Linear.Projection:perspective from linear-1.19.1.3, B

Percentage Accurate: 76.5% → 99.6%

Time: 8.6s

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: 76.5% 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: 99.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \frac{x \cdot 2}{x - y}\\ \mathbf{if}\;x \leq -5 \cdot 10^{-33}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-68}:\\ \;\;\;\;\frac{x \cdot 2}{\frac{x - y}{y}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (/ (* x 2.0) (- x y)))))
   (if (<= x -5e-33) t_0 (if (<= x 1.9e-68) (/ (* x 2.0) (/ (- x y) y)) t_0))))

C

double code(double x, double y) {
	double t_0 = y * ((x * 2.0) / (x - y));
	double tmp;
	if (x <= -5e-33) {
		tmp = t_0;
	} else if (x <= 1.9e-68) {
		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 = y * ((x * 2.0d0) / (x - y))
    if (x <= (-5d-33)) then
        tmp = t_0
    else if (x <= 1.9d-68) 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 = y * ((x * 2.0) / (x - y));
	double tmp;
	if (x <= -5e-33) {
		tmp = t_0;
	} else if (x <= 1.9e-68) {
		tmp = (x * 2.0) / ((x - y) / y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * ((x * 2.0) / (x - y))
	tmp = 0
	if x <= -5e-33:
		tmp = t_0
	elif x <= 1.9e-68:
		tmp = (x * 2.0) / ((x - y) / y)
	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 <= -5e-33)
		tmp = t_0;
	elseif (x <= 1.9e-68)
		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 = y * ((x * 2.0) / (x - y));
	tmp = 0.0;
	if (x <= -5e-33)
		tmp = t_0;
	elseif (x <= 1.9e-68)
		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[(y * N[(N[(x * 2.0), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5e-33], t$95$0, If[LessEqual[x, 1.9e-68], N[(N[(x * 2.0), $MachinePrecision] / N[(N[(x - y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -5.00000000000000028e-33 or 1.90000000000000019e-68 < x

   1. Initial program 80.0%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      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 -5.00000000000000028e-33 < x < 1.90000000000000019e-68

   1. Initial program 76.8%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-33}:\\ \;\;\;\;y \cdot \frac{x \cdot 2}{x - y}\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-68}:\\ \;\;\;\;\frac{x \cdot 2}{\frac{x - y}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x \cdot 2}{x - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{x - y} \cdot \left(x \cdot y\right)\\ \mathbf{if}\;x \leq -6.8 \cdot 10^{+170}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(y, \frac{y}{x}, y\right)\\ \mathbf{elif}\;x \leq -1.05 \cdot 10^{-140}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{-145}:\\ \;\;\;\;x \cdot -2\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+87}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;2 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (/ 2.0 (- x y)) (* x y))))
   (if (<= x -6.8e+170)
     (* 2.0 (fma y (/ y x) y))
     (if (<= x -1.05e-140)
       t_0
       (if (<= x 2.9e-145) (* x -2.0) (if (<= x 7e+87) t_0 (* 2.0 y)))))))

C

double code(double x, double y) {
	double t_0 = (2.0 / (x - y)) * (x * y);
	double tmp;
	if (x <= -6.8e+170) {
		tmp = 2.0 * fma(y, (y / x), y);
	} else if (x <= -1.05e-140) {
		tmp = t_0;
	} else if (x <= 2.9e-145) {
		tmp = x * -2.0;
	} else if (x <= 7e+87) {
		tmp = t_0;
	} else {
		tmp = 2.0 * y;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(2.0 / Float64(x - y)) * Float64(x * y))
	tmp = 0.0
	if (x <= -6.8e+170)
		tmp = Float64(2.0 * fma(y, Float64(y / x), y));
	elseif (x <= -1.05e-140)
		tmp = t_0;
	elseif (x <= 2.9e-145)
		tmp = Float64(x * -2.0);
	elseif (x <= 7e+87)
		tmp = t_0;
	else
		tmp = Float64(2.0 * y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(2.0 / N[(x - y), $MachinePrecision]), $MachinePrecision] * N[(x * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -6.8e+170], N[(2.0 * N[(y * N[(y / x), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.05e-140], t$95$0, If[LessEqual[x, 2.9e-145], N[(x * -2.0), $MachinePrecision], If[LessEqual[x, 7e+87], t$95$0, N[(2.0 * y), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -6.8000000000000003e170

