Numeric.LinearAlgebra.Util:formatSparse from hmatrix-0.16.1.5

Percentage Accurate: 100.0% → 100.0%

Time: 5.0s

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: 57.0% accurate, 10.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.1 \cdot 10^{+88}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+23}:\\ \;\;\;\;\frac{y}{x + y}\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{+100}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 1.72 \cdot 10^{+168}:\\ \;\;\;\;y \cdot \frac{1}{y - x}\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{+257}:\\ \;\;\;\;x \cdot \frac{\frac{-x}{y}}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -6.1e+88)
   (/ x y)
   (if (<= x 4e+23)
     (/ y (+ x y))
     (if (<= x 6.8e+100)
       (/ x y)
       (if (<= x 1.72e+168)
         (* y (/ 1.0 (- y x)))
         (if (<= x 3.4e+257) (* x (/ (/ (- x) y) (+ x y))) (/ x y)))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -6.1e+88) {
		tmp = x / y;
	} else if (x <= 4e+23) {
		tmp = y / (x + y);
	} else if (x <= 6.8e+100) {
		tmp = x / y;
	} else if (x <= 1.72e+168) {
		tmp = y * (1.0 / (y - x));
	} else if (x <= 3.4e+257) {
		tmp = x * ((-x / 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 (x <= (-6.1d+88)) then
        tmp = x / y
    else if (x <= 4d+23) then
        tmp = y / (x + y)
    else if (x <= 6.8d+100) then
        tmp = x / y
    else if (x <= 1.72d+168) then
        tmp = y * (1.0d0 / (y - x))
    else if (x <= 3.4d+257) then
        tmp = x * ((-x / 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 (x <= -6.1e+88) {
		tmp = x / y;
	} else if (x <= 4e+23) {
		tmp = y / (x + y);
	} else if (x <= 6.8e+100) {
		tmp = x / y;
	} else if (x <= 1.72e+168) {
		tmp = y * (1.0 / (y - x));
	} else if (x <= 3.4e+257) {
		tmp = x * ((-x / y) / (x + y));
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -6.1e+88:
		tmp = x / y
	elif x <= 4e+23:
		tmp = y / (x + y)
	elif x <= 6.8e+100:
		tmp = x / y
	elif x <= 1.72e+168:
		tmp = y * (1.0 / (y - x))
	elif x <= 3.4e+257:
		tmp = x * ((-x / y) / (x + y))
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -6.1e+88)
		tmp = Float64(x / y);
	elseif (x <= 4e+23)
		tmp = Float64(y / Float64(x + y));
	elseif (x <= 6.8e+100)
		tmp = Float64(x / y);
	elseif (x <= 1.72e+168)
		tmp = Float64(y * Float64(1.0 / Float64(y - x)));
	elseif (x <= 3.4e+257)
		tmp = Float64(x * Float64(Float64(Float64(-x) / y) / Float64(x + y)));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -6.1e+88)
		tmp = x / y;
	elseif (x <= 4e+23)
		tmp = y / (x + y);
	elseif (x <= 6.8e+100)
		tmp = x / y;
	elseif (x <= 1.72e+168)
		tmp = y * (1.0 / (y - x));
	elseif (x <= 3.4e+257)
		tmp = x * ((-x / y) / (x + y));
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -6.1e+88], N[(x / y), $MachinePrecision], If[LessEqual[x, 4e+23], N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 6.8e+100], N[(x / y), $MachinePrecision], If[LessEqual[x, 1.72e+168], N[(y * N[(1.0 / N[(y - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.4e+257], N[(x * N[(N[((-x) / y), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -6.0999999999999998e88 or 3.9999999999999997e23 < x < 6.79999999999999988e100 or 3.4000000000000002e257 < x

