Numeric.LinearAlgebra.Util:formatSparse from hmatrix-0.16.1.5

Percentage Accurate: 100.0% → 100.0%

Time: 5.4s

Alternatives: 6

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: 58.8% accurate, 11.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{\frac{x}{y}}{y - x}\\ \mathbf{if}\;y \leq -8 \cdot 10^{-117}:\\ \;\;\;\;\frac{y}{x + y}\\ \mathbf{elif}\;y \leq -5.8 \cdot 10^{-281}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 3.9 \cdot 10^{-193}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{-169}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{-125}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{1}{y - x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (/ (/ x y) (- y x)))))
   (if (<= y -8e-117)
     (/ y (+ x y))
     (if (<= y -5.8e-281)
       t_0
       (if (<= y 3.9e-193)
         (/ x y)
         (if (<= y 4.6e-169)
           t_0
           (if (<= y 6.5e-125) (/ x y) (* y (/ 1.0 (- y x))))))))))

C

double code(double x, double y) {
	double t_0 = x * ((x / y) / (y - x));
	double tmp;
	if (y <= -8e-117) {
		tmp = y / (x + y);
	} else if (y <= -5.8e-281) {
		tmp = t_0;
	} else if (y <= 3.9e-193) {
		tmp = x / y;
	} else if (y <= 4.6e-169) {
		tmp = t_0;
	} else if (y <= 6.5e-125) {
		tmp = x / y;
	} else {
		tmp = y * (1.0 / (y - x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * ((x / y) / (y - x))
    if (y <= (-8d-117)) then
        tmp = y / (x + y)
    else if (y <= (-5.8d-281)) then
        tmp = t_0
    else if (y <= 3.9d-193) then
        tmp = x / y
    else if (y <= 4.6d-169) then
        tmp = t_0
    else if (y <= 6.5d-125) then
        tmp = x / y
    else
        tmp = y * (1.0d0 / (y - x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x * ((x / y) / (y - x));
	double tmp;
	if (y <= -8e-117) {
		tmp = y / (x + y);
	} else if (y <= -5.8e-281) {
		tmp = t_0;
	} else if (y <= 3.9e-193) {
		tmp = x / y;
	} else if (y <= 4.6e-169) {
		tmp = t_0;
	} else if (y <= 6.5e-125) {
		tmp = x / y;
	} else {
		tmp = y * (1.0 / (y - x));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x * ((x / y) / (y - x))
	tmp = 0
	if y <= -8e-117:
		tmp = y / (x + y)
	elif y <= -5.8e-281:
		tmp = t_0
	elif y <= 3.9e-193:
		tmp = x / y
	elif y <= 4.6e-169:
		tmp = t_0
	elif y <= 6.5e-125:
		tmp = x / y
	else:
		tmp = y * (1.0 / (y - x))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x * Float64(Float64(x / y) / Float64(y - x)))
	tmp = 0.0
	if (y <= -8e-117)
		tmp = Float64(y / Float64(x + y));
	elseif (y <= -5.8e-281)
		tmp = t_0;
	elseif (y <= 3.9e-193)
		tmp = Float64(x / y);
	elseif (y <= 4.6e-169)
		tmp = t_0;
	elseif (y <= 6.5e-125)
		tmp = Float64(x / y);
	else
		tmp = Float64(y * Float64(1.0 / Float64(y - x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x * ((x / y) / (y - x));
	tmp = 0.0;
	if (y <= -8e-117)
		tmp = y / (x + y);
	elseif (y <= -5.8e-281)
		tmp = t_0;
	elseif (y <= 3.9e-193)
		tmp = x / y;
	elseif (y <= 4.6e-169)
		tmp = t_0;
	elseif (y <= 6.5e-125)
		tmp = x / y;
	else
		tmp = y * (1.0 / (y - x));
	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]}, If[LessEqual[y, -8e-117], N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -5.8e-281], t$95$0, If[LessEqual[y, 3.9e-193], N[(x / y), $MachinePrecision], If[LessEqual[y, 4.6e-169], t$95$0, If[LessEqual[y, 6.5e-125], N[(x / y), $MachinePrecision], N[(y * N[(1.0 / N[(y - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -8.00000000000000024e-117

