Linear.Projection:perspective from linear-1.19.1.3, B

Percentage Accurate: 77.8% → 99.0%

Time: 8.9s

Alternatives: 8

Speedup: 0.7×

Specification

Math

\[\begin{array}{l} \\ \frac{\left(x \cdot 2\right) \cdot y}{x - y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ (* (* x 2.0) y) (- x y)))

C

double code(double x, double y) {
	return ((x * 2.0) * y) / (x - y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((x * 2.0d0) * y) / (x - y)
end function

Java

public static double code(double x, double y) {
	return ((x * 2.0) * y) / (x - y);
}

Python

def code(x, y):
	return ((x * 2.0) * y) / (x - y)

Julia

function code(x, y)
	return Float64(Float64(Float64(x * 2.0) * y) / Float64(x - y))
end

MATLAB

function tmp = code(x, y)
	tmp = ((x * 2.0) * y) / (x - y);
end

Wolfram

code[x_, y_] := N[(N[(N[(x * 2.0), $MachinePrecision] * y), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision]

Initial Program: 77.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(x \cdot 2\right) \cdot y}{x - y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ (* (* x 2.0) y) (- x y)))

C

double code(double x, double y) {
	return ((x * 2.0) * y) / (x - y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((x * 2.0d0) * y) / (x - y)
end function

Java

public static double code(double x, double y) {
	return ((x * 2.0) * y) / (x - y);
}

Python

def code(x, y):
	return ((x * 2.0) * y) / (x - y)

Julia

function code(x, y)
	return Float64(Float64(Float64(x * 2.0) * y) / Float64(x - y))
end

MATLAB

function tmp = code(x, y)
	tmp = ((x * 2.0) * y) / (x - y);
end

Wolfram

code[x_, y_] := N[(N[(N[(x * 2.0), $MachinePrecision] * y), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(x \cdot 2\right) \cdot y}{x - y}\\ t_1 := y \cdot \frac{x \cdot 2}{x - y}\\ \mathbf{if}\;t\_0 \leq -0.04:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq -1 \cdot 10^{-303}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{y + y}{1 - \frac{y}{x}}\\ \mathbf{elif}\;t\_0 \leq 0.5:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (* x 2.0) y) (- x y))) (t_1 (* y (/ (* x 2.0) (- x y)))))
   (if (<= t_0 -0.04)
     t_1
     (if (<= t_0 -1e-303)
       t_0
       (if (<= t_0 0.0)
         (/ (+ y y) (- 1.0 (/ y x)))
         (if (<= t_0 0.5) t_0 t_1))))))

C

double code(double x, double y) {
	double t_0 = ((x * 2.0) * y) / (x - y);
	double t_1 = y * ((x * 2.0) / (x - y));
	double tmp;
	if (t_0 <= -0.04) {
		tmp = t_1;
	} else if (t_0 <= -1e-303) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = (y + y) / (1.0 - (y / x));
	} else if (t_0 <= 0.5) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((x * 2.0d0) * y) / (x - y)
    t_1 = y * ((x * 2.0d0) / (x - y))
    if (t_0 <= (-0.04d0)) then
        tmp = t_1
    else if (t_0 <= (-1d-303)) then
        tmp = t_0
    else if (t_0 <= 0.0d0) then
        tmp = (y + y) / (1.0d0 - (y / x))
    else if (t_0 <= 0.5d0) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = ((x * 2.0) * y) / (x - y);
	double t_1 = y * ((x * 2.0) / (x - y));
	double tmp;
	if (t_0 <= -0.04) {
		tmp = t_1;
	} else if (t_0 <= -1e-303) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = (y + y) / (1.0 - (y / x));
	} else if (t_0 <= 0.5) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = ((x * 2.0) * y) / (x - y)
	t_1 = y * ((x * 2.0) / (x - y))
	tmp = 0
	if t_0 <= -0.04:
		tmp = t_1
	elif t_0 <= -1e-303:
		tmp = t_0
	elif t_0 <= 0.0:
		tmp = (y + y) / (1.0 - (y / x))
	elif t_0 <= 0.5:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(x * 2.0) * y) / Float64(x - y))
	t_1 = Float64(y * Float64(Float64(x * 2.0) / Float64(x - y)))
	tmp = 0.0
	if (t_0 <= -0.04)
		tmp = t_1;
	elseif (t_0 <= -1e-303)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(y + y) / Float64(1.0 - Float64(y / x)));
	elseif (t_0 <= 0.5)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = ((x * 2.0) * y) / (x - y);
	t_1 = y * ((x * 2.0) / (x - y));
	tmp = 0.0;
	if (t_0 <= -0.04)
		tmp = t_1;
	elseif (t_0 <= -1e-303)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = (y + y) / (1.0 - (y / x));
	elseif (t_0 <= 0.5)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(x * 2.0), $MachinePrecision] * y), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y * N[(N[(x * 2.0), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.04], t$95$1, If[LessEqual[t$95$0, -1e-303], t$95$0, If[LessEqual[t$95$0, 0.0], N[(N[(y + y), $MachinePrecision] / N[(1.0 - N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.5], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (*.f64 x #s(literal 2 binary64)) y) (-.f64 x y)) < -0.0400000000000000008 or 0.5 < (/.f64 (*.f64 (*.f64 x #s(literal 2 binary64)) y) (-.f64 x y))

   1. Initial program 61.9%

      \[\frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(x \cdot 2\right) \cdot y}{x - y}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot 2\right) \cdot y}}{x - y} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(x \cdot 2\right)}}{x - y} \]

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   if -0.0400000000000000008 < (/.f64 (*.f64 (*.f64 x #s(literal 2 binary64)) y) (-.f64 x y)) < -9.99999999999999931e-304 or -0.0 < (/.f64 (*.f64 (*.f64 x #s(literal 2 binary64)) y) (-.f64 x y)) < 0.5

   1. Initial program 99.0%

      \[\frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
   2. Add Preprocessing

   if -9.99999999999999931e-304 < (/.f64 (*.f64 (*.f64 x #s(literal 2 binary64)) y) (-.f64 x y)) < -0.0

   1. Initial program 12.6%

      \[\frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(x \cdot 2\right) \cdot y}{x - y}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot 2\right) \cdot y}}{x - y} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(x \cdot 2\right)}}{x - y} \]

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. div-inv N/A

         \[\leadsto \frac{x \cdot 2}{\color{blue}{\left(x \cdot x - y \cdot y\right) \cdot \frac{1}{x + y}}} \cdot y \]

      4. difference-of-squares N/A

         \[\leadsto \frac{x \cdot 2}{\color{blue}{\left(\left(x + y\right) \cdot \left(x - y\right)\right)} \cdot \frac{1}{x + y}} \cdot y \]

