Numeric.LinearAlgebra.Util:formatSparse from hmatrix-0.16.1.5

Percentage Accurate: 100.0% → 100.0%

Time: 4.7s

Alternatives: 7

Speedup: 2.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. fabs-sub 100.0%

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

   2. div-fabs 100.0%

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

3. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 82.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{-58}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{1}{y}\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{-48}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left(y - x\right) \cdot \left(\frac{1}{y + x} \cdot \frac{y + x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.4e-58)
   (* (- y x) (/ 1.0 y))
   (if (<= y 1.2e-48)
     (fabs (/ x y))
     (* (- y x) (* (/ 1.0 (+ y x)) (/ (+ y x) y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.4e-58) {
		tmp = (y - x) * (1.0 / y);
	} else if (y <= 1.2e-48) {
		tmp = fabs((x / y));
	} else {
		tmp = (y - x) * ((1.0 / (y + x)) * ((y + x) / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-2.4d-58)) then
        tmp = (y - x) * (1.0d0 / y)
    else if (y <= 1.2d-48) then
        tmp = abs((x / y))
    else
        tmp = (y - x) * ((1.0d0 / (y + x)) * ((y + x) / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2.4e-58) {
		tmp = (y - x) * (1.0 / y);
	} else if (y <= 1.2e-48) {
		tmp = Math.abs((x / y));
	} else {
		tmp = (y - x) * ((1.0 / (y + x)) * ((y + x) / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.4e-58:
		tmp = (y - x) * (1.0 / y)
	elif y <= 1.2e-48:
		tmp = math.fabs((x / y))
	else:
		tmp = (y - x) * ((1.0 / (y + x)) * ((y + x) / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.4e-58)
		tmp = Float64(Float64(y - x) * Float64(1.0 / y));
	elseif (y <= 1.2e-48)
		tmp = abs(Float64(x / y));
	else
		tmp = Float64(Float64(y - x) * Float64(Float64(1.0 / Float64(y + x)) * Float64(Float64(y + x) / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.4e-58)
		tmp = (y - x) * (1.0 / y);
	elseif (y <= 1.2e-48)
		tmp = abs((x / y));
	else
		tmp = (y - x) * ((1.0 / (y + x)) * ((y + x) / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.4e-58], N[(N[(y - x), $MachinePrecision] * N[(1.0 / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.2e-48], N[Abs[N[(x / y), $MachinePrecision]], $MachinePrecision], N[(N[(y - x), $MachinePrecision] * N[(N[(1.0 / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(N[(y + x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.4000000000000001e-58

   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 90.2%

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

      3. fabs-sqr 90.2%

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

      4. add-sqr-sqrt 90.9%

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

      5. *-commutative 90.9%

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

      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 10.3%

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

   3. Applied egg-rr 10.3%

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

      1. flip-- 4.1%

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

      2. associate-*r/ 4.1%

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

      3. difference-of-squares 4.2%

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

      4. +-commutative 4.2%

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

      5. +-commutative 4.2%

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

   5. Applied egg-rr 4.2%

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

      1. flip-+ 3.7%

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

      2. associate-/r/ 3.7%

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

      3. associate-*r* 4.9%

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

      4. *-commutative 4.9%

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

      5. associate-*l/ 4.9%

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

      6. *-un-lft-identity 4.9%

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

      7. cancel-sign-sub-inv 4.9%

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

      8. fma-def 5.1%

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

   7. Applied egg-rr 5.1%

      \[\leadsto \color{blue}{\frac{\left(x - y\right) \cdot \frac{y + x}{y}}{\mathsf{fma}\left(y, y, \left(-x\right) \cdot x\right)} \cdot \left(y - x\right)} \]
   8. Step-by-step derivation

      1. *-commutative 5.1%

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

      2. associate-/l* 5.2%

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

      3. fma-udef 5.0%

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

      4. cancel-sign-sub-inv 5.0%

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

      5. +-commutative 5.0%

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

   9. Simplified 5.0%

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

       1. expm1-log1p-u 4.9%

          \[\leadsto \left(y - x\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x - y}{\frac{y \cdot y - x \cdot x}{\frac{x + y}{y}}}\right)\right)} \]

       2. expm1-udef 4.0%

          \[\leadsto \left(y - x\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{x - y}{\frac{y \cdot y - x \cdot x}{\frac{x + y}{y}}}\right)} - 1\right)} \]

   11. Applied egg-rr 5.6%

       \[\leadsto \left(y - x\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{x + y}{\left(x + y\right) \cdot y}\right)} - 1\right)} \]
   12. Step-by-step derivation

       1. expm1-def 38.6%

          \[\leadsto \left(y - x\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x + y}{\left(x + y\right) \cdot y}\right)\right)} \]

