Numeric.LinearAlgebra.Util:formatSparse from hmatrix-0.16.1.5

Percentage Accurate: 100.0% → 100.0%

Time: 4.2s

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: 58.1% accurate, 13.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{x + y}\\ \mathbf{if}\;x \leq -8.2 \cdot 10^{+52}:\\ \;\;\;\;x \cdot \frac{\frac{x}{y}}{y - x}\\ \mathbf{elif}\;x \leq -2.15 \cdot 10^{-10}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -1.6 \cdot 10^{-31}:\\ \;\;\;\;\frac{x \cdot \frac{x}{y}}{x + y}\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{+162}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (+ x y))))
   (if (<= x -8.2e+52)
     (* x (/ (/ x y) (- y x)))
     (if (<= x -2.15e-10)
       t_0
       (if (<= x -1.6e-31)
         (/ (* x (/ x y)) (+ x y))
         (if (<= x 2.4e+162) t_0 (+ (/ x y) -1.0)))))))

C

double code(double x, double y) {
	double t_0 = y / (x + y);
	double tmp;
	if (x <= -8.2e+52) {
		tmp = x * ((x / y) / (y - x));
	} else if (x <= -2.15e-10) {
		tmp = t_0;
	} else if (x <= -1.6e-31) {
		tmp = (x * (x / y)) / (x + y);
	} else if (x <= 2.4e+162) {
		tmp = t_0;
	} else {
		tmp = (x / y) + -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y / (x + y)
    if (x <= (-8.2d+52)) then
        tmp = x * ((x / y) / (y - x))
    else if (x <= (-2.15d-10)) then
        tmp = t_0
    else if (x <= (-1.6d-31)) then
        tmp = (x * (x / y)) / (x + y)
    else if (x <= 2.4d+162) then
        tmp = t_0
    else
        tmp = (x / y) + (-1.0d0)
    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 <= -8.2e+52) {
		tmp = x * ((x / y) / (y - x));
	} else if (x <= -2.15e-10) {
		tmp = t_0;
	} else if (x <= -1.6e-31) {
		tmp = (x * (x / y)) / (x + y);
	} else if (x <= 2.4e+162) {
		tmp = t_0;
	} else {
		tmp = (x / y) + -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y / (x + y)
	tmp = 0
	if x <= -8.2e+52:
		tmp = x * ((x / y) / (y - x))
	elif x <= -2.15e-10:
		tmp = t_0
	elif x <= -1.6e-31:
		tmp = (x * (x / y)) / (x + y)
	elif x <= 2.4e+162:
		tmp = t_0
	else:
		tmp = (x / y) + -1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y / Float64(x + y))
	tmp = 0.0
	if (x <= -8.2e+52)
		tmp = Float64(x * Float64(Float64(x / y) / Float64(y - x)));
	elseif (x <= -2.15e-10)
		tmp = t_0;
	elseif (x <= -1.6e-31)
		tmp = Float64(Float64(x * Float64(x / y)) / Float64(x + y));
	elseif (x <= 2.4e+162)
		tmp = t_0;
	else
		tmp = Float64(Float64(x / y) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y / (x + y);
	tmp = 0.0;
	if (x <= -8.2e+52)
		tmp = x * ((x / y) / (y - x));
	elseif (x <= -2.15e-10)
		tmp = t_0;
	elseif (x <= -1.6e-31)
		tmp = (x * (x / y)) / (x + y);
	elseif (x <= 2.4e+162)
		tmp = t_0;
	else
		tmp = (x / y) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -8.2e+52], N[(x * N[(N[(x / y), $MachinePrecision] / N[(y - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2.15e-10], t$95$0, If[LessEqual[x, -1.6e-31], N[(N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.4e+162], t$95$0, N[(N[(x / y), $MachinePrecision] + -1.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -8.1999999999999999e52

