);
	elseif (x <= 1.06e+216)
		tmp = t_0;
	else
		tmp = (x * x) / (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x * N[(N[(x / y), $MachinePrecision] / N[(y - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -6.5e+183], N[(x / y), $MachinePrecision], If[LessEqual[x, -2.4e+123], t$95$1, If[LessEqual[x, -1.8e+97], N[(N[(x / y), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[x, -3.3e-31], t$95$0, If[LessEqual[x, 6.9e+71], t$95$1, If[LessEqual[x, 1.35e+93], N[(N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.06e+216], t$95$0, N[(N[(x * x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if x < -6.49999999999999983e183

   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 0.0%

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

      3. fabs-sqr 0.0%

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

      4. add-sqr-sqrt 0.5%

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

      5. *-commutative 0.5%

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

      6. add-sqr-sqrt 0.3%

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

      7. fabs-sqr 0.3%

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

      8. add-sqr-sqrt 61.0%

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

   3. Applied egg-rr 61.0%

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

   4. Taylor expanded in y around 0 61.2%

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

   if -6.49999999999999983e183 < x < -2.39999999999999989e123 or -3.2999999999999999e-31 < x < 6.89999999999999975e71

   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 56.8%

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

      3. fabs-sqr 56.8%

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

      4. add-sqr-sqrt 57.6%

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

      5. *-commutative 57.6%

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

      6. add-sqr-sqrt 8.8%

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

      7. fabs-sqr 8.8%

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

      8. add-sqr-sqrt 13.6%

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

   3. Applied egg-rr 13.6%

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

      1. flip-- 9.5%

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

      2. associate-*r/ 9.5%

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

      3. +-commutative 9.5%

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

   5. Applied egg-rr 9.5%

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

   6. Taylor expanded in x around 0 1.8%

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

      1. unpow2 1.8%

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

      2. mul-1-neg 1.8%

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

      3. distribute-rgt-neg-out 1.8%

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

   8. Simplified 1.8%

      \[\leadsto \frac{\frac{1}{y} \cdot \color{blue}{\left(y \cdot \left(-y\right)\right)}}{y + x} \]
   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. neg-sub0 2.0%

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

      5. metadata-eval 2.0%

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

      6. sub-neg 2.0%

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

      7. metadata-eval 2.0%

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

      8. add-sqr-sqrt 1.1%

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

      9. sqrt-unprod 18.6%

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

      10. sqr-neg 18.6%

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

      11. sqrt-unprod 31.6%

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

      12. add-sqr-sqrt 73.4%

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

   10. Applied egg-rr 73.4%

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

       1. +-lft-identity 73.4%

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

   12. Simplified 73.4%

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

   if -2.39999999999999989e123 < x < -1.79999999999999983e97

   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 66.4%

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

      7. fabs-sqr 66.4%

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

      8. rem-square-sqrt 67.0%

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

      9. div-sub 67.0%

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

      10. sub-neg 67.0%

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

      11. *-inverses 67.0%

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

      12. metadata-eval 67.0%

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

      13. +-commutative 67.0%

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

   4. Simplified 67.0%

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

   if -1.79999999999999983e97 < x < -3.2999999999999999e-31 or 1.35e93 < x < 1.06e216