       2. expm1-log1p 47.5%

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

       3. associate-/r* 90.9%

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

       4. *-inverses 90.9%

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

   13. Simplified 90.9%

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

   if -2.4000000000000001e-58 < y < 1.2e-48

   1. Initial program 100.0%

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

      1. fabs-sub 100.0%

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

      2. div-fabs 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in y around 0 83.4%

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

      1. neg-mul-1 83.4%

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

      2. distribute-neg-frac 83.4%

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

   6. Simplified 83.4%

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

   if 1.2e-48 < 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 15.2%

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

      3. fabs-sqr 15.2%

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

      4. add-sqr-sqrt 16.6%

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

      5. *-commutative 16.6%

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

      6. add-sqr-sqrt 16.5%

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

      7. fabs-sqr 16.5%

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

      8. add-sqr-sqrt 16.6%

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

   3. Applied egg-rr 16.6%

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

      1. flip-- 6.4%

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

      2. associate-*r/ 6.3%

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

      3. difference-of-squares 6.4%

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

      4. +-commutative 6.4%

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

      5. +-commutative 6.4%

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

   5. Applied egg-rr 6.4%

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

      1. flip-+ 5.7%

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

      2. associate-/r/ 5.7%

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

      3. associate-*r* 6.8%

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

      4. *-commutative 6.8%

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

      5. associate-*l/ 6.8%

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

      6. *-un-lft-identity 6.8%

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

      7. cancel-sign-sub-inv 6.8%

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

      8. fma-def 7.1%

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

   7. Applied egg-rr 7.1%

      \[\leadsto \color{blue}{\frac{\left(x - y\right) \cdot \frac{y + x}{y}}{\mathsf{fma}\left(y, y, \left(-x\right) \cdot x\right)} \cdot \left(y - x\right)} \]
   8. Step-by-step derivation

      1. *-commutative 7.1%

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

      2. associate-/l* 7.3%

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

      3. fma-udef 7.0%

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

      4. cancel-sign-sub-inv 7.0%

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

      5. +-commutative 7.0%

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

   9. Simplified 7.0%

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

       1. *-un-lft-identity 7.0%

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

       2. associate-/r/ 7.0%

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

       3. +-commutative 7.0%

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

       4. flip-- 14.7%

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

       5. times-frac 16.6%

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

       6. sub-neg 16.6%

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

       7. mul-1-neg 16.6%

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

       8. add-sqr-sqrt 0.8%

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

       9. sqrt-unprod 1.8%

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

       10. mul-1-neg 1.8%

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

       11. mul-1-neg 1.8%

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

       12. sqr-neg 1.8%

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

       13. sqrt-prod 0.6%

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

       14. add-sqr-sqrt 12.8%

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

       15. +-commutative 12.8%

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

   11. Applied egg-rr 84.5%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{-58}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{1}{y}\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{-48}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left(y - x\right) \cdot \left(\frac{1}{y + x} \cdot \frac{y + x}{y}\right)\\ \end{array} \]

Alternative 3: 73.6% accurate, 18.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.4 \cdot 10^{-203} \lor \neg \left(y \leq 1.9 \cdot 10^{-297}\right):\\ \;\;\;\;\left(y - x\right) \cdot \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -8.4e-203) (not (<= y 1.9e-297)))
   (* (- y x) (/ 1.0 y))
   (/ x y)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -8.4e-203) || !(y <= 1.9e-297)) {
		tmp = (y - x) * (1.0 / 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 <= (-8.4d-203)) .or. (.not. (y <= 1.9d-297))) then
        tmp = (y - x) * (1.0d0 / y)
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -8.4e-203) || !(y <= 1.9e-297)) {
		tmp = (y - x) * (1.0 / y);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -8.4e-203) or not (y <= 1.9e-297):
		tmp = (y - x) * (1.0 / y)
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -8.4e-203) || !(y <= 1.9e-297))
		tmp = Float64(Float64(y - x) * Float64(1.0 / y));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -8.4e-203) || ~((y <= 1.9e-297)))
		tmp = (y - x) * (1.0 / y);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -8.4e-203], N[Not[LessEqual[y, 1.9e-297]], $MachinePrecision]], N[(N[(y - x), $MachinePrecision] * N[(1.0 / y), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -8.40000000000000008e-203 or 1.90000000000000002e-297 < y