   1. Initial program 69.1%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      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. Taylor expanded in x around inf

      \[\leadsto \color{blue}{2 \cdot y + 2 \cdot \frac{{y}^{2}}{x}} \]
   6. 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 97.9

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

   7. Applied rewrites 97.9%

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

   if -6.8000000000000003e170 < x < -1.05000000000000009e-140 or 2.89999999999999984e-145 < x < 6.99999999999999972e87

   1. Initial program 87.4%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 87.1

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

   4. Applied rewrites 87.1%

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

   if -1.05000000000000009e-140 < x < 2.89999999999999984e-145

   1. Initial program 72.1%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 93.3

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

   5. Applied rewrites 93.3%

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

   if 6.99999999999999972e87 < x

   1. Initial program 73.9%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{y \cdot 2} \]

      2. lower-*.f64 89.5

         \[\leadsto \color{blue}{y \cdot 2} \]

   5. Applied rewrites 89.5%

      \[\leadsto \color{blue}{y \cdot 2} \]
3. Recombined 4 regimes into one program.

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{+170}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(y, \frac{y}{x}, y\right)\\ \mathbf{elif}\;x \leq -1.05 \cdot 10^{-140}:\\ \;\;\;\;\frac{2}{x - y} \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{-145}:\\ \;\;\;\;x \cdot -2\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+87}:\\ \;\;\;\;\frac{2}{x - y} \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \frac{x \cdot 2}{x - y}\\ \mathbf{if}\;x \leq -5 \cdot 10^{+35}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+72}:\\ \;\;\;\;x \cdot \frac{2 \cdot y}{x - y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (/ (* x 2.0) (- x y)))))
   (if (<= x -5e+35) t_0 (if (<= x 5e+72) (* x (/ (* 2.0 y) (- x y))) t_0))))

C

double code(double x, double y) {
	double t_0 = y * ((x * 2.0) / (x - y));
	double tmp;
	if (x <= -5e+35) {
		tmp = t_0;
	} else if (x <= 5e+72) {
		tmp = x * ((2.0 * y) / (x - 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 = y * ((x * 2.0d0) / (x - y))
    if (x <= (-5d+35)) then
        tmp = t_0
    else if (x <= 5d+72) then
        tmp = x * ((2.0d0 * y) / (x - y))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y * ((x * 2.0) / (x - y));
	double tmp;
	if (x <= -5e+35) {
		tmp = t_0;
	} else if (x <= 5e+72) {
		tmp = x * ((2.0 * y) / (x - y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * ((x * 2.0) / (x - y))
	tmp = 0
	if x <= -5e+35:
		tmp = t_0
	elif x <= 5e+72:
		tmp = x * ((2.0 * y) / (x - y))
	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 <= -5e+35)
		tmp = t_0;
	elseif (x <= 5e+72)
		tmp = Float64(x * Float64(Float64(2.0 * y) / Float64(x - y)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * ((x * 2.0) / (x - y));
	tmp = 0.0;
	if (x <= -5e+35)
		tmp = t_0;
	elseif (x <= 5e+72)
		tmp = x * ((2.0 * y) / (x - y));
	else
		tmp = t_0;
	end
	tmp_2 = 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, -5e+35], t$95$0, If[LessEqual[x, 5e+72], N[(x * N[(N[(2.0 * y), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -5.00000000000000021e35 or 4.99999999999999992e72 < x

   1. Initial program 73.1%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      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 -5.00000000000000021e35 < x < 4.99999999999999992e72