   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 3.9%

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

      3. fabs-sqr 3.9%

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

      4. add-sqr-sqrt 4.4%

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

      5. *-commutative 4.4%

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

      6. add-sqr-sqrt 0.4%

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

      7. fabs-sqr 0.4%

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

      8. add-sqr-sqrt 37.5%

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

   3. Applied egg-rr 37.5%

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

      1. flip-- 22.7%

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

      2. associate-*r/ 17.1%

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

      3. +-commutative 17.1%

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

   5. Applied egg-rr 17.1%

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

   6. Taylor expanded in y around 0 17.5%

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

      1. unpow2 17.5%

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

      2. associate-/l* 21.4%

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

      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/ 17.5%

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

      2. *-un-lft-identity 17.5%

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

      3. associate-*l/ 17.5%

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

      4. frac-2neg 17.5%

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

      5. div-inv 17.5%

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

      6. associate-*l/ 17.5%

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

      7. *-un-lft-identity 17.5%

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

      8. distribute-neg-frac 17.5%

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

      9. add-sqr-sqrt 0.3%

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

      10. sqrt-unprod 0.4%

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

      11. sqr-neg 0.4%

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

      12. sqrt-unprod 0.2%

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

      13. add-sqr-sqrt 21.9%

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

      14. frac-2neg 21.9%

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

      15. associate-*r/ 27.5%

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

      16. distribute-neg-in 27.5%

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

      17. add-sqr-sqrt 0.2%

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

      18. sqrt-unprod 25.7%

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

      19. sqr-neg 25.7%

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

      20. sqrt-unprod 27.4%

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

      21. add-sqr-sqrt 27.7%

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

      22. sub-neg 27.7%

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

   10. Applied egg-rr 27.7%

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

       1. associate-*l* 53.2%

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

       2. associate-*r/ 53.3%

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

       3. *-rgt-identity 53.3%

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

   12. Simplified 53.3%

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

   if -8.1999999999999999e52 < x < -2.15000000000000007e-10 or -1.60000000000000009e-31 < x < 2.40000000000000009e162

   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 57.2%

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

      3. fabs-sqr 57.2%

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

      4. add-sqr-sqrt 58.1%

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

      5. *-commutative 58.1%

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

      6. add-sqr-sqrt 11.8%

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

      7. fabs-sqr 11.8%

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

      8. add-sqr-sqrt 18.4%

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

   3. Applied egg-rr 18.4%

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

      1. flip-- 12.4%

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

      2. associate-*r/ 10.7%

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

      3. +-commutative 10.7%

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

   5. Applied egg-rr 10.7%

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

   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* 1.8%

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

      2. lft-mult-inverse 1.8%

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

      3. *-un-lft-identity 1.8%

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

      4. neg-sub0 1.8%

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

      5. metadata-eval 1.8%

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

      6. sub-neg 1.8%

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

      7. metadata-eval 1.8%

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

      8. add-sqr-sqrt 1.0%

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

      9. sqrt-unprod 17.9%

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

      10. sqr-neg 17.9%

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

      11. sqrt-unprod 34.2%

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

      12. add-sqr-sqrt 70.7%

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

   10. Applied egg-rr 70.7%

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

       1. +-lft-identity 70.7%

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

   12. Simplified 70.7%

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

   if -2.15000000000000007e-10 < x < -1.60000000000000009e-31

   1. Initial program 99.8%

      \[\frac{\left|x - y\right|}{\left|y\right|} \]
   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 0.0%

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

      3. fabs-sqr 0.0%

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

      4. add-sqr-sqrt 1.0%

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

      5. *-commutative 1.0%

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

      6. add-sqr-sqrt 0.4%

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

      7. fabs-sqr 0.4%

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

      8. add-sqr-sqrt 75.2%

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

   3. Applied egg-rr 75.2%

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

      1. flip-- 75.1%

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

      2. associate-*r/ 74.8%

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

      3. +-commutative 74.8%

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

   5. Applied egg-rr 74.8%

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

   6. Taylor expanded in y around 0 74.9%

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

      1. unpow2 74.9%

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

      2. associate-/l* 75.1%

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

      3. associate-/r/ 75.4%

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

   8. Simplified 75.4%

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

   if 2.40000000000000009e162 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around -inf 100.0%