   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 50.6%

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

      3. fabs-sqr 50.6%

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

      4. add-sqr-sqrt 51.3%

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

      5. *-commutative 51.3%

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

      6. add-sqr-sqrt 13.1%

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

      7. fabs-sqr 13.1%

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

      8. add-sqr-sqrt 18.9%

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

   3. Applied egg-rr 18.9%

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

      1. flip-- 13.5%

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

      2. associate-*r/ 13.4%

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

      3. +-commutative 13.4%

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

   5. Applied egg-rr 13.4%

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

   6. Taylor expanded in y around 0 14.5%

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

      1. unpow2 14.5%

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

      2. associate-/l* 14.7%

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

      3. associate-/r/ 14.7%

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

   8. Simplified 14.7%

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

      1. associate-*l/ 14.5%

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

      2. *-un-lft-identity 14.5%

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

      3. associate-*l/ 14.5%

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

      4. frac-2neg 14.5%

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

      5. div-inv 14.5%

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

      6. associate-*l/ 14.5%

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

      7. *-un-lft-identity 14.5%

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

      8. distribute-neg-frac 14.5%

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

      9. add-sqr-sqrt 8.4%

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

      10. sqrt-unprod 20.5%

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

      11. sqr-neg 20.5%

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

      12. sqrt-unprod 12.1%

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

      13. add-sqr-sqrt 36.9%

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

      14. frac-2neg 36.9%

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

      15. associate-*r/ 38.8%

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

      16. distribute-neg-in 38.8%

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

      17. add-sqr-sqrt 14.0%

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

      18. sqrt-unprod 39.0%

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

      19. sqr-neg 39.0%

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

      20. sqrt-unprod 25.1%

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

      21. add-sqr-sqrt 39.3%

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

      22. sub-neg 39.3%

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

   10. Applied egg-rr 39.3%

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

       1. associate-*l* 52.6%

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

       2. associate-*r/ 52.8%

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

       3. *-rgt-identity 52.8%

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

   12. Simplified 52.8%

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

   if 6.89999999999999975e71 < x < 1.35e93

   1. Initial program 100.0%

      \[\frac{\left|x - y\right|}{\left|y\right|} \]
   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 86.8%

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

      3. fabs-sqr 86.8%

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

      4. add-sqr-sqrt 87.3%

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

      5. *-commutative 87.3%

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

      6. add-sqr-sqrt 74.8%

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

      7. fabs-sqr 74.8%

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

      8. add-sqr-sqrt 75.0%

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

   3. Applied egg-rr 75.0%

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

      1. flip-- 75.0%

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

      2. associate-*r/ 74.9%

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

      3. +-commutative 74.9%

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

   5. Applied egg-rr 74.9%

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

   6. Taylor expanded in y around 0 77.4%

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

      1. unpow2 77.4%

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

      2. associate-/l* 77.4%

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

      3. associate-/r/ 77.6%

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

   8. Simplified 77.6%

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

   if 1.06e216 < x

   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 94.5%

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

      3. fabs-sqr 94.5%

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

      4. add-sqr-sqrt 94.7%

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

      5. *-commutative 94.7%

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

      6. add-sqr-sqrt 42.2%

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

      7. fabs-sqr 42.2%

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

      8. add-sqr-sqrt 42.3%

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

   3. Applied egg-rr 42.3%

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

      1. flip-- 32.8%

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

      2. associate-*r/ 32.8%

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

      3. +-commutative 32.8%

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

   5. Applied egg-rr 32.8%

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

   6. Taylor expanded in y around 0 33.0%

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

      1. unpow2 33.0%

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

      2. associate-/l* 33.2%

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

      3. associate-/r/ 33.2%

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

   8. Simplified 33.2%

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

   9. Taylor expanded in x around 0 60.3%

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

       1. unpow2 60.3%

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

       2. unpow2 60.3%

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

   11. Simplified 60.3%

       \[\leadsto \color{blue}{\frac{x \cdot x}{y \cdot y}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{+183}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -2.4 \cdot 10^{+123}:\\ \;\;\;\;\frac{y}{x + y}\\ \mathbf{elif}\;x \leq -1.8 \cdot 10^{+97}:\\ \;\;\;\;\frac{x}{y} + -1\\ \mathbf{elif}\;x \leq -3.3 \cdot 10^{-31}:\\ \;\;\;\;x \cdot \frac{\frac{x}{y}}{y - x}\\ \mathbf{elif}\;x \leq 6.9 \cdot 10^{+71}:\\ \;\;\;\;\frac{y}{x + y}\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+93}:\\ \;\;\;\;\frac{x \cdot \frac{x}{y}}{x + y}\\ \mathbf{elif}\;x \leq 1.06 \cdot 10^{+216}:\\ \;\;\;\;x \cdot \frac{\frac{x}{y}}{y - x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot x}{y \cdot y}\\ \end{array} \]

Alternative 4: 56.7% accurate, 13.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{x + y}\\ \mathbf{if}\;x \leq -1.65 \cdot 10^{+184}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 1.16 \cdot 10^{+69}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 8 \cdot 10^{+89}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 5.3 \cdot 10^{+171}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot x}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (+ x y))))
   (if (<= x -1.65e+184)
     (/ x y)
     (if (<= x 1.16e+69)
       t_0
       (if (<= x 8e+89)
         (/ x y)
         (if (<= x 5.3e+171) t_0 (/ (* x x) (* y y))))))))