      5. lift--.f64 N/A

         \[\leadsto \frac{x \cdot 2}{\left(\left(x + y\right) \cdot \color{blue}{\left(x - y\right)}\right) \cdot \frac{1}{x + y}} \cdot y \]

      6. associate-*l* N/A

         \[\leadsto \frac{x \cdot 2}{\color{blue}{\left(x + y\right) \cdot \left(\left(x - y\right) \cdot \frac{1}{x + y}\right)}} \cdot y \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{x \cdot 2}{\color{blue}{\left(x + y\right) \cdot \left(\left(x - y\right) \cdot \frac{1}{x + y}\right)}} \cdot y \]

      8. lower-+.f64 N/A

         \[\leadsto \frac{x \cdot 2}{\color{blue}{\left(x + y\right)} \cdot \left(\left(x - y\right) \cdot \frac{1}{x + y}\right)} \cdot y \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{x \cdot 2}{\left(x + y\right) \cdot \color{blue}{\left(\left(x - y\right) \cdot \frac{1}{x + y}\right)}} \cdot y \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{x \cdot 2}{\left(x + y\right) \cdot \left(\left(x - y\right) \cdot \color{blue}{\frac{1}{x + y}}\right)} \cdot y \]

      11. lower-+.f64 99.7

          \[\leadsto \frac{x \cdot 2}{\left(x + y\right) \cdot \left(\left(x - y\right) \cdot \frac{1}{\color{blue}{x + y}}\right)} \cdot y \]

   6. Applied rewrites 99.7%

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

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot 2}{\left(x + y\right) \cdot \left(\left(x - y\right) \cdot \frac{1}{x + y}\right)} \cdot y} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot 2}{\left(x + y\right) \cdot \left(\left(x - y\right) \cdot \frac{1}{x + y}\right)}} \cdot y \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\left(x \cdot 2\right) \cdot y}{\left(x + y\right) \cdot \left(\left(x - y\right) \cdot \frac{1}{x + y}\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot 2\right)} \cdot y}{\left(x + y\right) \cdot \left(\left(x - y\right) \cdot \frac{1}{x + y}\right)} \]

      5. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{x \cdot \left(2 \cdot y\right)}}{\left(x + y\right) \cdot \left(\left(x - y\right) \cdot \frac{1}{x + y}\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(2 \cdot y\right)}}{\left(x + y\right) \cdot \left(\left(x - y\right) \cdot \frac{1}{x + y}\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(2 \cdot y\right) \cdot x}}{\left(x + y\right) \cdot \left(\left(x - y\right) \cdot \frac{1}{x + y}\right)} \]

      8. associate-/l* N/A

         \[\leadsto \color{blue}{\left(2 \cdot y\right) \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x - y\right) \cdot \frac{1}{x + y}\right)}} \]

      9. lift-*.f64 N/A

         \[\leadsto \left(2 \cdot y\right) \cdot \frac{x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x - y\right) \cdot \frac{1}{x + y}\right)}} \]

      10. lift-*.f64 N/A

          \[\leadsto \left(2 \cdot y\right) \cdot \frac{x}{\left(x + y\right) \cdot \color{blue}{\left(\left(x - y\right) \cdot \frac{1}{x + y}\right)}} \]

      11. *-commutative N/A

          \[\leadsto \left(2 \cdot y\right) \cdot \frac{x}{\left(x + y\right) \cdot \color{blue}{\left(\frac{1}{x + y} \cdot \left(x - y\right)\right)}} \]

      12. associate-*r* N/A

          \[\leadsto \left(2 \cdot y\right) \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) \cdot \frac{1}{x + y}\right) \cdot \left(x - y\right)}} \]

      13. lift-/.f64 N/A

          \[\leadsto \left(2 \cdot y\right) \cdot \frac{x}{\left(\left(x + y\right) \cdot \color{blue}{\frac{1}{x + y}}\right) \cdot \left(x - y\right)} \]

      14. rgt-mult-inverse N/A

          \[\leadsto \left(2 \cdot y\right) \cdot \frac{x}{\color{blue}{1} \cdot \left(x - y\right)} \]

      15. *-lft-identity N/A

          \[\leadsto \left(2 \cdot y\right) \cdot \frac{x}{\color{blue}{x - y}} \]

      16. clear-num N/A

          \[\leadsto \left(2 \cdot y\right) \cdot \color{blue}{\frac{1}{\frac{x - y}{x}}} \]

   8. Applied rewrites 99.9%

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

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{y \cdot 2}}{1 - \frac{y}{x}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{2 \cdot y}}{1 - \frac{y}{x}} \]

      3. count-2 N/A

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

      4. lift-+.f64 99.9

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

   10. Applied rewrites 99.9%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot 2\right) \cdot y}{x - y} \leq -0.04:\\ \;\;\;\;y \cdot \frac{x \cdot 2}{x - y}\\ \mathbf{elif}\;\frac{\left(x \cdot 2\right) \cdot y}{x - y} \leq -1 \cdot 10^{-303}:\\ \;\;\;\;\frac{\left(x \cdot 2\right) \cdot y}{x - y}\\ \mathbf{elif}\;\frac{\left(x \cdot 2\right) \cdot y}{x - y} \leq 0:\\ \;\;\;\;\frac{y + y}{1 - \frac{y}{x}}\\ \mathbf{elif}\;\frac{\left(x \cdot 2\right) \cdot y}{x - y} \leq 0.5:\\ \;\;\;\;\frac{\left(x \cdot 2\right) \cdot y}{x - y}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x \cdot 2}{x - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(x \cdot 2\right) \cdot y}{x - y}\\ t_1 := y \cdot \frac{x \cdot 2}{x - y}\\ \mathbf{if}\;t\_0 \leq -0.04:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq -1 \cdot 10^{-303}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0.5:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (* x 2.0) y) (- x y))) (t_1 (* y (/ (* x 2.0) (- x y)))))
   (if (<= t_0 -0.04)
     t_1
     (if (<= t_0 -1e-303)
       t_0
       (if (<= t_0 0.0) t_1 (if (<= t_0 0.5) t_0 t_1))))))

