Linear.Projection:perspective from linear-1.19.1.3, B

Percentage Accurate: 77.6% → 93.9%

Time: 6.7s

Alternatives: 6

Speedup: 0.8×

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: 77.6% 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: 93.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(x \cdot 2\right) \cdot y}{x - y}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;x \cdot \frac{2 \cdot y}{x - y}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-70}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2}{\frac{x - y}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (* x 2.0) y) (- x y))))
   (if (<= t_0 0.0)
     (* x (/ (* 2.0 y) (- x y)))
     (if (<= t_0 5e-70) t_0 (/ (* x 2.0) (/ (- x y) y))))))

C

double code(double x, double y) {
	double t_0 = ((x * 2.0) * y) / (x - y);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = x * ((2.0 * y) / (x - y));
	} else if (t_0 <= 5e-70) {
		tmp = t_0;
	} else {
		tmp = (x * 2.0) / ((x - y) / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((x * 2.0d0) * y) / (x - y)
    if (t_0 <= 0.0d0) then
        tmp = x * ((2.0d0 * y) / (x - y))
    else if (t_0 <= 5d-70) then
        tmp = t_0
    else
        tmp = (x * 2.0d0) / ((x - y) / y)
    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 tmp;
	if (t_0 <= 0.0) {
		tmp = x * ((2.0 * y) / (x - y));
	} else if (t_0 <= 5e-70) {
		tmp = t_0;
	} else {
		tmp = (x * 2.0) / ((x - y) / y);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = ((x * 2.0) * y) / (x - y)
	tmp = 0
	if t_0 <= 0.0:
		tmp = x * ((2.0 * y) / (x - y))
	elif t_0 <= 5e-70:
		tmp = t_0
	else:
		tmp = (x * 2.0) / ((x - y) / y)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(x * 2.0) * y) / Float64(x - y))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(x * Float64(Float64(2.0 * y) / Float64(x - y)));
	elseif (t_0 <= 5e-70)
		tmp = t_0;
	else
		tmp = Float64(Float64(x * 2.0) / Float64(Float64(x - y) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = ((x * 2.0) * y) / (x - y);
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = x * ((2.0 * y) / (x - y));
	elseif (t_0 <= 5e-70)
		tmp = t_0;
	else
		tmp = (x * 2.0) / ((x - y) / y);
	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]}, If[LessEqual[t$95$0, 0.0], N[(x * N[(N[(2.0 * y), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e-70], t$95$0, N[(N[(x * 2.0), $MachinePrecision] / N[(N[(x - y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 73.0%

      \[\frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
   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. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. --lowering--.f64 95.8

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

   4. Applied egg-rr 95.8%

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

   if 0.0 < (/.f64 (*.f64 (*.f64 x #s(literal 2 binary64)) y) (-.f64 x y)) < 4.9999999999999998e-70

   1. Initial program 98.4%

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

   if 4.9999999999999998e-70 < (/.f64 (*.f64 (*.f64 x #s(literal 2 binary64)) y) (-.f64 x y))

   1. Initial program 73.8%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. --lowering--.f64 99.9

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

   4. Applied egg-rr 99.9%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot 2\right) \cdot y}{x - y} \leq 0:\\ \;\;\;\;x \cdot \frac{2 \cdot y}{x - y}\\ \mathbf{elif}\;\frac{\left(x \cdot 2\right) \cdot y}{x - y} \leq 5 \cdot 10^{-70}:\\ \;\;\;\;\frac{\left(x \cdot 2\right) \cdot y}{x - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2}{\frac{x - y}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 94.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(x \cdot 2\right) \cdot y}{x - y}\\ t_1 := x \cdot \frac{2 \cdot y}{x - y}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{-65}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (* x 2.0) y) (- x y))) (t_1 (* x (/ (* 2.0 y) (- x y)))))
   (if (<= t_0 0.0) t_1 (if (<= t_0 1e-65) t_0 t_1))))

C

double code(double x, double y) {
	double t_0 = ((x * 2.0) * y) / (x - y);
	double t_1 = x * ((2.0 * y) / (x - y));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 1e-65) {
		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 = x * ((2.0d0 * y) / (x - y))
    if (t_0 <= 0.0d0) then
        tmp = t_1
    else if (t_0 <= 1d-65) 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 = x * ((2.0 * y) / (x - y));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 1e-65) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = ((x * 2.0) * y) / (x - y)
	t_1 = x * ((2.0 * y) / (x - y))
	tmp = 0
	if t_0 <= 0.0:
		tmp = t_1
	elif t_0 <= 1e-65:
		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(x * Float64(Float64(2.0 * y) / Float64(x - y)))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 1e-65)
		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 = x * ((2.0 * y) / (x - y));
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 1e-65)
		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[(x * N[(N[(2.0 * y), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], t$95$1, If[LessEqual[t$95$0, 1e-65], 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)) < 0.0 or 9.99999999999999923e-66 < (/.f64 (*.f64 (*.f64 x #s(literal 2 binary64)) y) (-.f64 x y))

   1. Initial program 73.0%

      \[\frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
   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. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. --lowering--.f64 96.9

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

   4. Applied egg-rr 96.9%

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

   if 0.0 < (/.f64 (*.f64 (*.f64 x #s(literal 2 binary64)) y) (-.f64 x y)) < 9.99999999999999923e-66