   1. Initial program 100.0%

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

      1. div-inv 99.7%

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

      2. add-sqr-sqrt 48.9%

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

      3. fabs-sqr 48.9%

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

      4. add-sqr-sqrt 49.8%

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

      5. *-commutative 49.8%

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

      6. add-sqr-sqrt 14.5%

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

      7. fabs-sqr 14.5%

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

      8. add-sqr-sqrt 21.7%

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

   3. Applied egg-rr 21.7%

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

      1. flip-- 14.7%

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

      2. associate-*r/ 14.3%

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

      3. difference-of-squares 14.4%

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

      4. +-commutative 14.4%

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

      5. +-commutative 14.4%

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

   5. Applied egg-rr 14.4%

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

      1. flip-+ 11.6%

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

      2. associate-/r/ 11.6%

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

      3. associate-*r* 12.3%

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

      4. *-commutative 12.3%

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

      5. associate-*l/ 12.3%

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

      6. *-un-lft-identity 12.3%

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

      7. cancel-sign-sub-inv 12.3%

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

      8. fma-def 12.5%

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

   7. Applied egg-rr 12.5%

      \[\leadsto \color{blue}{\frac{\left(x - y\right) \cdot \frac{y + x}{y}}{\mathsf{fma}\left(y, y, \left(-x\right) \cdot x\right)} \cdot \left(y - x\right)} \]
   8. Step-by-step derivation

      1. *-commutative 12.5%

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

      2. associate-/l* 12.0%

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

      3. fma-udef 11.8%

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

      4. cancel-sign-sub-inv 11.8%

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

      5. +-commutative 11.8%

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

   9. Simplified 11.8%

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

       1. expm1-log1p-u 5.7%

          \[\leadsto \left(y - x\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x - y}{\frac{y \cdot y - x \cdot x}{\frac{x + y}{y}}}\right)\right)} \]

       2. expm1-udef 4.7%

          \[\leadsto \left(y - x\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{x - y}{\frac{y \cdot y - x \cdot x}{\frac{x + y}{y}}}\right)} - 1\right)} \]

   11. Applied egg-rr 20.3%

       \[\leadsto \left(y - x\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{x + y}{\left(x + y\right) \cdot y}\right)} - 1\right)} \]
   12. Step-by-step derivation

       1. expm1-def 39.8%

          \[\leadsto \left(y - x\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x + y}{\left(x + y\right) \cdot y}\right)\right)} \]

       2. expm1-log1p 50.6%

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

       3. associate-/r* 79.3%

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

       4. *-inverses 79.3%

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

   13. Simplified 79.3%

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

   if -8.40000000000000008e-203 < y < 1.90000000000000002e-297

   1. Initial program 100.0%

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

      1. div-inv 99.9%

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

      2. add-sqr-sqrt 44.6%

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

      3. fabs-sqr 44.6%

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

      4. add-sqr-sqrt 44.9%

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

      5. *-commutative 44.9%

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

      6. add-sqr-sqrt 6.9%

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

      7. fabs-sqr 6.9%

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

      8. add-sqr-sqrt 62.2%

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

   3. Applied egg-rr 62.2%

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

   4. Taylor expanded in y around 0 62.3%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.4 \cdot 10^{-203} \lor \neg \left(y \leq 1.9 \cdot 10^{-297}\right):\\ \;\;\;\;\left(y - x\right) \cdot \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 4: 57.9% accurate, 22.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9.5 \cdot 10^{-58} \lor \neg \left(y \leq 8 \cdot 10^{-49}\right):\\ \;\;\;\;\frac{y}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -9.5e-58) (not (<= y 8e-49))) (/ y (+ y x)) (/ x y)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -9.5e-58) || !(y <= 8e-49)) {
		tmp = y / (y + x);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-9.5d-58)) .or. (.not. (y <= 8d-49))) then
        tmp = y / (y + x)
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -9.5e-58) || !(y <= 8e-49)) {
		tmp = y / (y + x);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -9.5e-58) or not (y <= 8e-49):
		tmp = y / (y + x)
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -9.5e-58) || !(y <= 8e-49))
		tmp = Float64(y / Float64(y + x));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -9.5e-58) || ~((y <= 8e-49)))
		tmp = y / (y + x);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -9.5e-58], N[Not[LessEqual[y, 8e-49]], $MachinePrecision]], N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -9.4999999999999994e-58 or 7.99999999999999949e-49 < 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 49.9%