   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 87.7%

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

      3. fabs-sqr 87.7%

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

      4. add-sqr-sqrt 88.5%

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

      5. *-commutative 88.5%

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

      6. add-sqr-sqrt 0.0%

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

      7. fabs-sqr 0.0%

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

      8. add-sqr-sqrt 12.6%

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

   3. Applied egg-rr 12.6%

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

      1. flip-- 4.5%

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

      2. associate-*r/ 4.5%

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

      3. +-commutative 4.5%

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

   5. Applied egg-rr 4.5%

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

   6. Taylor expanded in x around 0 1.5%

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

      1. unpow2 1.5%

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

      2. mul-1-neg 1.5%

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

      3. distribute-rgt-neg-out 1.5%

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

   8. Simplified 1.5%

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

      1. associate-*r* 1.9%

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

      2. lft-mult-inverse 1.9%

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

      3. *-un-lft-identity 1.9%

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

      4. neg-sub0 1.9%

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

      5. metadata-eval 1.9%

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

      6. sub-neg 1.9%

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

      7. metadata-eval 1.9%

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

      8. add-sqr-sqrt 1.9%

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

      9. sqrt-unprod 1.5%

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

      10. sqr-neg 1.5%

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

      11. sqrt-unprod 0.0%

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

      12. add-sqr-sqrt 78.2%

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

   10. Applied egg-rr 78.2%

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

       1. +-lft-identity 78.2%

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

   12. Simplified 78.2%

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

   if -8.00000000000000024e-117 < y < -5.7999999999999998e-281 or 3.8999999999999999e-193 < y < 4.6000000000000002e-169

   1. Initial program 99.9%

      \[\frac{\left|x - y\right|}{\left|y\right|} \]
   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 62.0%

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

      3. fabs-sqr 62.0%

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

      4. add-sqr-sqrt 62.2%

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

      5. *-commutative 62.2%

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

      6. add-sqr-sqrt 0.0%

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

      7. fabs-sqr 0.0%

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

      8. add-sqr-sqrt 22.7%

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

   3. Applied egg-rr 22.7%

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

      1. flip-- 20.9%

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

      2. associate-*r/ 18.9%

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

      3. +-commutative 18.9%

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

   5. Applied egg-rr 18.9%

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

   6. Taylor expanded in y around 0 19.1%

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

      1. unpow2 19.1%

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

      2. associate-/l* 21.3%

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

      3. associate-/r/ 21.4%

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

   8. Simplified 21.4%

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

      1. associate-*l/ 19.1%

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

      2. *-un-lft-identity 19.1%

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

      3. associate-*l/ 19.1%

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

      4. frac-2neg 19.1%

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

      5. div-inv 19.0%

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

      6. associate-*l/ 19.0%

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

      7. *-un-lft-identity 19.0%

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

      8. distribute-neg-frac 19.0%

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

      9. add-sqr-sqrt 0.0%

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

      10. sqrt-unprod 23.4%

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

      11. sqr-neg 23.4%

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

      12. sqrt-unprod 38.7%

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

      13. add-sqr-sqrt 54.3%

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

      14. frac-2neg 54.3%

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

      15. associate-*r/ 60.7%

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

      16. distribute-neg-in 60.7%

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

      17. add-sqr-sqrt 45.2%

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

      18. sqrt-unprod 60.6%

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

      19. sqr-neg 60.6%

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

      20. sqrt-unprod 15.6%

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

      21. add-sqr-sqrt 60.6%

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

      22. sub-neg 60.6%

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

   10. Applied egg-rr 60.6%

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

       1. associate-*l* 62.7%

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

       2. associate-*r/ 62.7%

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

       3. *-rgt-identity 62.7%

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

   12. Simplified 62.7%

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

   if -5.7999999999999998e-281 < y < 3.8999999999999999e-193 or 4.6000000000000002e-169 < y < 6.4999999999999999e-125