   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 30.5%

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

      3. fabs-sqr 30.5%

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

      4. add-sqr-sqrt 31.0%

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

      5. *-commutative 31.0%

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

      6. add-sqr-sqrt 21.5%

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

      7. fabs-sqr 21.5%

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

      8. add-sqr-sqrt 60.2%

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

   3. Applied egg-rr 60.2%

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

   4. Taylor expanded in y around 0 60.8%

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

   if -6.0999999999999998e88 < x < 3.9999999999999997e23

   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 48.5%

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

      3. fabs-sqr 48.5%

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

      4. add-sqr-sqrt 49.4%

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

      5. *-commutative 49.4%

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

      6. add-sqr-sqrt 8.3%

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

      7. fabs-sqr 8.3%

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

      8. add-sqr-sqrt 13.8%

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

   3. Applied egg-rr 13.8%

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

      1. flip-- 12.3%

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

      2. associate-*r/ 12.3%

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

      3. +-commutative 12.3%

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

   5. Applied egg-rr 12.3%

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

   6. Taylor expanded in y around inf 1.9%

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

      1. neg-mul-1 1.9%

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

   8. Simplified 1.9%

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

      1. *-un-lft-identity 1.9%

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

      2. add-sqr-sqrt 0.9%

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

      3. sqrt-unprod 20.5%

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

      4. sqr-neg 20.5%

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

      5. sqrt-unprod 37.6%

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

      6. times-frac 37.6%

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

   10. Applied egg-rr 37.6%

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

       1. /-rgt-identity 37.6%

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

       2. associate-*r/ 37.6%

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

       3. rem-square-sqrt 73.9%

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

   12. Simplified 73.9%

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

   if 6.79999999999999988e100 < x < 1.7200000000000001e168

   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 99.4%

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

      3. fabs-sqr 99.4%

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

      4. add-sqr-sqrt 99.6%

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

      5. *-commutative 99.6%

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

      6. add-sqr-sqrt 33.2%

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

      7. fabs-sqr 33.2%

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

      8. add-sqr-sqrt 34.3%

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

   3. Applied egg-rr 34.3%

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

      1. flip-- 33.8%

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

      2. associate-*r/ 26.2%

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

      3. +-commutative 26.2%

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

   5. Applied egg-rr 26.2%

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

   6. Taylor expanded in y around inf 2.4%

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

      1. neg-mul-1 2.4%

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

   8. Simplified 2.4%

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

      1. frac-2neg 2.4%

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

      2. div-inv 2.4%

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

      3. remove-double-neg 2.4%

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

      4. distribute-neg-in 2.4%

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

      5. add-sqr-sqrt 1.9%

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

      6. sqrt-unprod 3.2%

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

      7. sqr-neg 3.2%

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

      8. sqrt-unprod 0.5%

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

      9. add-sqr-sqrt 51.2%

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

      10. sub-neg 51.2%

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

   10. Applied egg-rr 51.2%

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

   if 1.7200000000000001e168 < x < 3.4000000000000002e257

   1. Initial program 99.9%

      \[\frac{\left|x - y\right|}{\left|y\right|} \]
   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 95.7%