C

double code(double x, double y) {
	double t_0 = ((x * 2.0) * y) / (x - y);
	double t_1 = y * ((x * 2.0) / (x - y));
	double tmp;
	if (t_0 <= -0.04) {
		tmp = t_1;
	} else if (t_0 <= -1e-303) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 0.5) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((x * 2.0d0) * y) / (x - y)
    t_1 = y * ((x * 2.0d0) / (x - y))
    if (t_0 <= (-0.04d0)) then
        tmp = t_1
    else if (t_0 <= (-1d-303)) then
        tmp = t_0
    else if (t_0 <= 0.0d0) then
        tmp = t_1
    else if (t_0 <= 0.5d0) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = ((x * 2.0) * y) / (x - y);
	double t_1 = y * ((x * 2.0) / (x - y));
	double tmp;
	if (t_0 <= -0.04) {
		tmp = t_1;
	} else if (t_0 <= -1e-303) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 0.5) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = ((x * 2.0) * y) / (x - y)
	t_1 = y * ((x * 2.0) / (x - y))
	tmp = 0
	if t_0 <= -0.04:
		tmp = t_1
	elif t_0 <= -1e-303:
		tmp = t_0
	elif t_0 <= 0.0:
		tmp = t_1
	elif t_0 <= 0.5:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(x * 2.0) * y) / Float64(x - y))
	t_1 = Float64(y * Float64(Float64(x * 2.0) / Float64(x - y)))
	tmp = 0.0
	if (t_0 <= -0.04)
		tmp = t_1;
	elseif (t_0 <= -1e-303)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 0.5)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = ((x * 2.0) * y) / (x - y);
	t_1 = y * ((x * 2.0) / (x - y));
	tmp = 0.0;
	if (t_0 <= -0.04)
		tmp = t_1;
	elseif (t_0 <= -1e-303)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 0.5)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(x * 2.0), $MachinePrecision] * y), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y * N[(N[(x * 2.0), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.04], t$95$1, If[LessEqual[t$95$0, -1e-303], t$95$0, If[LessEqual[t$95$0, 0.0], t$95$1, If[LessEqual[t$95$0, 0.5], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (*.f64 x #s(literal 2 binary64)) y) (-.f64 x y)) < -0.0400000000000000008 or -9.99999999999999931e-304 < (/.f64 (*.f64 (*.f64 x #s(literal 2 binary64)) y) (-.f64 x y)) < -0.0 or 0.5 < (/.f64 (*.f64 (*.f64 x #s(literal 2 binary64)) y) (-.f64 x y))

   1. Initial program 46.7%

      \[\frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(x \cdot 2\right) \cdot y}{x - y}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot 2\right) \cdot y}}{x - y} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(x \cdot 2\right)}}{x - y} \]

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   if -0.0400000000000000008 < (/.f64 (*.f64 (*.f64 x #s(literal 2 binary64)) y) (-.f64 x y)) < -9.99999999999999931e-304 or -0.0 < (/.f64 (*.f64 (*.f64 x #s(literal 2 binary64)) y) (-.f64 x y)) < 0.5

   1. Initial program 99.0%

      \[\frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot 2\right) \cdot y}{x - y} \leq -0.04:\\ \;\;\;\;y \cdot \frac{x \cdot 2}{x - y}\\ \mathbf{elif}\;\frac{\left(x \cdot 2\right) \cdot y}{x - y} \leq -1 \cdot 10^{-303}:\\ \;\;\;\;\frac{\left(x \cdot 2\right) \cdot y}{x - y}\\ \mathbf{elif}\;\frac{\left(x \cdot 2\right) \cdot y}{x - y} \leq 0:\\ \;\;\;\;y \cdot \frac{x \cdot 2}{x - y}\\ \mathbf{elif}\;\frac{\left(x \cdot 2\right) \cdot y}{x - y} \leq 0.5:\\ \;\;\;\;\frac{\left(x \cdot 2\right) \cdot y}{x - y}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x \cdot 2}{x - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \frac{x \cdot 2}{x - y}\\ \mathbf{if}\;x \leq -1.15 \cdot 10^{-174}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-186}:\\ \;\;\;\;-2 \cdot \mathsf{fma}\left(x, \frac{x}{y}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (/ (* x 2.0) (- x y)))))
   (if (<= x -1.15e-174)
     t_0
     (if (<= x 1.25e-186) (* -2.0 (fma x (/ x y) x)) t_0))))

C

double code(double x, double y) {
	double t_0 = y * ((x * 2.0) / (x - y));
	double tmp;
	if (x <= -1.15e-174) {
		tmp = t_0;
	} else if (x <= 1.25e-186) {
		tmp = -2.0 * fma(x, (x / y), x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(y * Float64(Float64(x * 2.0) / Float64(x - y)))
	tmp = 0.0
	if (x <= -1.15e-174)
		tmp = t_0;
	elseif (x <= 1.25e-186)
		tmp = Float64(-2.0 * fma(x, Float64(x / y), x));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(N[(x * 2.0), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.15e-174], t$95$0, If[LessEqual[x, 1.25e-186], N[(-2.0 * N[(x * N[(x / y), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.1499999999999999e-174 or 1.25e-186 < x

   1. Initial program 81.0%

      \[\frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(x \cdot 2\right) \cdot y}{x - y}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot 2\right) \cdot y}}{x - y} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(x \cdot 2\right)}}{x - y} \]

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 96.4

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

   4. Applied rewrites 96.4%

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

   if -1.1499999999999999e-174 < x < 1.25e-186

   1. Initial program 75.0%

      \[\frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. sub-neg N/A

         \[\leadsto x \cdot \color{blue}{\left(-2 \cdot \frac{x}{y} + \left(\mathsf{neg}\left(2\right)\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto x \cdot \left(-2 \cdot \frac{x}{y} + \color{blue}{-2}\right) \]

      3. distribute-lft-in N/A

         \[\leadsto \color{blue}{x \cdot \left(-2 \cdot \frac{x}{y}\right) + x \cdot -2} \]

      4. *-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(\frac{x}{y} \cdot -2\right)} + x \cdot -2 \]

      5. associate-*r* N/A

         \[\leadsto \color{blue}{\left(x \cdot \frac{x}{y}\right) \cdot -2} + x \cdot -2 \]

      6. associate-/l* N/A

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

      7. unpow2 N/A

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

      8. distribute-rgt-out N/A

         \[\leadsto \color{blue}{-2 \cdot \left(\frac{{x}^{2}}{y} + x\right)} \]

      9. +-commutative N/A

         \[\leadsto -2 \cdot \color{blue}{\left(x + \frac{{x}^{2}}{y}\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{-2 \cdot \left(x + \frac{{x}^{2}}{y}\right)} \]

      11. +-commutative N/A

          \[\leadsto -2 \cdot \color{blue}{\left(\frac{{x}^{2}}{y} + x\right)} \]

      12. unpow2 N/A

          \[\leadsto -2 \cdot \left(\frac{\color{blue}{x \cdot x}}{y} + x\right) \]

      13. associate-/l* N/A

          \[\leadsto -2 \cdot \left(\color{blue}{x \cdot \frac{x}{y}} + x\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto -2 \cdot \color{blue}{\mathsf{fma}\left(x, \frac{x}{y}, x\right)} \]

      15. lower-/.f64 93.4

          \[\leadsto -2 \cdot \mathsf{fma}\left(x, \color{blue}{\frac{x}{y}}, x\right) \]