   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 61.2%

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

      3. fabs-sqr 61.2%

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

      4. add-sqr-sqrt 61.6%

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

      5. *-commutative 61.6%

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

      6. add-sqr-sqrt 58.8%

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

      7. fabs-sqr 58.8%

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

      8. add-sqr-sqrt 74.3%

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

   3. Applied egg-rr 74.3%

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

   4. Taylor expanded in y around 0 74.8%

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

   if 6.4999999999999999e-125 < y

   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 19.8%

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

      3. fabs-sqr 19.8%

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

      4. add-sqr-sqrt 21.0%

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

      5. *-commutative 21.0%

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

      6. add-sqr-sqrt 21.0%

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

      7. fabs-sqr 21.0%

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

      8. add-sqr-sqrt 21.0%

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

   3. Applied egg-rr 21.0%

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

      1. flip-- 8.0%

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

      2. associate-*r/ 8.0%

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

      3. +-commutative 8.0%

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

   5. Applied egg-rr 8.0%

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

   6. Taylor expanded in y around inf 2.3%

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

      1. neg-mul-1 2.3%

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

   8. Simplified 2.3%

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

      1. frac-2neg 2.3%

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

      2. div-inv 2.3%

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

      3. remove-double-neg 2.3%

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

      4. distribute-neg-in 2.3%

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

      5. add-sqr-sqrt 0.0%

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

      6. sqrt-unprod 34.0%

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

      7. sqr-neg 34.0%

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

      8. sqrt-unprod 61.9%

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

      9. add-sqr-sqrt 62.4%

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

      10. sub-neg 62.4%

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

   10. Applied egg-rr 62.4%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{-117}:\\ \;\;\;\;\frac{y}{x + y}\\ \mathbf{elif}\;y \leq -5.8 \cdot 10^{-281}:\\ \;\;\;\;x \cdot \frac{\frac{x}{y}}{y - x}\\ \mathbf{elif}\;y \leq 3.9 \cdot 10^{-193}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{-169}:\\ \;\;\;\;x \cdot \frac{\frac{x}{y}}{y - x}\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{-125}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{1}{y - x}\\ \end{array} \]

Alternative 3: 58.5% accurate, 18.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{-135}:\\ \;\;\;\;\frac{y}{x + y}\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{-125}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{1}{y - x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -8e-135)
   (/ y (+ x y))
   (if (<= y 4.5e-125) (/ x y) (* y (/ 1.0 (- y x))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -8e-135) {
		tmp = y / (x + y);
	} else if (y <= 4.5e-125) {
		tmp = x / y;
	} else {
		tmp = y * (1.0 / (y - x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-8d-135)) then
        tmp = y / (x + y)
    else if (y <= 4.5d-125) then
        tmp = x / y
    else
        tmp = y * (1.0d0 / (y - x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -8e-135) {
		tmp = y / (x + y);
	} else if (y <= 4.5e-125) {
		tmp = x / y;
	} else {
		tmp = y * (1.0 / (y - x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -8e-135:
		tmp = y / (x + y)
	elif y <= 4.5e-125:
		tmp = x / y
	else:
		tmp = y * (1.0 / (y - x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -8e-135)
		tmp = Float64(y / Float64(x + y));
	elseif (y <= 4.5e-125)
		tmp = Float64(x / y);
	else
		tmp = Float64(y * Float64(1.0 / Float64(y - x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -8e-135)
		tmp = y / (x + y);
	elseif (y <= 4.5e-125)
		tmp = x / y;
	else
		tmp = y * (1.0 / (y - x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -8e-135], N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.5e-125], N[(x / y), $MachinePrecision], N[(y * N[(1.0 / N[(y - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.0000000000000003e-135