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

      3. fabs-sqr 95.7%

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

      4. add-sqr-sqrt 96.0%

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

      5. *-commutative 96.0%

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

      6. add-sqr-sqrt 27.0%

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

      7. fabs-sqr 27.0%

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

      8. add-sqr-sqrt 27.5%

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

   3. Applied egg-rr 27.5%

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

      1. flip-- 15.8%

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

      2. associate-*r/ 15.8%

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

      3. +-commutative 15.8%

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

   5. Applied egg-rr 15.8%

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

   6. Taylor expanded in x around inf 16.2%

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

      1. unpow2 16.2%

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

   8. Simplified 16.2%

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

      1. frac-2neg 16.2%

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

      2. distribute-frac-neg 16.2%

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

      3. *-commutative 16.2%

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

      4. div-inv 16.2%

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

      5. frac-2neg 16.2%

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

      6. add-sqr-sqrt 0.2%

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

      7. sqrt-unprod 0.1%

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

      8. sqr-neg 0.1%

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

      9. sqrt-unprod 0.1%

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

      10. add-sqr-sqrt 25.1%

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

      11. distribute-neg-frac 25.1%

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

      12. associate-*l/ 53.8%

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

      13. frac-2neg 53.8%

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

      14. associate-/l* 64.8%

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

      15. associate-/r/ 64.7%

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

   10. Applied egg-rr 64.7%

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

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.1 \cdot 10^{+88}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+23}:\\ \;\;\;\;\frac{y}{x + y}\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{+100}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 1.72 \cdot 10^{+168}:\\ \;\;\;\;y \cdot \frac{1}{y - x}\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{+257}:\\ \;\;\;\;x \cdot \frac{\frac{-x}{y}}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 3: 56.0% accurate, 10.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{+91}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+23}:\\ \;\;\;\;\frac{y}{x + y}\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{+103}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{+169}:\\ \;\;\;\;y \cdot \frac{1}{y - x}\\ \mathbf{elif}\;x \leq 7.8 \cdot 10^{+251}:\\ \;\;\;\;\frac{\frac{x}{\frac{y}{x}}}{y - x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -3e+91)
   (/ x y)
   (if (<= x 4e+23)
     (/ y (+ x y))
     (if (<= x 1.6e+103)
       (/ x y)
       (if (<= x 3.5e+169)
         (* y (/ 1.0 (- y x)))
         (if (<= x 7.8e+251) (/ (/ x (/ y x)) (- y x)) (/ x y)))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -3e+91) {
		tmp = x / y;
	} else if (x <= 4e+23) {
		tmp = y / (x + y);
	} else if (x <= 1.6e+103) {
		tmp = x / y;
	} else if (x <= 3.5e+169) {
		tmp = y * (1.0 / (y - x));
	} else if (x <= 7.8e+251) {
		tmp = (x / (y / x)) / (y - x);
	} 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 (x <= (-3d+91)) then
        tmp = x / y
    else if (x <= 4d+23) then
        tmp = y / (x + y)
    else if (x <= 1.6d+103) then
        tmp = x / y
    else if (x <= 3.5d+169) then
        tmp = y * (1.0d0 / (y - x))
    else if (x <= 7.8d+251) then
        tmp = (x / (y / x)) / (y - x)
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -3e+91) {
		tmp = x / y;
	} else if (x <= 4e+23) {
		tmp = y / (x + y);
	} else if (x <= 1.6e+103) {
		tmp = x / y;
	} else if (x <= 3.5e+169) {
		tmp = y * (1.0 / (y - x));
	} else if (x <= 7.8e+251) {
		tmp = (x / (y / x)) / (y - x);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -3e+91:
		tmp = x / y
	elif x <= 4e+23:
		tmp = y / (x + y)
	elif x <= 1.6e+103:
		tmp = x / y
	elif x <= 3.5e+169:
		tmp = y * (1.0 / (y - x))
	elif x <= 7.8e+251:
		tmp = (x / (y / x)) / (y - x)
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -3e+91)
		tmp = Float64(x / y);
	elseif (x <= 4e+23)
		tmp = Float64(y / Float64(x + y));
	elseif (x <= 1.6e+103)
		tmp = Float64(x / y);
	elseif (x <= 3.5e+169)
		tmp = Float64(y * Float64(1.0 / Float64(y - x)));
	elseif (x <= 7.8e+251)
		tmp = Float64(Float64(x / Float64(y / x)) / Float64(y - x));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -3e+91)
		tmp = x / y;
	elseif (x <= 4e+23)
		tmp = y / (x + y);
	elseif (x <= 1.6e+103)
		tmp = x / y;
	elseif (x <= 3.5e+169)
		tmp = y * (1.0 / (y - x));
	elseif (x <= 7.8e+251)
		tmp = (x / (y / x)) / (y - x);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -3e+91], N[(x / y), $MachinePrecision], If[LessEqual[x, 4e+23], N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.6e+103], N[(x / y), $MachinePrecision], If[LessEqual[x, 3.5e+169], N[(y * N[(1.0 / N[(y - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 7.8e+251], N[(N[(x / N[(y / x), $MachinePrecision]), $MachinePrecision] / N[(y - x), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -3.00000000000000006e91 or 3.9999999999999997e23 < x < 1.59999999999999996e103 or 7.79999999999999951e251 < x