   5. Applied rewrites 93.4%

      \[\leadsto \color{blue}{-2 \cdot \mathsf{fma}\left(x, \frac{x}{y}, x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.15 \cdot 10^{-174}:\\ \;\;\;\;y \cdot \frac{x \cdot 2}{x - y}\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-186}:\\ \;\;\;\;-2 \cdot \mathsf{fma}\left(x, \frac{x}{y}, x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x \cdot 2}{x - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.1 \cdot 10^{+56}:\\ \;\;\;\;y + y\\ \mathbf{elif}\;x \leq 1.72 \cdot 10^{-121}:\\ \;\;\;\;-2 \cdot \mathsf{fma}\left(x, \frac{x}{y}, x\right)\\ \mathbf{else}:\\ \;\;\;\;y + \mathsf{fma}\left(\frac{y}{x}, y + y, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -5.1e+56)
   (+ y y)
   (if (<= x 1.72e-121)
     (* -2.0 (fma x (/ x y) x))
     (+ y (fma (/ y x) (+ y y) y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -5.1e+56) {
		tmp = y + y;
	} else if (x <= 1.72e-121) {
		tmp = -2.0 * fma(x, (x / y), x);
	} else {
		tmp = y + fma((y / x), (y + y), y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -5.1e+56)
		tmp = Float64(y + y);
	elseif (x <= 1.72e-121)
		tmp = Float64(-2.0 * fma(x, Float64(x / y), x));
	else
		tmp = Float64(y + fma(Float64(y / x), Float64(y + y), y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -5.1e+56], N[(y + y), $MachinePrecision], If[LessEqual[x, 1.72e-121], N[(-2.0 * N[(x * N[(x / y), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[(y + N[(N[(y / x), $MachinePrecision] * N[(y + y), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.1000000000000002e56

   1. Initial program 73.6%

      \[\frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(x \cdot 2\right) \cdot y}{x - y}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot 2\right) \cdot y}}{x - y} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(x \cdot 2\right)}}{x - y} \]

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\left(x \cdot 2\right) \cdot y}{x - y}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot 2\right)} \cdot y}{x - y} \]

      5. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x - y}{\left(x \cdot 2\right) \cdot y}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x - y}{\left(x \cdot 2\right) \cdot y}}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{x - y}{\left(x \cdot 2\right) \cdot y}}} \]

      8. associate-*l* N/A

         \[\leadsto \frac{1}{\frac{x - y}{\color{blue}{x \cdot \left(2 \cdot y\right)}}} \]

      9. *-commutative N/A

         \[\leadsto \frac{1}{\frac{x - y}{x \cdot \color{blue}{\left(y \cdot 2\right)}}} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{1}{\frac{x - y}{x \cdot \color{blue}{\left(y \cdot 2\right)}}} \]

      11. lower-*.f64 73.4

          \[\leadsto \frac{1}{\frac{x - y}{\color{blue}{x \cdot \left(y \cdot 2\right)}}} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{1}{\frac{x - y}{x \cdot \color{blue}{\left(y \cdot 2\right)}}} \]

      13. *-commutative N/A

          \[\leadsto \frac{1}{\frac{x - y}{x \cdot \color{blue}{\left(2 \cdot y\right)}}} \]

      14. lower-*.f64 73.4

          \[\leadsto \frac{1}{\frac{x - y}{x \cdot \color{blue}{\left(2 \cdot y\right)}}} \]

   6. Applied rewrites 73.4%

      \[\leadsto \color{blue}{\frac{1}{\frac{x - y}{x \cdot \left(2 \cdot y\right)}}} \]

   7. Taylor expanded in x around inf

      \[\leadsto \color{blue}{2 \cdot y} \]
   8. Step-by-step derivation

      1. count-2 N/A

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

      2. lower-+.f64 80.1

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

   9. Applied rewrites 80.1%

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

   if -5.1000000000000002e56 < x < 1.72000000000000007e-121

   1. Initial program 81.3%

      \[\frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. sub-neg N/A

         \[\leadsto x \cdot \color{blue}{\left(-2 \cdot \frac{x}{y} + \left(\mathsf{neg}\left(2\right)\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto x \cdot \left(-2 \cdot \frac{x}{y} + \color{blue}{-2}\right) \]

      3. distribute-lft-in N/A

         \[\leadsto \color{blue}{x \cdot \left(-2 \cdot \frac{x}{y}\right) + x \cdot -2} \]

      4. *-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(\frac{x}{y} \cdot -2\right)} + x \cdot -2 \]

      5. associate-*r* N/A

         \[\leadsto \color{blue}{\left(x \cdot \frac{x}{y}\right) \cdot -2} + x \cdot -2 \]

      6. associate-/l* N/A

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

      7. unpow2 N/A

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

      8. distribute-rgt-out N/A

         \[\leadsto \color{blue}{-2 \cdot \left(\frac{{x}^{2}}{y} + x\right)} \]

      9. +-commutative N/A

         \[\leadsto -2 \cdot \color{blue}{\left(x + \frac{{x}^{2}}{y}\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{-2 \cdot \left(x + \frac{{x}^{2}}{y}\right)} \]

      11. +-commutative N/A

          \[\leadsto -2 \cdot \color{blue}{\left(\frac{{x}^{2}}{y} + x\right)} \]

      12. unpow2 N/A

          \[\leadsto -2 \cdot \left(\frac{\color{blue}{x \cdot x}}{y} + x\right) \]

      13. associate-/l* N/A

          \[\leadsto -2 \cdot \left(\color{blue}{x \cdot \frac{x}{y}} + x\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto -2 \cdot \color{blue}{\mathsf{fma}\left(x, \frac{x}{y}, x\right)} \]

      15. lower-/.f64 79.9

          \[\leadsto -2 \cdot \mathsf{fma}\left(x, \color{blue}{\frac{x}{y}}, x\right) \]

   5. Applied rewrites 79.9%

      \[\leadsto \color{blue}{-2 \cdot \mathsf{fma}\left(x, \frac{x}{y}, x\right)} \]

   if 1.72000000000000007e-121 < x

   1. Initial program 82.1%

      \[\frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(x \cdot 2\right) \cdot y}{x - y}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot 2\right) \cdot y}}{x - y} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(x \cdot 2\right)}}{x - y} \]

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\left(x \cdot 2\right) \cdot y}{x - y}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot 2\right)} \cdot y}{x - y} \]

      5. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x - y}{\left(x \cdot 2\right) \cdot y}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x - y}{\left(x \cdot 2\right) \cdot y}}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{x - y}{\left(x \cdot 2\right) \cdot y}}} \]

      8. associate-*l* N/A

         \[\leadsto \frac{1}{\frac{x - y}{\color{blue}{x \cdot \left(2 \cdot y\right)}}} \]