C

double code(double x, double y) {
	double t_0 = y / (x + y);
	double tmp;
	if (x <= -1.65e+184) {
		tmp = x / y;
	} else if (x <= 1.16e+69) {
		tmp = t_0;
	} else if (x <= 8e+89) {
		tmp = x / y;
	} else if (x <= 5.3e+171) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = y / (x + y)
    if (x <= (-1.65d+184)) then
        tmp = x / y
    else if (x <= 1.16d+69) then
        tmp = t_0
    else if (x <= 8d+89) then
        tmp = x / y
    else if (x <= 5.3d+171) then
        tmp = t_0
    else
        tmp = (x * x) / (y * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y / (x + y);
	double tmp;
	if (x <= -1.65e+184) {
		tmp = x / y;
	} else if (x <= 1.16e+69) {
		tmp = t_0;
	} else if (x <= 8e+89) {
		tmp = x / y;
	} else if (x <= 5.3e+171) {
		tmp = t_0;
	} else {
		tmp = (x * x) / (y * y);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y / (x + y)
	tmp = 0
	if x <= -1.65e+184:
		tmp = x / y
	elif x <= 1.16e+69:
		tmp = t_0
	elif x <= 8e+89:
		tmp = x / y
	elif x <= 5.3e+171:
		tmp = t_0
	else:
		tmp = (x * x) / (y * y)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y / Float64(x + y))
	tmp = 0.0
	if (x <= -1.65e+184)
		tmp = Float64(x / y);
	elseif (x <= 1.16e+69)
		tmp = t_0;
	elseif (x <= 8e+89)
		tmp = Float64(x / y);
	elseif (x <= 5.3e+171)
		tmp = t_0;
	else
		tmp = Float64(Float64(x * x) / Float64(y * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y / (x + y);
	tmp = 0.0;
	if (x <= -1.65e+184)
		tmp = x / y;
	elseif (x <= 1.16e+69)
		tmp = t_0;
	elseif (x <= 8e+89)
		tmp = x / y;
	elseif (x <= 5.3e+171)
		tmp = t_0;
	else
		tmp = (x * x) / (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.65e+184], N[(x / y), $MachinePrecision], If[LessEqual[x, 1.16e+69], t$95$0, If[LessEqual[x, 8e+89], N[(x / y), $MachinePrecision], If[LessEqual[x, 5.3e+171], t$95$0, N[(N[(x * x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.6499999999999999e184 or 1.16000000000000005e69 < x < 7.99999999999999996e89

   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 17.1%

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

      3. fabs-sqr 17.1%

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

      4. add-sqr-sqrt 17.6%

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

      5. *-commutative 17.6%

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

      6. add-sqr-sqrt 17.5%

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

      7. fabs-sqr 17.5%

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

      8. add-sqr-sqrt 65.6%

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

   3. Applied egg-rr 65.6%

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

   4. Taylor expanded in y around 0 66.0%

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

   if -1.6499999999999999e184 < x < 1.16000000000000005e69 or 7.99999999999999996e89 < x < 5.29999999999999982e171

   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 52.1%

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

      3. fabs-sqr 52.1%

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

      4. add-sqr-sqrt 53.0%

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

      5. *-commutative 53.0%

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

      6. add-sqr-sqrt 8.1%

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

      7. fabs-sqr 8.1%

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

      8. add-sqr-sqrt 15.4%

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

   3. Applied egg-rr 15.4%

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

      1. flip-- 12.2%

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

      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 2.0%

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

      1. unpow2 2.0%

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

      2. mul-1-neg 2.0%

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

      3. distribute-rgt-neg-out 2.0%

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

   8. Simplified 2.0%

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

      1. associate-*r* 2.2%

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

      2. lft-mult-inverse 2.2%

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

      3. *-un-lft-identity 2.2%

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

      4. neg-sub0 2.2%

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

      5. metadata-eval 2.2%

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

      6. sub-neg 2.2%

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

      7. metadata-eval 2.2%

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

      8. add-sqr-sqrt 1.1%

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

      9. sqrt-unprod 15.7%

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

      10. sqr-neg 15.7%

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

      11. sqrt-unprod 27.1%

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

      12. add-sqr-sqrt 62.9%

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

   10. Applied egg-rr 62.9%

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

       1. +-lft-identity 62.9%

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

   12. Simplified 62.9%

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

   if 5.29999999999999982e171 < x

   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 93.2%

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

      3. fabs-sqr 93.2%

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

      4. add-sqr-sqrt 93.5%

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

      5. *-commutative 93.5%

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

      6. add-sqr-sqrt 42.0%

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

      7. fabs-sqr 42.0%

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

      8. add-sqr-sqrt 42.2%

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

   3. Applied egg-rr 42.2%

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

      1. flip-- 27.1%

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

      2. associate-*r/ 27.1%

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

      3. +-commutative 27.1%

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

   5. Applied egg-rr 27.1%

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

   6. Taylor expanded in y around 0 27.4%

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

      1. unpow2 27.4%

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

      2. associate-/l* 27.8%

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

      3. associate-/r/ 27.8%

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

   8. Simplified 27.8%

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

   9. Taylor expanded in x around 0 48.0%

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

       1. unpow2 48.0%

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

       2. unpow2 48.0%

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

   11. Simplified 48.0%

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

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{+184}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 1.16 \cdot 10^{+69}:\\ \;\;\;\;\frac{y}{x + y}\\ \mathbf{elif}\;x \leq 8 \cdot 10^{+89}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 5.3 \cdot 10^{+171}:\\ \;\;\;\;\frac{y}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot x}{y \cdot y}\\ \end{array} \]