   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 30.5%

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

      3. fabs-sqr 30.5%

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

      4. add-sqr-sqrt 31.0%

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

      5. *-commutative 31.0%

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

      6. add-sqr-sqrt 21.5%

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

      7. fabs-sqr 21.5%

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

      8. add-sqr-sqrt 60.2%

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

   3. Applied egg-rr 60.2%

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

   4. Taylor expanded in y around 0 60.8%

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

   if -3.00000000000000006e91 < x < 3.9999999999999997e23

   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 48.5%

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

      3. fabs-sqr 48.5%

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

      4. add-sqr-sqrt 49.4%

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

      5. *-commutative 49.4%

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

      6. add-sqr-sqrt 8.3%

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

      7. fabs-sqr 8.3%

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

      8. add-sqr-sqrt 13.8%

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

   3. Applied egg-rr 13.8%

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

      1. flip-- 12.3%

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

      2. associate-*r/ 12.3%

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

      3. +-commutative 12.3%

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

   5. Applied egg-rr 12.3%

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

   6. Taylor expanded in y around inf 1.9%

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

      1. neg-mul-1 1.9%

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

   8. Simplified 1.9%

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

      1. *-un-lft-identity 1.9%

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

      2. add-sqr-sqrt 0.9%

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

      3. sqrt-unprod 20.5%

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

      4. sqr-neg 20.5%

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

      5. sqrt-unprod 37.6%

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

      6. times-frac 37.6%

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

   10. Applied egg-rr 37.6%

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

       1. /-rgt-identity 37.6%

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

       2. associate-*r/ 37.6%

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

       3. rem-square-sqrt 73.9%

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

   12. Simplified 73.9%

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

   if 1.59999999999999996e103 < x < 3.50000000000000019e169

   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 99.4%

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

      3. fabs-sqr 99.4%

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

      4. add-sqr-sqrt 99.6%

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

      5. *-commutative 99.6%

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

      6. add-sqr-sqrt 33.2%

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

      7. fabs-sqr 33.2%

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

      8. add-sqr-sqrt 34.3%

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

   3. Applied egg-rr 34.3%

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

      1. flip-- 33.8%

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

      2. associate-*r/ 26.2%

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

      3. +-commutative 26.2%

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

   5. Applied egg-rr 26.2%

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

   6. Taylor expanded in y around inf 2.4%

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

      1. neg-mul-1 2.4%

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

   8. Simplified 2.4%

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

      1. frac-2neg 2.4%

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

      2. div-inv 2.4%

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

      3. remove-double-neg 2.4%

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

      4. distribute-neg-in 2.4%

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

      5. add-sqr-sqrt 1.9%

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

      6. sqrt-unprod 3.2%

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

      7. sqr-neg 3.2%

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

      8. sqrt-unprod 0.5%

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

      9. add-sqr-sqrt 51.2%

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

      10. sub-neg 51.2%

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

   10. Applied egg-rr 51.2%

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

   if 3.50000000000000019e169 < x < 7.79999999999999951e251

   1. Initial program 99.9%

      \[\frac{\left|x - y\right|}{\left|y\right|} \]
   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 95.7%