      9. *-commutative N/A

         \[\leadsto \frac{1}{\frac{x - y}{x \cdot \color{blue}{\left(y \cdot 2\right)}}} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{1}{\frac{x - y}{x \cdot \color{blue}{\left(y \cdot 2\right)}}} \]

      11. lower-*.f64 81.8

          \[\leadsto \frac{1}{\frac{x - y}{\color{blue}{x \cdot \left(y \cdot 2\right)}}} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{1}{\frac{x - y}{x \cdot \color{blue}{\left(y \cdot 2\right)}}} \]

      13. *-commutative N/A

          \[\leadsto \frac{1}{\frac{x - y}{x \cdot \color{blue}{\left(2 \cdot y\right)}}} \]

      14. lower-*.f64 81.8

          \[\leadsto \frac{1}{\frac{x - y}{x \cdot \color{blue}{\left(2 \cdot y\right)}}} \]

   6. Applied rewrites 81.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{x - y}{x \cdot \left(2 \cdot y\right)}}} \]

   7. Taylor expanded in x around inf

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

      1. distribute-lft-out N/A

         \[\leadsto \color{blue}{2 \cdot \left(y + \frac{{y}^{2}}{x}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{2 \cdot \left(y + \frac{{y}^{2}}{x}\right)} \]

      3. +-commutative N/A

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{{y}^{2}}{x} + y\right)} \]

      4. unpow2 N/A

         \[\leadsto 2 \cdot \left(\frac{\color{blue}{y \cdot y}}{x} + y\right) \]

      5. associate-/l* N/A

         \[\leadsto 2 \cdot \left(\color{blue}{y \cdot \frac{y}{x}} + y\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\mathsf{fma}\left(y, \frac{y}{x}, y\right)} \]

      7. lower-/.f64 76.2

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, \color{blue}{\frac{y}{x}}, y\right) \]

   9. Applied rewrites 76.2%

      \[\leadsto \color{blue}{2 \cdot \mathsf{fma}\left(y, \frac{y}{x}, y\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 76.2%

       \[\leadsto \mathsf{fma}\left(\frac{y}{x}, y + y, y\right) + \color{blue}{y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.1 \cdot 10^{+56}:\\ \;\;\;\;y + y\\ \mathbf{elif}\;x \leq 1.72 \cdot 10^{-121}:\\ \;\;\;\;-2 \cdot \mathsf{fma}\left(x, \frac{x}{y}, x\right)\\ \mathbf{else}:\\ \;\;\;\;y + \mathsf{fma}\left(\frac{y}{x}, y + y, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.1 \cdot 10^{+56}:\\ \;\;\;\;y + y\\ \mathbf{elif}\;x \leq 1.72 \cdot 10^{-121}:\\ \;\;\;\;-2 \cdot \mathsf{fma}\left(x, \frac{x}{y}, x\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(y, \frac{y}{x}, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -5.1e+56)
   (+ y y)
   (if (<= x 1.72e-121) (* -2.0 (fma x (/ x y) x)) (* 2.0 (fma y (/ y x) y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -5.1e+56) {
		tmp = y + y;
	} else if (x <= 1.72e-121) {
		tmp = -2.0 * fma(x, (x / y), x);
	} else {
		tmp = 2.0 * fma(y, (y / x), y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -5.1e+56)
		tmp = Float64(y + y);
	elseif (x <= 1.72e-121)
		tmp = Float64(-2.0 * fma(x, Float64(x / y), x));
	else
		tmp = Float64(2.0 * fma(y, Float64(y / x), y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -5.1e+56], N[(y + y), $MachinePrecision], If[LessEqual[x, 1.72e-121], N[(-2.0 * N[(x * N[(x / y), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(y * N[(y / x), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.1000000000000002e56

   1. Initial program 73.6%

      \[\frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(x \cdot 2\right) \cdot y}{x - y}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot 2\right) \cdot y}}{x - y} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(x \cdot 2\right)}}{x - y} \]

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\left(x \cdot 2\right) \cdot y}{x - y}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot 2\right)} \cdot y}{x - y} \]

      5. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x - y}{\left(x \cdot 2\right) \cdot y}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x - y}{\left(x \cdot 2\right) \cdot y}}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{x - y}{\left(x \cdot 2\right) \cdot y}}} \]

      8. associate-*l* N/A

         \[\leadsto \frac{1}{\frac{x - y}{\color{blue}{x \cdot \left(2 \cdot y\right)}}} \]

      9. *-commutative N/A

         \[\leadsto \frac{1}{\frac{x - y}{x \cdot \color{blue}{\left(y \cdot 2\right)}}} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{1}{\frac{x - y}{x \cdot \color{blue}{\left(y \cdot 2\right)}}} \]

      11. lower-*.f64 73.4

          \[\leadsto \frac{1}{\frac{x - y}{\color{blue}{x \cdot \left(y \cdot 2\right)}}} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{1}{\frac{x - y}{x \cdot \color{blue}{\left(y \cdot 2\right)}}} \]

      13. *-commutative N/A

          \[\leadsto \frac{1}{\frac{x - y}{x \cdot \color{blue}{\left(2 \cdot y\right)}}} \]

      14. lower-*.f64 73.4

          \[\leadsto \frac{1}{\frac{x - y}{x \cdot \color{blue}{\left(2 \cdot y\right)}}} \]

   6. Applied rewrites 73.4%

      \[\leadsto \color{blue}{\frac{1}{\frac{x - y}{x \cdot \left(2 \cdot y\right)}}} \]

   7. Taylor expanded in x around inf

      \[\leadsto \color{blue}{2 \cdot y} \]
   8. Step-by-step derivation

      1. count-2 N/A

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

      2. lower-+.f64 80.1

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

   9. Applied rewrites 80.1%

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

   if -5.1000000000000002e56 < x < 1.72000000000000007e-121

   1. Initial program 81.3%

      \[\frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. sub-neg N/A

         \[\leadsto x \cdot \color{blue}{\left(-2 \cdot \frac{x}{y} + \left(\mathsf{neg}\left(2\right)\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto x \cdot \left(-2 \cdot \frac{x}{y} + \color{blue}{-2}\right) \]

      3. distribute-lft-in N/A

         \[\leadsto \color{blue}{x \cdot \left(-2 \cdot \frac{x}{y}\right) + x \cdot -2} \]

      4. *-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(\frac{x}{y} \cdot -2\right)} + x \cdot -2 \]

      5. associate-*r* N/A

         \[\leadsto \color{blue}{\left(x \cdot \frac{x}{y}\right) \cdot -2} + x \cdot -2 \]

      6. associate-/l* N/A

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

      7. unpow2 N/A

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

      8. distribute-rgt-out N/A

         \[\leadsto \color{blue}{-2 \cdot \left(\frac{{x}^{2}}{y} + x\right)} \]