   1. Initial program 82.7%

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

      1. associate-*l* N/A

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

      2. lift--.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+35}:\\ \;\;\;\;y \cdot \frac{x \cdot 2}{x - y}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+72}:\\ \;\;\;\;x \cdot \frac{2 \cdot y}{x - y}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x \cdot 2}{x - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{+174}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(y, \frac{y}{x}, y\right)\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+87}:\\ \;\;\;\;x \cdot \frac{2 \cdot y}{x - y}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -8.5e+174)
   (* 2.0 (fma y (/ y x) y))
   (if (<= x 7e+87) (* x (/ (* 2.0 y) (- x y))) (* 2.0 y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -8.5e+174) {
		tmp = 2.0 * fma(y, (y / x), y);
	} else if (x <= 7e+87) {
		tmp = x * ((2.0 * y) / (x - y));
	} else {
		tmp = 2.0 * y;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -8.5e+174)
		tmp = Float64(2.0 * fma(y, Float64(y / x), y));
	elseif (x <= 7e+87)
		tmp = Float64(x * Float64(Float64(2.0 * y) / Float64(x - y)));
	else
		tmp = Float64(2.0 * y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -8.5e+174], N[(2.0 * N[(y * N[(y / x), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 7e+87], N[(x * N[(N[(2.0 * y), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.5000000000000007e174

   1. Initial program 69.1%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      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. Taylor expanded in x around inf

      \[\leadsto \color{blue}{2 \cdot y + 2 \cdot \frac{{y}^{2}}{x}} \]
   6. 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 97.9

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

   7. Applied rewrites 97.9%

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

   if -8.5000000000000007e174 < x < 6.99999999999999972e87

   1. Initial program 81.5%

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

      1. associate-*l* N/A

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

      2. lift--.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 98.7

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

   4. Applied rewrites 98.7%

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

   if 6.99999999999999972e87 < x

   1. Initial program 73.9%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{y \cdot 2} \]

      2. lower-*.f64 89.5

         \[\leadsto \color{blue}{y \cdot 2} \]

   5. Applied rewrites 89.5%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{+174}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(y, \frac{y}{x}, y\right)\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+87}:\\ \;\;\;\;x \cdot \frac{2 \cdot y}{x - y}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{+174}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(y, \frac{y}{x}, y\right)\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+87}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{2}{x - y}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -8.5e+174)
   (* 2.0 (fma y (/ y x) y))
   (if (<= x 7e+87) (* x (* y (/ 2.0 (- x y)))) (* 2.0 y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -8.5e+174) {
		tmp = 2.0 * fma(y, (y / x), y);
	} else if (x <= 7e+87) {
		tmp = x * (y * (2.0 / (x - y)));
	} else {
		tmp = 2.0 * y;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -8.5e+174)
		tmp = Float64(2.0 * fma(y, Float64(y / x), y));
	elseif (x <= 7e+87)
		tmp = Float64(x * Float64(y * Float64(2.0 / Float64(x - y))));
	else
		tmp = Float64(2.0 * y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -8.5e+174], N[(2.0 * N[(y * N[(y / x), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 7e+87], N[(x * N[(y * N[(2.0 / N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.5000000000000007e174

   1. Initial program 69.1%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      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. Taylor expanded in x around inf

      \[\leadsto \color{blue}{2 \cdot y + 2 \cdot \frac{{y}^{2}}{x}} \]
   6. 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 97.9

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

   7. Applied rewrites 97.9%

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

   if -8.5000000000000007e174 < x < 6.99999999999999972e87

   1. Initial program 81.5%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      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 86.3

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

   4. Applied rewrites 86.3%

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

      1. lift--.f64 N/A

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

      2. associate-/l* N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 98.4

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

   6. Applied rewrites 98.4%

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

   if 6.99999999999999972e87 < x

   1. Initial program 73.9%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{y \cdot 2} \]

      2. lower-*.f64 89.5

         \[\leadsto \color{blue}{y \cdot 2} \]