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

      3. fabs-sqr 95.7%

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

      4. add-sqr-sqrt 96.0%

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

      5. *-commutative 96.0%

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

      6. add-sqr-sqrt 27.0%

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

      7. fabs-sqr 27.0%

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

      8. add-sqr-sqrt 27.5%

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

   3. Applied egg-rr 27.5%

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

      1. flip-- 15.8%

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

      2. associate-*r/ 15.8%

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

      3. +-commutative 15.8%

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

   5. Applied egg-rr 15.8%

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

   6. Taylor expanded in x around inf 16.2%

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

      1. unpow2 16.2%

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

   8. Simplified 16.2%

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

      1. frac-2neg 16.2%

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

      2. div-inv 16.2%

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

      3. associate-*l/ 16.2%

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

      4. *-un-lft-identity 16.2%

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

      5. distribute-neg-frac 16.2%

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

      6. add-sqr-sqrt 15.9%

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

      7. sqrt-unprod 39.9%

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

      8. sqr-neg 39.9%

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

      9. sqrt-unprod 25.0%

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

      10. add-sqr-sqrt 25.1%

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

      11. frac-2neg 25.1%

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

      12. associate-*r/ 53.7%

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

      13. distribute-neg-in 53.7%

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

      14. add-sqr-sqrt 53.6%

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

      15. sqrt-unprod 36.2%

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

      16. sqr-neg 36.2%

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

      17. sqrt-unprod 0.3%

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

      18. add-sqr-sqrt 51.7%

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

      19. sub-neg 51.7%

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

   10. Applied egg-rr 51.7%

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

       1. associate-*r/ 51.8%

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

       2. *-rgt-identity 51.8%

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

       3. associate-*r/ 25.2%

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

       4. associate-/l* 51.7%

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

   12. Simplified 51.7%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{+91}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+23}:\\ \;\;\;\;\frac{y}{x + y}\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{+103}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{+169}:\\ \;\;\;\;y \cdot \frac{1}{y - x}\\ \mathbf{elif}\;x \leq 7.8 \cdot 10^{+251}:\\ \;\;\;\;\frac{\frac{x}{\frac{y}{x}}}{y - x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 4: 56.5% accurate, 11.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{x + y}\\ \mathbf{if}\;x \leq -9.8 \cdot 10^{+91}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+22}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+99}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+115}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{+236}:\\ \;\;\;\;\frac{x \cdot x}{y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (+ x y))))
   (if (<= x -9.8e+91)
     (/ x y)
     (if (<= x 3e+22)
       t_0
       (if (<= x 3e+99)
         (/ x y)
         (if (<= x 7.5e+115)
           t_0
           (if (<= x 3.4e+236) (/ (* x x) (* y y)) (/ x y))))))))

C

double code(double x, double y) {
	double t_0 = y / (x + y);
	double tmp;
	if (x <= -9.8e+91) {
		tmp = x / y;
	} else if (x <= 3e+22) {
		tmp = t_0;
	} else if (x <= 3e+99) {
		tmp = x / y;
	} else if (x <= 7.5e+115) {
		tmp = t_0;
	} else if (x <= 3.4e+236) {
		tmp = (x * x) / (y * 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) :: t_0
    real(8) :: tmp
    t_0 = y / (x + y)
    if (x <= (-9.8d+91)) then
        tmp = x / y
    else if (x <= 3d+22) then
        tmp = t_0
    else if (x <= 3d+99) then
        tmp = x / y
    else if (x <= 7.5d+115) then
        tmp = t_0
    else if (x <= 3.4d+236) then
        tmp = (x * x) / (y * y)
    else
        tmp = x / 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 <= -9.8e+91) {
		tmp = x / y;
	} else if (x <= 3e+22) {
		tmp = t_0;
	} else if (x <= 3e+99) {
		tmp = x / y;
	} else if (x <= 7.5e+115) {
		tmp = t_0;
	} else if (x <= 3.4e+236) {
		tmp = (x * x) / (y * y);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y / (x + y)
	tmp = 0
	if x <= -9.8e+91:
		tmp = x / y
	elif x <= 3e+22:
		tmp = t_0
	elif x <= 3e+99:
		tmp = x / y
	elif x <= 7.5e+115:
		tmp = t_0
	elif x <= 3.4e+236:
		tmp = (x * x) / (y * y)
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y / Float64(x + y))
	tmp = 0.0
	if (x <= -9.8e+91)
		tmp = Float64(x / y);
	elseif (x <= 3e+22)
		tmp = t_0;
	elseif (x <= 3e+99)
		tmp = Float64(x / y);
	elseif (x <= 7.5e+115)
		tmp = t_0;
	elseif (x <= 3.4e+236)
		tmp = Float64(Float64(x * x) / Float64(y * y));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y / (x + y);
	tmp = 0.0;
	if (x <= -9.8e+91)
		tmp = x / y;
	elseif (x <= 3e+22)
		tmp = t_0;
	elseif (x <= 3e+99)
		tmp = x / y;
	elseif (x <= 7.5e+115)
		tmp = t_0;
	elseif (x <= 3.4e+236)
		tmp = (x * x) / (y * y);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -9.8e+91], N[(x / y), $MachinePrecision], If[LessEqual[x, 3e+22], t$95$0, If[LessEqual[x, 3e+99], N[(x / y), $MachinePrecision], If[LessEqual[x, 7.5e+115], t$95$0, If[LessEqual[x, 3.4e+236], N[(N[(x * x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -9.8000000000000006e91 or 3e22 < x < 3.00000000000000014e99 or 3.40000000000000007e236 < x