      9. +-commutative N/A

         \[\leadsto -2 \cdot \color{blue}{\left(x + \frac{{x}^{2}}{y}\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{-2 \cdot \left(x + \frac{{x}^{2}}{y}\right)} \]

      11. +-commutative N/A

          \[\leadsto -2 \cdot \color{blue}{\left(\frac{{x}^{2}}{y} + x\right)} \]

      12. unpow2 N/A

          \[\leadsto -2 \cdot \left(\frac{\color{blue}{x \cdot x}}{y} + x\right) \]

      13. associate-/l* N/A

          \[\leadsto -2 \cdot \left(\color{blue}{x \cdot \frac{x}{y}} + x\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto -2 \cdot \color{blue}{\mathsf{fma}\left(x, \frac{x}{y}, x\right)} \]

      15. lower-/.f64 79.9

          \[\leadsto -2 \cdot \mathsf{fma}\left(x, \color{blue}{\frac{x}{y}}, x\right) \]

   5. Applied rewrites 79.9%

      \[\leadsto \color{blue}{-2 \cdot \mathsf{fma}\left(x, \frac{x}{y}, x\right)} \]

   if 1.72000000000000007e-121 < x

   1. Initial program 82.1%

      \[\frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(x \cdot 2\right) \cdot y}{x - y}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot 2\right) \cdot y}}{x - y} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(x \cdot 2\right)}}{x - y} \]

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\left(x \cdot 2\right) \cdot y}{x - y}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot 2\right)} \cdot y}{x - y} \]

      5. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x - y}{\left(x \cdot 2\right) \cdot y}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x - y}{\left(x \cdot 2\right) \cdot y}}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{x - y}{\left(x \cdot 2\right) \cdot y}}} \]

      8. associate-*l* N/A

         \[\leadsto \frac{1}{\frac{x - y}{\color{blue}{x \cdot \left(2 \cdot y\right)}}} \]

      9. *-commutative N/A

         \[\leadsto \frac{1}{\frac{x - y}{x \cdot \color{blue}{\left(y \cdot 2\right)}}} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{1}{\frac{x - y}{x \cdot \color{blue}{\left(y \cdot 2\right)}}} \]

      11. lower-*.f64 81.8

          \[\leadsto \frac{1}{\frac{x - y}{\color{blue}{x \cdot \left(y \cdot 2\right)}}} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{1}{\frac{x - y}{x \cdot \color{blue}{\left(y \cdot 2\right)}}} \]

      13. *-commutative N/A

          \[\leadsto \frac{1}{\frac{x - y}{x \cdot \color{blue}{\left(2 \cdot y\right)}}} \]

      14. lower-*.f64 81.8

          \[\leadsto \frac{1}{\frac{x - y}{x \cdot \color{blue}{\left(2 \cdot y\right)}}} \]

   6. Applied rewrites 81.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{x - y}{x \cdot \left(2 \cdot y\right)}}} \]

   7. Taylor expanded in x around inf

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

      1. distribute-lft-out N/A

         \[\leadsto \color{blue}{2 \cdot \left(y + \frac{{y}^{2}}{x}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{2 \cdot \left(y + \frac{{y}^{2}}{x}\right)} \]

      3. +-commutative N/A

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{{y}^{2}}{x} + y\right)} \]

      4. unpow2 N/A

         \[\leadsto 2 \cdot \left(\frac{\color{blue}{y \cdot y}}{x} + y\right) \]

      5. associate-/l* N/A

         \[\leadsto 2 \cdot \left(\color{blue}{y \cdot \frac{y}{x}} + y\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\mathsf{fma}\left(y, \frac{y}{x}, y\right)} \]

      7. lower-/.f64 76.2

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, \color{blue}{\frac{y}{x}}, y\right) \]

   9. Applied rewrites 76.2%

      \[\leadsto \color{blue}{2 \cdot \mathsf{fma}\left(y, \frac{y}{x}, y\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 72.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.1 \cdot 10^{+56}:\\ \;\;\;\;y + y\\ \mathbf{elif}\;x \leq 1.72 \cdot 10^{-121}:\\ \;\;\;\;x \cdot -2\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(y, \frac{y}{x}, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -5.1e+56)
   (+ y y)
   (if (<= x 1.72e-121) (* x -2.0) (* 2.0 (fma y (/ y x) y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -5.1e+56) {
		tmp = y + y;
	} else if (x <= 1.72e-121) {
		tmp = x * -2.0;
	} else {
		tmp = 2.0 * fma(y, (y / x), y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -5.1e+56)
		tmp = Float64(y + y);
	elseif (x <= 1.72e-121)
		tmp = Float64(x * -2.0);
	else
		tmp = Float64(2.0 * fma(y, Float64(y / x), y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -5.1e+56], N[(y + y), $MachinePrecision], If[LessEqual[x, 1.72e-121], N[(x * -2.0), $MachinePrecision], N[(2.0 * N[(y * N[(y / x), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.1000000000000002e56

   1. Initial program 73.6%

      \[\frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(x \cdot 2\right) \cdot y}{x - y}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot 2\right) \cdot y}}{x - y} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(x \cdot 2\right)}}{x - y} \]

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\left(x \cdot 2\right) \cdot y}{x - y}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot 2\right)} \cdot y}{x - y} \]

      5. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x - y}{\left(x \cdot 2\right) \cdot y}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x - y}{\left(x \cdot 2\right) \cdot y}}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{x - y}{\left(x \cdot 2\right) \cdot y}}} \]

      8. associate-*l* N/A

         \[\leadsto \frac{1}{\frac{x - y}{\color{blue}{x \cdot \left(2 \cdot y\right)}}} \]

      9. *-commutative N/A

         \[\leadsto \frac{1}{\frac{x - y}{x \cdot \color{blue}{\left(y \cdot 2\right)}}} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{1}{\frac{x - y}{x \cdot \color{blue}{\left(y \cdot 2\right)}}} \]

      11. lower-*.f64 73.4

          \[\leadsto \frac{1}{\frac{x - y}{\color{blue}{x \cdot \left(y \cdot 2\right)}}} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{1}{\frac{x - y}{x \cdot \color{blue}{\left(y \cdot 2\right)}}} \]

      13. *-commutative N/A

          \[\leadsto \frac{1}{\frac{x - y}{x \cdot \color{blue}{\left(2 \cdot y\right)}}} \]

      14. lower-*.f64 73.4

          \[\leadsto \frac{1}{\frac{x - y}{x \cdot \color{blue}{\left(2 \cdot y\right)}}} \]

   6. Applied rewrites 73.4%

      \[\leadsto \color{blue}{\frac{1}{\frac{x - y}{x \cdot \left(2 \cdot y\right)}}} \]