   5. Applied rewrites 89.5%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{+174}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(y, \frac{y}{x}, y\right)\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+87}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{2}{x - y}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{-82}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(y, \frac{y}{x}, y\right)\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-6}:\\ \;\;\;\;x \cdot -2\\ \mathbf{else}:\\ \;\;\;\;2 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.7e-82)
   (* 2.0 (fma y (/ y x) y))
   (if (<= x 2e-6) (* x -2.0) (* 2.0 y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.7e-82) {
		tmp = 2.0 * fma(y, (y / x), y);
	} else if (x <= 2e-6) {
		tmp = x * -2.0;
	} else {
		tmp = 2.0 * y;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.7e-82)
		tmp = Float64(2.0 * fma(y, Float64(y / x), y));
	elseif (x <= 2e-6)
		tmp = Float64(x * -2.0);
	else
		tmp = Float64(2.0 * y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.7e-82], N[(2.0 * N[(y * N[(y / x), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2e-6], N[(x * -2.0), $MachinePrecision], N[(2.0 * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.69999999999999988e-82

   1. Initial program 76.9%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      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. Taylor expanded in x around inf

      \[\leadsto \color{blue}{2 \cdot y + 2 \cdot \frac{{y}^{2}}{x}} \]
   6. 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 75.6

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

   7. Applied rewrites 75.6%

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

   if -1.69999999999999988e-82 < x < 1.99999999999999991e-6

   1. Initial program 78.6%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 82.0

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

   5. Applied rewrites 82.0%

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

   if 1.99999999999999991e-6 < x

   1. Initial program 81.2%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{y \cdot 2} \]

      2. lower-*.f64 83.8

         \[\leadsto \color{blue}{y \cdot 2} \]

   5. Applied rewrites 83.8%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{-82}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(y, \frac{y}{x}, y\right)\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-6}:\\ \;\;\;\;x \cdot -2\\ \mathbf{else}:\\ \;\;\;\;2 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{-82}:\\ \;\;\;\;2 \cdot y\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-6}:\\ \;\;\;\;x \cdot -2\\ \mathbf{else}:\\ \;\;\;\;2 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.7e-82) (* 2.0 y) (if (<= x 2e-6) (* x -2.0) (* 2.0 y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.7e-82) {
		tmp = 2.0 * y;
	} else if (x <= 2e-6) {
		tmp = x * -2.0;
	} else {
		tmp = 2.0 * 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 <= (-1.7d-82)) then
        tmp = 2.0d0 * y
    else if (x <= 2d-6) then
        tmp = x * (-2.0d0)
    else
        tmp = 2.0d0 * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.7e-82) {
		tmp = 2.0 * y;
	} else if (x <= 2e-6) {
		tmp = x * -2.0;
	} else {
		tmp = 2.0 * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.7e-82:
		tmp = 2.0 * y
	elif x <= 2e-6:
		tmp = x * -2.0
	else:
		tmp = 2.0 * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.7e-82)
		tmp = Float64(2.0 * y);
	elseif (x <= 2e-6)
		tmp = Float64(x * -2.0);
	else
		tmp = Float64(2.0 * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.7e-82)
		tmp = 2.0 * y;
	elseif (x <= 2e-6)
		tmp = x * -2.0;
	else
		tmp = 2.0 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.7e-82], N[(2.0 * y), $MachinePrecision], If[LessEqual[x, 2e-6], N[(x * -2.0), $MachinePrecision], N[(2.0 * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.69999999999999988e-82 or 1.99999999999999991e-6 < x

   1. Initial program 78.8%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{y \cdot 2} \]

      2. lower-*.f64 78.7

         \[\leadsto \color{blue}{y \cdot 2} \]

   5. Applied rewrites 78.7%

      \[\leadsto \color{blue}{y \cdot 2} \]

   if -1.69999999999999988e-82 < x < 1.99999999999999991e-6

   1. Initial program 78.6%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 82.0

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

   5. Applied rewrites 82.0%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{-82}:\\ \;\;\;\;2 \cdot y\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-6}:\\ \;\;\;\;x \cdot -2\\ \mathbf{else}:\\ \;\;\;\;2 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 50.4% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ x \cdot -2 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.7%

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

3. Taylor expanded in x around 0

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

   1. lower-*.f64 48.9

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

5. Applied rewrites 48.9%

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

6. Final simplification 48.9%

   \[\leadsto x \cdot -2 \]
7. Add Preprocessing

Developer Target 1: 99.4% 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 2024219 
(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)))