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

      1. fabs-neg 100.0%

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

      2. mul-1-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. fabs-sub 100.0%

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

      5. fabs-div 100.0%

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

      6. rem-square-sqrt 48.1%

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

      7. fabs-sqr 48.1%

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

      8. rem-square-sqrt 48.7%

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

      9. div-sub 48.7%

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

      10. sub-neg 48.7%

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

      11. *-inverses 48.7%

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

      12. metadata-eval 48.7%

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

      13. +-commutative 48.7%

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

   4. Simplified 48.7%

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

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.2 \cdot 10^{+52}:\\ \;\;\;\;x \cdot \frac{\frac{x}{y}}{y - x}\\ \mathbf{elif}\;x \leq -2.15 \cdot 10^{-10}:\\ \;\;\;\;\frac{y}{x + y}\\ \mathbf{elif}\;x \leq -1.6 \cdot 10^{-31}:\\ \;\;\;\;\frac{x \cdot \frac{x}{y}}{x + y}\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{+162}:\\ \;\;\;\;\frac{y}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + -1\\ \end{array} \]

Alternative 3: 58.0% accurate, 15.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{+52}:\\ \;\;\;\;x \cdot \frac{\frac{x}{y}}{y - x}\\ \mathbf{elif}\;x \leq -9.5 \cdot 10^{-11} \lor \neg \left(x \leq -1.65 \cdot 10^{-31}\right) \land x \leq 5.4 \cdot 10^{+163}:\\ \;\;\;\;\frac{y}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -8.5e+52)
   (* x (/ (/ x y) (- y x)))
   (if (or (<= x -9.5e-11) (and (not (<= x -1.65e-31)) (<= x 5.4e+163)))
     (/ y (+ x y))
     (+ (/ x y) -1.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -8.5e+52) {
		tmp = x * ((x / y) / (y - x));
	} else if ((x <= -9.5e-11) || (!(x <= -1.65e-31) && (x <= 5.4e+163))) {
		tmp = y / (x + y);
	} else {
		tmp = (x / y) + -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-8.5d+52)) then
        tmp = x * ((x / y) / (y - x))
    else if ((x <= (-9.5d-11)) .or. (.not. (x <= (-1.65d-31))) .and. (x <= 5.4d+163)) then
        tmp = y / (x + y)
    else
        tmp = (x / y) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -8.5e+52) {
		tmp = x * ((x / y) / (y - x));
	} else if ((x <= -9.5e-11) || (!(x <= -1.65e-31) && (x <= 5.4e+163))) {
		tmp = y / (x + y);
	} else {
		tmp = (x / y) + -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -8.5e+52:
		tmp = x * ((x / y) / (y - x))
	elif (x <= -9.5e-11) or (not (x <= -1.65e-31) and (x <= 5.4e+163)):
		tmp = y / (x + y)
	else:
		tmp = (x / y) + -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -8.5e+52)
		tmp = Float64(x * Float64(Float64(x / y) / Float64(y - x)));
	elseif ((x <= -9.5e-11) || (!(x <= -1.65e-31) && (x <= 5.4e+163)))
		tmp = Float64(y / Float64(x + y));
	else
		tmp = Float64(Float64(x / y) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -8.5e+52)
		tmp = x * ((x / y) / (y - x));
	elseif ((x <= -9.5e-11) || (~((x <= -1.65e-31)) && (x <= 5.4e+163)))
		tmp = y / (x + y);
	else
		tmp = (x / y) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -8.5e+52], N[(x * N[(N[(x / y), $MachinePrecision] / N[(y - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[x, -9.5e-11], And[N[Not[LessEqual[x, -1.65e-31]], $MachinePrecision], LessEqual[x, 5.4e+163]]], N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.49999999999999994e52