   1. Initial program 99.9%

      \[\frac{\left|x - y\right|}{\left|y\right|} \]
   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 35.6%

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

      3. fabs-sqr 35.6%

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

      4. add-sqr-sqrt 36.1%

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

      5. *-commutative 36.1%

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

      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 57.1%

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

   3. Applied egg-rr 57.1%

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

   4. Taylor expanded in y around 0 57.6%

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

   if -9.8000000000000006e91 < x < 3e22 or 3.00000000000000014e99 < x < 7.4999999999999997e115

   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 50.2%

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

      3. fabs-sqr 50.2%

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

      4. add-sqr-sqrt 51.1%

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

      5. *-commutative 51.1%

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

      6. add-sqr-sqrt 8.7%

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

      7. fabs-sqr 8.7%

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

      8. add-sqr-sqrt 14.0%

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

   3. Applied egg-rr 14.0%

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

      1. flip-- 12.5%

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

      2. associate-*r/ 11.9%

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

      3. +-commutative 11.9%

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

   5. Applied egg-rr 11.9%

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

   6. Taylor expanded in y around inf 1.9%

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

      1. neg-mul-1 1.9%

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

   8. Simplified 1.9%

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

      1. *-un-lft-identity 1.9%

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

      2. add-sqr-sqrt 0.9%

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

      3. sqrt-unprod 19.8%

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

      4. sqr-neg 19.8%

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

      5. sqrt-unprod 36.3%

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

      6. times-frac 36.4%

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

   10. Applied egg-rr 36.4%

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

       1. /-rgt-identity 36.4%

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

       2. associate-*r/ 36.3%

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

       3. rem-square-sqrt 74.1%

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

   12. Simplified 74.1%

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

   if 7.4999999999999997e115 < x < 3.40000000000000007e236

   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 95.8%

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

      3. fabs-sqr 95.8%

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

      4. add-sqr-sqrt 96.1%

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

      5. *-commutative 96.1%

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

      6. add-sqr-sqrt 33.3%

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

      7. fabs-sqr 33.3%

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

      8. add-sqr-sqrt 33.9%

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

   3. Applied egg-rr 33.9%

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

      1. flip-- 26.2%

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

      2. associate-*r/ 26.2%

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

      3. +-commutative 26.2%

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

   5. Applied egg-rr 26.2%

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

   6. Taylor expanded in x around inf 26.8%

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

      1. unpow2 26.8%

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

   8. Simplified 26.8%

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

   9. Taylor expanded in y around inf 42.9%

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

       1. unpow2 42.9%

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

       2. unpow2 42.9%

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

   11. Simplified 42.9%

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

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.8 \cdot 10^{+91}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+22}:\\ \;\;\;\;\frac{y}{x + y}\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+99}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+115}:\\ \;\;\;\;\frac{y}{x + y}\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{+236}:\\ \;\;\;\;\frac{x \cdot x}{y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 5: 57.7% accurate, 22.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{+90}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{+23}:\\ \;\;\;\;\frac{y}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.2e+90) (/ x y) (if (<= x 3.8e+23) (/ y (+ x y)) (/ x y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.2e+90) {
		tmp = x / y;
	} else if (x <= 3.8e+23) {
		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 (x <= (-2.2d+90)) then
        tmp = x / y