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

      3. fabs-sqr 49.9%

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

      4. add-sqr-sqrt 50.9%

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

      5. *-commutative 50.9%

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

      6. add-sqr-sqrt 8.9%

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

      7. fabs-sqr 8.9%

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

      8. add-sqr-sqrt 13.7%

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

   3. Applied egg-rr 13.7%

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

      1. flip-- 5.3%

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

      2. associate-*r/ 5.3%

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

      3. difference-of-squares 5.4%

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

      4. +-commutative 5.4%

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

      5. +-commutative 5.4%

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

   5. Applied egg-rr 5.4%

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

   6. Taylor expanded in y around inf 1.8%

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

      1. *-commutative 1.8%

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

      2. neg-mul-1 1.8%

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

      3. +-commutative 1.8%

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

      4. sub-neg 1.8%

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

      5. fma-def 1.8%

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

      6. sub-neg 1.8%

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

      7. add-sqr-sqrt 0.8%

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

      8. sqrt-unprod 21.4%

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

      9. sqr-neg 21.4%

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

      10. sqrt-prod 38.9%

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

      11. add-sqr-sqrt 79.4%

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

   8. Applied egg-rr 79.4%

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

      1. fma-udef 79.4%

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

      2. *-commutative 79.4%

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

      3. mul-1-neg 79.4%

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

      4. neg-sub0 79.4%

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

      5. associate--r- 79.4%

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

      6. neg-sub0 79.4%

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

      7. neg-mul-1 79.4%

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

      8. associate--r+ 79.5%

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

      9. +-inverses 79.5%

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

      10. neg-sub0 79.5%

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

      11. mul-1-neg 79.5%

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

      12. associate-*r* 79.5%

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

      13. metadata-eval 79.5%

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

      14. *-lft-identity 79.5%

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

   10. Simplified 79.5%

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

   if -9.4999999999999994e-58 < y < 7.99999999999999949e-49

   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 46.5%

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

      3. fabs-sqr 46.5%

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

      4. add-sqr-sqrt 47.1%

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

      5. *-commutative 47.1%

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

      6. add-sqr-sqrt 19.9%

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

      7. fabs-sqr 19.9%

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

      8. add-sqr-sqrt 42.6%

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

   3. Applied egg-rr 42.6%

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

   4. Taylor expanded in y around 0 43.3%

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

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.5 \cdot 10^{-58} \lor \neg \left(y \leq 8 \cdot 10^{-49}\right):\\ \;\;\;\;\frac{y}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 5: 2.6% accurate, 68.3× speedup

Math

\[\begin{array}{l} \\ \frac{0}{x} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ 0.0 x))

C

double code(double x, double y) {
	return 0.0 / x;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.0d0 / x
end function

Java

public static double code(double x, double y) {
	return 0.0 / x;
}

Python

def code(x, y):
	return 0.0 / x

Julia

function code(x, y)
	return Float64(0.0 / x)
end

MATLAB

function tmp = code(x, y)
	tmp = 0.0 / x;
end

Wolfram

code[x_, y_] := N[(0.0 / x), $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 48.4%

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

   3. fabs-sqr 48.4%

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

   4. add-sqr-sqrt 49.3%

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

   5. *-commutative 49.3%

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

   6. add-sqr-sqrt 13.7%

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

   7. fabs-sqr 13.7%

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

   8. add-sqr-sqrt 26.2%

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

3. Applied egg-rr 26.2%

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

   1. flip-- 19.0%

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

   2. associate-*r/ 17.6%

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

   3. difference-of-squares 17.7%

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

   4. +-commutative 17.7%

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

   5. +-commutative 17.7%

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

5. Applied egg-rr 17.7%

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

6. Taylor expanded in y around inf 2.0%

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

7. Taylor expanded in y around 0 2.7%

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

   1. +-commutative 2.7%

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

   2. mul-1-neg 2.7%

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

   3. sub-neg 2.7%

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

   4. +-inverses 2.7%

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

9. Simplified 2.7%

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

10. Final simplification 2.7%

    \[\leadsto \frac{0}{x} \]

Alternative 6: 27.7% 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 48.4%

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

   3. fabs-sqr 48.4%

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

   4. add-sqr-sqrt 49.3%

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

   5. *-commutative 49.3%

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

   6. add-sqr-sqrt 13.7%

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

   7. fabs-sqr 13.7%

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

   8. add-sqr-sqrt 26.2%

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

3. Applied egg-rr 26.2%

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

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 7: 1.3% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

public static double code(double x, double y) {
	return -1.0;
}

Python

def code(x, y):
	return -1.0

Julia

function code(x, y)
	return -1.0
end

MATLAB

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

Wolfram

code[x_, y_] := -1.0

Derivation

1. Initial program 100.0%

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

   1. div-inv 99.7%

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

   2. add-sqr-sqrt 48.4%

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

   3. fabs-sqr 48.4%

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

   4. add-sqr-sqrt 49.3%

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

   5. *-commutative 49.3%

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

   6. add-sqr-sqrt 13.7%

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

   7. fabs-sqr 13.7%

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

   8. add-sqr-sqrt 26.2%

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

3. Applied egg-rr 26.2%

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

4. Taylor expanded in y around inf 1.3%

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

5. Final simplification 1.3%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023238 
(FPCore (x y)
  :name "Numeric.LinearAlgebra.Util:formatSparse from hmatrix-0.16.1.5"
  :precision binary64
  (/ (fabs (- x y)) (fabs y)))