   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 3.9%

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

      3. fabs-sqr 3.9%

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

      4. add-sqr-sqrt 4.4%

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

      5. *-commutative 4.4%

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

      6. add-sqr-sqrt 0.4%

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

      7. fabs-sqr 0.4%

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

      8. add-sqr-sqrt 37.5%

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

   3. Applied egg-rr 37.5%

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

      1. flip-- 22.7%

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

      2. associate-*r/ 17.1%

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

      3. +-commutative 17.1%

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

   5. Applied egg-rr 17.1%

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

   6. Taylor expanded in y around 0 17.5%

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

      1. unpow2 17.5%

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

      2. associate-/l* 21.4%

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

      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/ 17.5%

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

      2. *-un-lft-identity 17.5%

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

      3. associate-*l/ 17.5%

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

      4. frac-2neg 17.5%

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

      5. div-inv 17.5%

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

      6. associate-*l/ 17.5%

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

      7. *-un-lft-identity 17.5%

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

      8. distribute-neg-frac 17.5%

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

      9. add-sqr-sqrt 0.3%

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

      10. sqrt-unprod 0.4%

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

      11. sqr-neg 0.4%

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

      12. sqrt-unprod 0.2%

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

      13. add-sqr-sqrt 21.9%

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

      14. frac-2neg 21.9%

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

      15. associate-*r/ 27.5%

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

      16. distribute-neg-in 27.5%

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

      17. add-sqr-sqrt 0.2%

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

      18. sqrt-unprod 25.7%

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

      19. sqr-neg 25.7%

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

      20. sqrt-unprod 27.4%

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

      21. add-sqr-sqrt 27.7%

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

      22. sub-neg 27.7%

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

   10. Applied egg-rr 27.7%

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

       1. associate-*l* 53.2%

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

       2. associate-*r/ 53.3%

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

       3. *-rgt-identity 53.3%

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

   12. Simplified 53.3%

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

   if -8.49999999999999994e52 < x < -9.49999999999999951e-11 or -1.65e-31 < x < 5.39999999999999998e163

   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 57.2%

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

      3. fabs-sqr 57.2%

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

      4. add-sqr-sqrt 58.1%

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

      5. *-commutative 58.1%

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

      6. add-sqr-sqrt 11.8%

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

      7. fabs-sqr 11.8%

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

      8. add-sqr-sqrt 18.4%

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

   3. Applied egg-rr 18.4%

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

      1. flip-- 12.4%

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

      2. associate-*r/ 10.7%

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

      3. +-commutative 10.7%

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

   5. Applied egg-rr 10.7%

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

   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* 1.8%

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

      2. lft-mult-inverse 1.8%

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

      3. *-un-lft-identity 1.8%

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

      4. neg-sub0 1.8%

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

      5. metadata-eval 1.8%

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

      6. sub-neg 1.8%

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

      7. metadata-eval 1.8%

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

      8. add-sqr-sqrt 1.0%

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

      9. sqrt-unprod 17.9%

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

      10. sqr-neg 17.9%

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

      11. sqrt-unprod 34.2%

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

      12. add-sqr-sqrt 70.7%

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

   10. Applied egg-rr 70.7%

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

       1. +-lft-identity 70.7%

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

   12. Simplified 70.7%

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

   if -9.49999999999999951e-11 < x < -1.65e-31 or 5.39999999999999998e163 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around -inf 100.0%

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

      1. fabs-neg 100.0%

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

      2. mul-1-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. fabs-sub 100.0%

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

      5. fabs-div 100.0%

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

      6. rem-square-sqrt 53.7%

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

      7. fabs-sqr 53.7%

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

      8. rem-square-sqrt 54.4%

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

      9. div-sub 54.5%

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

      10. sub-neg 54.5%

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

      11. *-inverses 54.5%

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

      12. metadata-eval 54.5%

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

      13. +-commutative 54.5%

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

   4. Simplified 54.5%

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

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{+52}:\\ \;\;\;\;x \cdot \frac{\frac{x}{y}}{y - x}\\ \mathbf{elif}\;x \leq -9.5 \cdot 10^{-11} \lor \neg \left(x \leq -1.65 \cdot 10^{-31}\right) \land x \leq 5.4 \cdot 10^{+163}:\\ \;\;\;\;\frac{y}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + -1\\ \end{array} \]