    else if (x <= 3.8d+23) 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 (x <= -2.2e+90) {
		tmp = x / y;
	} else if (x <= 3.8e+23) {
		tmp = y / (x + y);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.2e+90:
		tmp = x / y
	elif x <= 3.8e+23:
		tmp = y / (x + y)
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.2e+90)
		tmp = Float64(x / y);
	elseif (x <= 3.8e+23)
		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 (x <= -2.2e+90)
		tmp = x / y;
	elseif (x <= 3.8e+23)
		tmp = y / (x + y);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.2e+90], N[(x / y), $MachinePrecision], If[LessEqual[x, 3.8e+23], N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.1999999999999999e90 or 3.79999999999999975e23 < x

   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 52.8%

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

      3. fabs-sqr 52.8%

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

      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 24.0%

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

      7. fabs-sqr 24.0%

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

      8. add-sqr-sqrt 50.0%

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

   3. Applied egg-rr 50.0%

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

   4. Taylor expanded in y around 0 50.4%

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

   if -2.1999999999999999e90 < x < 3.79999999999999975e23

   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 48.5%

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

      3. fabs-sqr 48.5%

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

      4. add-sqr-sqrt 49.4%

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

      5. *-commutative 49.4%

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

      6. add-sqr-sqrt 8.3%

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

      7. fabs-sqr 8.3%

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

      8. add-sqr-sqrt 13.8%

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

   3. Applied egg-rr 13.8%

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

      1. flip-- 12.3%

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

      2. associate-*r/ 12.3%

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

      3. +-commutative 12.3%

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

   5. Applied egg-rr 12.3%

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

   6. Taylor expanded in y around inf 1.9%

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

      1. neg-mul-1 1.9%

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

   8. Simplified 1.9%

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

      1. *-un-lft-identity 1.9%

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

      2. add-sqr-sqrt 0.9%

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

      3. sqrt-unprod 20.5%

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

      4. sqr-neg 20.5%

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

      5. sqrt-unprod 37.6%

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

      6. times-frac 37.6%

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

   10. Applied egg-rr 37.6%

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

       1. /-rgt-identity 37.6%

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

       2. associate-*r/ 37.6%

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

       3. rem-square-sqrt 73.9%

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

   12. Simplified 73.9%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{+90}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{+23}:\\ \;\;\;\;\frac{y}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 6: 26.2% 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 50.4%

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

   3. fabs-sqr 50.4%

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

   4. add-sqr-sqrt 51.1%

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

   5. *-commutative 51.1%

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

   6. add-sqr-sqrt 15.2%

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

   7. fabs-sqr 15.2%

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

   8. add-sqr-sqrt 29.7%

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

3. Applied egg-rr 29.7%

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

4. Taylor expanded in y around 0 30.9%

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

5. Final simplification 30.9%

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

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 50.4%

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

   3. fabs-sqr 50.4%

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

   4. add-sqr-sqrt 51.1%

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

   5. *-commutative 51.1%

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

   6. add-sqr-sqrt 15.2%

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

   7. fabs-sqr 15.2%

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

   8. add-sqr-sqrt 29.7%

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

3. Applied egg-rr 29.7%

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

4. Taylor expanded in y around inf 1.3%

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

5. Final simplification 1.3%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023257 
(FPCore (x y)
  :name "Numeric.LinearAlgebra.Util:formatSparse from hmatrix-0.16.1.5"
  :precision binary64
  (/ (fabs (- x y)) (fabs y)))