   7. Taylor expanded in x around inf

      \[\leadsto \color{blue}{2 \cdot y} \]
   8. Step-by-step derivation

      1. count-2 N/A

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

      2. lower-+.f64 80.1

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

   9. Applied rewrites 80.1%

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

   if -5.1000000000000002e56 < x < 1.72000000000000007e-121

   1. Initial program 81.3%

      \[\frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-2 \cdot x} \]
   4. Step-by-step derivation

      1. lower-*.f64 79.7

         \[\leadsto \color{blue}{-2 \cdot x} \]

   5. Applied rewrites 79.7%

      \[\leadsto \color{blue}{-2 \cdot x} \]

   if 1.72000000000000007e-121 < x

   1. Initial program 82.1%

      \[\frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(x \cdot 2\right) \cdot y}{x - y}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot 2\right) \cdot y}}{x - y} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(x \cdot 2\right)}}{x - y} \]

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\left(x \cdot 2\right) \cdot y}{x - y}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot 2\right)} \cdot y}{x - y} \]

      5. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x - y}{\left(x \cdot 2\right) \cdot y}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x - y}{\left(x \cdot 2\right) \cdot y}}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{x - y}{\left(x \cdot 2\right) \cdot y}}} \]

      8. associate-*l* N/A

         \[\leadsto \frac{1}{\frac{x - y}{\color{blue}{x \cdot \left(2 \cdot y\right)}}} \]

      9. *-commutative N/A

         \[\leadsto \frac{1}{\frac{x - y}{x \cdot \color{blue}{\left(y \cdot 2\right)}}} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{1}{\frac{x - y}{x \cdot \color{blue}{\left(y \cdot 2\right)}}} \]

      11. lower-*.f64 81.8

          \[\leadsto \frac{1}{\frac{x - y}{\color{blue}{x \cdot \left(y \cdot 2\right)}}} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{1}{\frac{x - y}{x \cdot \color{blue}{\left(y \cdot 2\right)}}} \]

      13. *-commutative N/A

          \[\leadsto \frac{1}{\frac{x - y}{x \cdot \color{blue}{\left(2 \cdot y\right)}}} \]

      14. lower-*.f64 81.8

          \[\leadsto \frac{1}{\frac{x - y}{x \cdot \color{blue}{\left(2 \cdot y\right)}}} \]

   6. Applied rewrites 81.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{x - y}{x \cdot \left(2 \cdot y\right)}}} \]

   7. Taylor expanded in x around inf

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

      1. distribute-lft-out N/A

         \[\leadsto \color{blue}{2 \cdot \left(y + \frac{{y}^{2}}{x}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{2 \cdot \left(y + \frac{{y}^{2}}{x}\right)} \]

      3. +-commutative N/A

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{{y}^{2}}{x} + y\right)} \]

      4. unpow2 N/A

         \[\leadsto 2 \cdot \left(\frac{\color{blue}{y \cdot y}}{x} + y\right) \]

      5. associate-/l* N/A

         \[\leadsto 2 \cdot \left(\color{blue}{y \cdot \frac{y}{x}} + y\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\mathsf{fma}\left(y, \frac{y}{x}, y\right)} \]

      7. lower-/.f64 76.2

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, \color{blue}{\frac{y}{x}}, y\right) \]

   9. Applied rewrites 76.2%

      \[\leadsto \color{blue}{2 \cdot \mathsf{fma}\left(y, \frac{y}{x}, y\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.1 \cdot 10^{+56}:\\ \;\;\;\;y + y\\ \mathbf{elif}\;x \leq 1.72 \cdot 10^{-121}:\\ \;\;\;\;x \cdot -2\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(y, \frac{y}{x}, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.1 \cdot 10^{+56}:\\ \;\;\;\;y + y\\ \mathbf{elif}\;x \leq 1.72 \cdot 10^{-121}:\\ \;\;\;\;x \cdot -2\\ \mathbf{else}:\\ \;\;\;\;y + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -5.1e+56) (+ y y) (if (<= x 1.72e-121) (* x -2.0) (+ y y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -5.1e+56) {
		tmp = y + y;
	} else if (x <= 1.72e-121) {
		tmp = x * -2.0;
	} else {
		tmp = y + y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-5.1d+56)) then
        tmp = y + y
    else if (x <= 1.72d-121) then
        tmp = x * (-2.0d0)
    else
        tmp = y + y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -5.1e+56) {
		tmp = y + y;
	} else if (x <= 1.72e-121) {
		tmp = x * -2.0;
	} else {
		tmp = y + y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -5.1e+56:
		tmp = y + y
	elif x <= 1.72e-121:
		tmp = x * -2.0
	else:
		tmp = y + y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -5.1e+56)
		tmp = Float64(y + y);
	elseif (x <= 1.72e-121)
		tmp = Float64(x * -2.0);
	else
		tmp = Float64(y + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -5.1e+56)
		tmp = y + y;
	elseif (x <= 1.72e-121)
		tmp = x * -2.0;
	else
		tmp = y + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -5.1e+56], N[(y + y), $MachinePrecision], If[LessEqual[x, 1.72e-121], N[(x * -2.0), $MachinePrecision], N[(y + y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -5.1000000000000002e56 or 1.72000000000000007e-121 < x

   1. Initial program 78.5%

      \[\frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(x \cdot 2\right) \cdot y}{x - y}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot 2\right) \cdot y}}{x - y} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(x \cdot 2\right)}}{x - y} \]

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\left(x \cdot 2\right) \cdot y}{x - y}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot 2\right)} \cdot y}{x - y} \]

      5. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x - y}{\left(x \cdot 2\right) \cdot y}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x - y}{\left(x \cdot 2\right) \cdot y}}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{x - y}{\left(x \cdot 2\right) \cdot y}}} \]

      8. associate-*l* N/A

         \[\leadsto \frac{1}{\frac{x - y}{\color{blue}{x \cdot \left(2 \cdot y\right)}}} \]

      9. *-commutative N/A

         \[\leadsto \frac{1}{\frac{x - y}{x \cdot \color{blue}{\left(y \cdot 2\right)}}} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{1}{\frac{x - y}{x \cdot \color{blue}{\left(y \cdot 2\right)}}} \]

      11. lower-*.f64 78.3

          \[\leadsto \frac{1}{\frac{x - y}{\color{blue}{x \cdot \left(y \cdot 2\right)}}} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{1}{\frac{x - y}{x \cdot \color{blue}{\left(y \cdot 2\right)}}} \]

      13. *-commutative N/A

          \[\leadsto \frac{1}{\frac{x - y}{x \cdot \color{blue}{\left(2 \cdot y\right)}}} \]

      14. lower-*.f64 78.3

          \[\leadsto \frac{1}{\frac{x - y}{x \cdot \color{blue}{\left(2 \cdot y\right)}}} \]