   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 88.0%

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

      3. fabs-sqr 88.0%

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

      4. add-sqr-sqrt 88.9%

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

      5. *-commutative 88.9%

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

      6. add-sqr-sqrt 0.0%

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

      7. fabs-sqr 0.0%

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

      8. add-sqr-sqrt 12.2%

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

   3. Applied egg-rr 12.2%

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

      1. flip-- 4.4%

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

      2. associate-*r/ 4.4%

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

      3. +-commutative 4.4%

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

   5. Applied egg-rr 4.4%

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

   6. Taylor expanded in x around 0 1.6%

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

      1. unpow2 1.6%

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

      2. mul-1-neg 1.6%

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

      3. distribute-rgt-neg-out 1.6%

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

   8. Simplified 1.6%

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

      1. associate-*r* 1.9%

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

      2. lft-mult-inverse 1.9%

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

      3. *-un-lft-identity 1.9%

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

      4. neg-sub0 1.9%

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

      5. metadata-eval 1.9%

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

      6. sub-neg 1.9%

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

      7. metadata-eval 1.9%

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

      8. add-sqr-sqrt 1.9%

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

      9. sqrt-unprod 1.6%

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

      10. sqr-neg 1.6%

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

      11. sqrt-unprod 0.0%

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

      12. add-sqr-sqrt 76.5%

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

   10. Applied egg-rr 76.5%

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

       1. +-lft-identity 76.5%

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

   12. Simplified 76.5%

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

   if -8.0000000000000003e-135 < y < 4.50000000000000012e-125

   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 60.2%

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

      3. fabs-sqr 60.2%

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

      4. add-sqr-sqrt 60.5%

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

      5. *-commutative 60.5%

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

      6. add-sqr-sqrt 28.3%

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

      7. fabs-sqr 28.3%

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

      8. add-sqr-sqrt 48.4%

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

   3. Applied egg-rr 48.4%

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

   4. Taylor expanded in y around 0 48.7%

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

   if 4.50000000000000012e-125 < y

   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 19.8%

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

      3. fabs-sqr 19.8%

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

      4. add-sqr-sqrt 21.0%

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

      5. *-commutative 21.0%

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

      6. add-sqr-sqrt 21.0%

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

      7. fabs-sqr 21.0%

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

      8. add-sqr-sqrt 21.0%

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

   3. Applied egg-rr 21.0%

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

      1. flip-- 8.0%

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

      2. associate-*r/ 8.0%

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

      3. +-commutative 8.0%

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

   5. Applied egg-rr 8.0%

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

   6. Taylor expanded in y around inf 2.3%

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

      1. neg-mul-1 2.3%

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

   8. Simplified 2.3%

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

      1. frac-2neg 2.3%

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

      2. div-inv 2.3%

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

      3. remove-double-neg 2.3%

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

      4. distribute-neg-in 2.3%

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

      5. add-sqr-sqrt 0.0%

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

      6. sqrt-unprod 34.0%

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

      7. sqr-neg 34.0%

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

      8. sqrt-unprod 61.9%

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

      9. add-sqr-sqrt 62.4%

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

      10. sub-neg 62.4%

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

   10. Applied egg-rr 62.4%

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

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{-135}:\\ \;\;\;\;\frac{y}{x + y}\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{-125}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{1}{y - x}\\ \end{array} \]

Alternative 4: 58.8% accurate, 22.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.2 \cdot 10^{-136} \lor \neg \left(y \leq 7.8 \cdot 10^{-125}\right):\\ \;\;\;\;\frac{y}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -8.2e-136) (not (<= y 7.8e-125))) (/ y (+ x y)) (/ x y)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -8.2e-136) || !(y <= 7.8e-125)) {
		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 <= (-8.2d-136)) .or. (.not. (y <= 7.8d-125))) 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 <= -8.2e-136) || !(y <= 7.8e-125)) {
		tmp = y / (x + y);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -8.2e-136) or not (y <= 7.8e-125):
		tmp = y / (x + y)
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -8.2e-136) || !(y <= 7.8e-125))
		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 <= -8.2e-136) || ~((y <= 7.8e-125)))
		tmp = y / (x + y);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -8.2e-136], N[Not[LessEqual[y, 7.8e-125]], $MachinePrecision]], N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -8.20000000000000051e-136 or 7.79999999999999965e-125 < 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 58.8%

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

      3. fabs-sqr 58.8%

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

      4. add-sqr-sqrt 59.8%

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

      5. *-commutative 59.8%

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

      6. add-sqr-sqrt 9.0%

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

      7. fabs-sqr 9.0%

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

      8. add-sqr-sqrt 16.0%

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

   3. Applied egg-rr 16.0%

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

      1. flip-- 5.9%

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

      2. associate-*r/ 5.9%

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

      3. +-commutative 5.9%

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

   5. Applied egg-rr 5.9%

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

   6. Taylor expanded in x around 0 1.7%

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

      1. unpow2 1.7%

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

      2. mul-1-neg 1.7%

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

      3. distribute-rgt-neg-out 1.7%

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

   8. Simplified 1.7%

      \[\leadsto \frac{\frac{1}{y} \cdot \color{blue}{\left(y \cdot \left(-y\right)\right)}}{y + x} \]
   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 15.4%

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

      10. sqr-neg 15.4%

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

      11. sqrt-unprod 26.4%

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

      12. add-sqr-sqrt 70.5%

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

   10. Applied egg-rr 70.5%

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

       1. +-lft-identity 70.5%

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

   12. Simplified 70.5%

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

   if -8.20000000000000051e-136 < y < 7.79999999999999965e-125

   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 60.2%

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

      3. fabs-sqr 60.2%

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

      4. add-sqr-sqrt 60.5%

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

      5. *-commutative 60.5%

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

      6. add-sqr-sqrt 28.3%

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

      7. fabs-sqr 28.3%

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

      8. add-sqr-sqrt 48.4%

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

   3. Applied egg-rr 48.4%

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

   4. Taylor expanded in y around 0 48.7%

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

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.2 \cdot 10^{-136} \lor \neg \left(y \leq 7.8 \cdot 10^{-125}\right):\\ \;\;\;\;\frac{y}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 5: 26.8% 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 59.3%

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

   3. fabs-sqr 59.3%

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

   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 15.1%

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

   7. fabs-sqr 15.1%

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

   8. add-sqr-sqrt 26.3%

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

3. Applied egg-rr 26.3%

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

4. Taylor expanded in y around 0 27.4%

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

5. Final simplification 27.4%

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

Alternative 6: 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 59.3%

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

   3. fabs-sqr 59.3%

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

   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 15.1%

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

   7. fabs-sqr 15.1%

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

   8. add-sqr-sqrt 26.3%

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

3. Applied egg-rr 26.3%

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

4. Taylor expanded in y around inf 1.3%

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

5. Final simplification 1.3%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023275 
(FPCore (x y)
  :name "Numeric.LinearAlgebra.Util:formatSparse from hmatrix-0.16.1.5"
  :precision binary64
  (/ (fabs (- x y)) (fabs y)))