Alternative 5: 57.2% accurate, 22.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.65 \cdot 10^{-83} \lor \neg \left(y \leq 6 \cdot 10^{-182}\right):\\ \;\;\;\;\frac{y}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -2.65e-83) (not (<= y 6e-182))) (/ y (+ x y)) (/ x y)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -2.65e-83) || !(y <= 6e-182)) {
		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 <= (-2.65d-83)) .or. (.not. (y <= 6d-182))) 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 <= -2.65e-83) || !(y <= 6e-182)) {
		tmp = y / (x + y);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -2.65e-83) or not (y <= 6e-182):
		tmp = y / (x + y)
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -2.65e-83) || !(y <= 6e-182))
		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 <= -2.65e-83) || ~((y <= 6e-182)))
		tmp = y / (x + y);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -2.65e-83], N[Not[LessEqual[y, 6e-182]], $MachinePrecision]], N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.65e-83 or 6.0000000000000003e-182 < 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 50.9%

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

      3. fabs-sqr 50.9%

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

      4. add-sqr-sqrt 51.8%

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

      5. *-commutative 51.8%

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

      6. add-sqr-sqrt 11.0%

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

      7. fabs-sqr 11.0%

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

      8. add-sqr-sqrt 17.3%

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

   3. Applied egg-rr 17.3%

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

      1. flip-- 11.1%

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

      2. associate-*r/ 11.0%

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

      3. +-commutative 11.0%

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

   5. Applied egg-rr 11.0%

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

   6. Taylor expanded in x around 0 1.8%

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

      1. unpow2 1.8%

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

      2. mul-1-neg 1.8%

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

      3. distribute-rgt-neg-out 1.8%

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

   8. Simplified 1.8%

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

      1. associate-*r* 2.1%

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

      2. lft-mult-inverse 2.1%

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

      3. *-un-lft-identity 2.1%

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

      4. neg-sub0 2.1%

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

      5. metadata-eval 2.1%

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

      6. sub-neg 2.1%

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

      7. metadata-eval 2.1%

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

      8. add-sqr-sqrt 0.9%

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

      9. sqrt-unprod 16.4%

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

      10. sqr-neg 16.4%

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

      11. sqrt-unprod 30.2%

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

      12. add-sqr-sqrt 62.2%

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

   10. Applied egg-rr 62.2%

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

       1. +-lft-identity 62.2%

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

   12. Simplified 62.2%

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

   if -2.65e-83 < y < 6.0000000000000003e-182

   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 59.7%

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

      3. fabs-sqr 59.7%

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

      4. add-sqr-sqrt 60.0%

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

      5. *-commutative 60.0%

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

      6. add-sqr-sqrt 19.9%

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

      7. fabs-sqr 19.9%

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

      8. add-sqr-sqrt 45.0%

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

   3. Applied egg-rr 45.0%

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

   4. Taylor expanded in y around 0 45.6%

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

4. Final simplification 58.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.65 \cdot 10^{-83} \lor \neg \left(y \leq 6 \cdot 10^{-182}\right):\\ \;\;\;\;\frac{y}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 6: 26.1% 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 99.7%

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

   2. add-sqr-sqrt 53.1%

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

   3. fabs-sqr 53.1%

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

   4. add-sqr-sqrt 53.9%

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

   5. *-commutative 53.9%

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

   6. add-sqr-sqrt 13.3%

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

   7. fabs-sqr 13.3%

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

   8. add-sqr-sqrt 24.3%

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

3. Applied egg-rr 24.3%

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

4. Taylor expanded in y around 0 25.3%

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

5. Final simplification 25.3%

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

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. 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 53.1%

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

   3. fabs-sqr 53.1%

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

   4. add-sqr-sqrt 53.9%

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

   5. *-commutative 53.9%

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

   6. add-sqr-sqrt 13.3%

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

   7. fabs-sqr 13.3%

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

   8. add-sqr-sqrt 24.3%

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

3. Applied egg-rr 24.3%

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

4. Taylor expanded in y around inf 1.3%

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

5. Final simplification 1.3%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023199 
(FPCore (x y)
  :name "Numeric.LinearAlgebra.Util:formatSparse from hmatrix-0.16.1.5"
  :precision binary64
  (/ (fabs (- x y)) (fabs y)))