   6. Applied rewrites 78.3%

      \[\leadsto \color{blue}{\frac{1}{\frac{x - y}{x \cdot \left(2 \cdot y\right)}}} \]

   7. Taylor expanded in x around inf

      \[\leadsto \color{blue}{2 \cdot y} \]
   8. Step-by-step derivation

      1. count-2 N/A

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

      2. lower-+.f64 77.8

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

   9. Applied rewrites 77.8%

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

   if -5.1000000000000002e56 < x < 1.72000000000000007e-121

   1. Initial program 81.3%

      \[\frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-2 \cdot x} \]
   4. Step-by-step derivation

      1. lower-*.f64 79.7

         \[\leadsto \color{blue}{-2 \cdot x} \]

   5. Applied rewrites 79.7%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.1 \cdot 10^{+56}:\\ \;\;\;\;y + y\\ \mathbf{elif}\;x \leq 1.72 \cdot 10^{-121}:\\ \;\;\;\;x \cdot -2\\ \mathbf{else}:\\ \;\;\;\;y + y\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 50.3% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ y + y \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ y y))

C

double code(double x, double y) {
	return y + y;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = y + y
end function

Java

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

Python

def code(x, y):
	return y + y

Julia

function code(x, y)
	return Float64(y + y)
end

MATLAB

function tmp = code(x, y)
	tmp = y + y;
end

Wolfram

code[x_, y_] := N[(y + y), $MachinePrecision]

Derivation

1. Initial program 79.8%

   \[\frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\left(x \cdot 2\right) \cdot y}{x - y}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(x \cdot 2\right) \cdot y}}{x - y} \]

   3. *-commutative N/A

      \[\leadsto \frac{\color{blue}{y \cdot \left(x \cdot 2\right)}}{x - y} \]

   4. associate-/l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 88.7

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

4. Applied rewrites 88.7%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

      \[\leadsto \color{blue}{\frac{\left(x \cdot 2\right) \cdot y}{x - y}} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(x \cdot 2\right)} \cdot y}{x - y} \]

   5. clear-num N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{x - y}{\left(x \cdot 2\right) \cdot y}}} \]

   6. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{x - y}{\left(x \cdot 2\right) \cdot y}}} \]

   7. lower-/.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{x - y}{\left(x \cdot 2\right) \cdot y}}} \]

   8. associate-*l* N/A

      \[\leadsto \frac{1}{\frac{x - y}{\color{blue}{x \cdot \left(2 \cdot y\right)}}} \]

   9. *-commutative N/A

      \[\leadsto \frac{1}{\frac{x - y}{x \cdot \color{blue}{\left(y \cdot 2\right)}}} \]

   10. lift-*.f64 N/A

       \[\leadsto \frac{1}{\frac{x - y}{x \cdot \color{blue}{\left(y \cdot 2\right)}}} \]

   11. lower-*.f64 79.6

       \[\leadsto \frac{1}{\frac{x - y}{\color{blue}{x \cdot \left(y \cdot 2\right)}}} \]

   12. lift-*.f64 N/A

       \[\leadsto \frac{1}{\frac{x - y}{x \cdot \color{blue}{\left(y \cdot 2\right)}}} \]

   13. *-commutative N/A

       \[\leadsto \frac{1}{\frac{x - y}{x \cdot \color{blue}{\left(2 \cdot y\right)}}} \]

   14. lower-*.f64 79.6

       \[\leadsto \frac{1}{\frac{x - y}{x \cdot \color{blue}{\left(2 \cdot y\right)}}} \]

6. Applied rewrites 79.6%

   \[\leadsto \color{blue}{\frac{1}{\frac{x - y}{x \cdot \left(2 \cdot y\right)}}} \]

7. Taylor expanded in x around inf

   \[\leadsto \color{blue}{2 \cdot y} \]
8. Step-by-step derivation

   1. count-2 N/A

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

   2. lower-+.f64 52.2

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

9. Applied rewrites 52.2%

   \[\leadsto \color{blue}{y + y} \]
10. Add Preprocessing

Developer Target 1: 99.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2 \cdot x}{x - y} \cdot y\\ \mathbf{if}\;x < -1.7210442634149447 \cdot 10^{+81}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x < 83645045635564430:\\ \;\;\;\;\frac{x \cdot 2}{\frac{x - y}{y}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (/ (* 2.0 x) (- x y)) y)))
   (if (< x -1.7210442634149447e+81)
     t_0
     (if (< x 83645045635564430.0) (/ (* x 2.0) (/ (- x y) y)) t_0))))

C

double code(double x, double y) {
	double t_0 = ((2.0 * x) / (x - y)) * y;
	double tmp;
	if (x < -1.7210442634149447e+81) {
		tmp = t_0;
	} else if (x < 83645045635564430.0) {
		tmp = (x * 2.0) / ((x - y) / y);
	} else {
		tmp = t_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 = ((2.0d0 * x) / (x - y)) * y
    if (x < (-1.7210442634149447d+81)) then
        tmp = t_0
    else if (x < 83645045635564430.0d0) then
        tmp = (x * 2.0d0) / ((x - y) / y)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = ((2.0 * x) / (x - y)) * y;
	double tmp;
	if (x < -1.7210442634149447e+81) {
		tmp = t_0;
	} else if (x < 83645045635564430.0) {
		tmp = (x * 2.0) / ((x - y) / y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = ((2.0 * x) / (x - y)) * y
	tmp = 0
	if x < -1.7210442634149447e+81:
		tmp = t_0
	elif x < 83645045635564430.0:
		tmp = (x * 2.0) / ((x - y) / y)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(2.0 * x) / Float64(x - y)) * y)
	tmp = 0.0
	if (x < -1.7210442634149447e+81)
		tmp = t_0;
	elseif (x < 83645045635564430.0)
		tmp = Float64(Float64(x * 2.0) / Float64(Float64(x - y) / y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = ((2.0 * x) / (x - y)) * y;
	tmp = 0.0;
	if (x < -1.7210442634149447e+81)
		tmp = t_0;
	elseif (x < 83645045635564430.0)
		tmp = (x * 2.0) / ((x - y) / y);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(2.0 * x), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]}, If[Less[x, -1.7210442634149447e+81], t$95$0, If[Less[x, 83645045635564430.0], N[(N[(x * 2.0), $MachinePrecision] / N[(N[(x - y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Reproduce

herbie shell --seed 2024238 
(FPCore (x y)
  :name "Linear.Projection:perspective from linear-1.19.1.3, B"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< x -1721044263414944700000000000000000000000000000000000000000000000000000000000000000) (* (/ (* 2 x) (- x y)) y) (if (< x 83645045635564430) (/ (* x 2) (/ (- x y) y)) (* (/ (* 2 x) (- x y)) y))))

  (/ (* (* x 2.0) y) (- x y)))