Alternative 4: 57.3% accurate, 22.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{-39} \lor \neg \left(y \leq 1.5 \cdot 10^{-12}\right):\\ \;\;\;\;\frac{y}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.8e-39) (not (<= y 1.5e-12))) (/ y (+ x y)) (+ (/ x y) -1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.8e-39) || !(y <= 1.5e-12)) {
		tmp = y / (x + y);
	} else {
		tmp = (x / y) + -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-1.8d-39)) .or. (.not. (y <= 1.5d-12))) then
        tmp = y / (x + y)
    else
        tmp = (x / y) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.8e-39) || !(y <= 1.5e-12)) {
		tmp = y / (x + y);
	} else {
		tmp = (x / y) + -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.8e-39) or not (y <= 1.5e-12):
		tmp = y / (x + y)
	else:
		tmp = (x / y) + -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.8e-39) || !(y <= 1.5e-12))
		tmp = Float64(y / Float64(x + y));
	else
		tmp = Float64(Float64(x / y) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.8e-39) || ~((y <= 1.5e-12)))
		tmp = y / (x + y);
	else
		tmp = (x / y) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.8e-39], N[Not[LessEqual[y, 1.5e-12]], $MachinePrecision]], N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.8e-39 or 1.5000000000000001e-12 < 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 51.2%

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

      3. fabs-sqr 51.2%

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

      4. add-sqr-sqrt 52.2%

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

      5. *-commutative 52.2%

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

      6. add-sqr-sqrt 6.1%

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

      7. fabs-sqr 6.1%

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

      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-- 4.7%

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

      2. associate-*r/ 4.7%

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

      3. +-commutative 4.7%

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

   5. Applied egg-rr 4.7%

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

   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* 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 15.7%

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

      10. sqr-neg 15.7%

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

      11. sqrt-unprod 33.7%

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

      12. add-sqr-sqrt 73.6%

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

   10. Applied egg-rr 73.6%

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

       1. +-lft-identity 73.6%

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

   12. Simplified 73.6%

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

   if -1.8e-39 < y < 1.5000000000000001e-12

   1. Initial program 100.0%

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

   2. Taylor expanded in x around -inf 100.0%

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

      1. fabs-neg 100.0%

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

      2. mul-1-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. fabs-sub 100.0%

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

      5. fabs-div 100.0%

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

      6. rem-square-sqrt 44.0%

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

      7. fabs-sqr 44.0%

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

      8. rem-square-sqrt 44.7%

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

      9. div-sub 44.7%

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

      10. sub-neg 44.7%

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

      11. *-inverses 44.7%

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

      12. metadata-eval 44.7%

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

      13. +-commutative 44.7%

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

   4. Simplified 44.7%

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

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{-39} \lor \neg \left(y \leq 1.5 \cdot 10^{-12}\right):\\ \;\;\;\;\frac{y}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + -1\\ \end{array} \]

Alternative 5: 25.9% 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 49.6%

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

   3. fabs-sqr 49.6%

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

   4. add-sqr-sqrt 50.4%

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

   5. *-commutative 50.4%

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

   6. add-sqr-sqrt 13.2%

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

   7. fabs-sqr 13.2%

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

   8. add-sqr-sqrt 27.4%

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

3. Applied egg-rr 27.4%

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

4. Taylor expanded in y around 0 27.8%

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

5. Final simplification 27.8%

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

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 49.6%

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

   3. fabs-sqr 49.6%

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

   4. add-sqr-sqrt 50.4%

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

   5. *-commutative 50.4%

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

   6. add-sqr-sqrt 13.2%

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

   7. fabs-sqr 13.2%

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

   8. add-sqr-sqrt 27.4%

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

3. Applied egg-rr 27.4%

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

4. Taylor expanded in y around inf 1.3%

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

5. Final simplification 1.3%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023297 
(FPCore (x y)
  :name "Numeric.LinearAlgebra.Util:formatSparse from hmatrix-0.16.1.5"
  :precision binary64
  (/ (fabs (- x y)) (fabs y)))