   1. Initial program 98.5%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot 2\right) \cdot y}{x - y} \leq 0:\\ \;\;\;\;x \cdot \frac{2 \cdot y}{x - y}\\ \mathbf{elif}\;\frac{\left(x \cdot 2\right) \cdot y}{x - y} \leq 10^{-65}:\\ \;\;\;\;\frac{\left(x \cdot 2\right) \cdot y}{x - y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{2 \cdot y}{x - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.12 \cdot 10^{-39}:\\ \;\;\;\;2 \cdot y\\ \mathbf{elif}\;x \leq 4.9 \cdot 10^{+27}:\\ \;\;\;\;x \cdot -2\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(y, \frac{y}{x}, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.12e-39)
   (* 2.0 y)
   (if (<= x 4.9e+27) (* x -2.0) (* 2.0 (fma y (/ y x) y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.12e-39) {
		tmp = 2.0 * y;
	} else if (x <= 4.9e+27) {
		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 <= -1.12e-39)
		tmp = Float64(2.0 * y);
	elseif (x <= 4.9e+27)
		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, -1.12e-39], N[(2.0 * y), $MachinePrecision], If[LessEqual[x, 4.9e+27], 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 < -1.12e-39

   1. Initial program 76.6%

      \[\frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
   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. *-lowering-*.f64 71.0

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

   5. Simplified 71.0%

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

   if -1.12e-39 < x < 4.90000000000000015e27

   1. Initial program 81.2%

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

   3. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 76.9

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

   5. Simplified 76.9%

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

   if 4.90000000000000015e27 < x

   1. Initial program 78.0%

      \[\frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
   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. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. --lowering--.f64 85.0

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

   4. Applied egg-rr 85.0%

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

   5. Taylor expanded in y around 0

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

      1. distribute-rgt-in N/A

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

      2. associate-*l* N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*l/ N/A

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-/l* N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. /-lowering-/.f64 83.5

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

   7. Simplified 83.5%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.12 \cdot 10^{-39}:\\ \;\;\;\;2 \cdot y\\ \mathbf{elif}\;x \leq 4.9 \cdot 10^{+27}:\\ \;\;\;\;x \cdot -2\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(y, \frac{y}{x}, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.65 \cdot 10^{+197}:\\ \;\;\;\;x \cdot \frac{2 \cdot y}{x - y}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(y, \frac{y}{x}, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.65e+197) (* x (/ (* 2.0 y) (- x y))) (* 2.0 (fma y (/ y x) y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.65e+197) {
		tmp = x * ((2.0 * y) / (x - y));
	} else {
		tmp = 2.0 * fma(y, (y / x), y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.65e+197)
		tmp = Float64(x * Float64(Float64(2.0 * y) / Float64(x - y)));
	else
		tmp = Float64(2.0 * fma(y, Float64(y / x), y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.65e+197], N[(x * N[(N[(2.0 * y), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(y * N[(y / x), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.6499999999999998e197

   1. Initial program 81.4%

      \[\frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
   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. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. --lowering--.f64 95.4

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

   4. Applied egg-rr 95.4%

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

   if 1.6499999999999998e197 < x

   1. Initial program 56.6%

      \[\frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
   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. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. --lowering--.f64 69.8

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

   4. Applied egg-rr 69.8%

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

   5. Taylor expanded in y around 0

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

      1. distribute-rgt-in N/A

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

      2. associate-*l* N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*l/ N/A

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-/l* N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. /-lowering-/.f64 95.5

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

   7. Simplified 95.5%

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

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.65 \cdot 10^{+197}:\\ \;\;\;\;x \cdot \frac{2 \cdot y}{x - y}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(y, \frac{y}{x}, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{-39}:\\ \;\;\;\;2 \cdot y\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{+27}:\\ \;\;\;\;x \cdot -2\\ \mathbf{else}:\\ \;\;\;\;2 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.1e-39) (* 2.0 y) (if (<= x 3.8e+27) (* x -2.0) (* 2.0 y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.1e-39) {
		tmp = 2.0 * y;
	} else if (x <= 3.8e+27) {
		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 <= (-1.1d-39)) then
        tmp = 2.0d0 * y
    else if (x <= 3.8d+27) 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 <= -1.1e-39) {
		tmp = 2.0 * y;
	} else if (x <= 3.8e+27) {
		tmp = x * -2.0;
	} else {
		tmp = 2.0 * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.1e-39:
		tmp = 2.0 * y
	elif x <= 3.8e+27:
		tmp = x * -2.0
	else:
		tmp = 2.0 * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.1e-39)
		tmp = Float64(2.0 * y);
	elseif (x <= 3.8e+27)
		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 <= -1.1e-39)
		tmp = 2.0 * y;
	elseif (x <= 3.8e+27)
		tmp = x * -2.0;
	else
		tmp = 2.0 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.1e-39], N[(2.0 * y), $MachinePrecision], If[LessEqual[x, 3.8e+27], N[(x * -2.0), $MachinePrecision], N[(2.0 * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.1e-39 or 3.80000000000000022e27 < x

   1. Initial program 77.3%

      \[\frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
   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. *-lowering-*.f64 75.8

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

   5. Simplified 75.8%

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

   if -1.1e-39 < x < 3.80000000000000022e27

   1. Initial program 81.2%

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

   3. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 76.9

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

   5. Simplified 76.9%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{-39}:\\ \;\;\;\;2 \cdot y\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{+27}:\\ \;\;\;\;x \cdot -2\\ \mathbf{else}:\\ \;\;\;\;2 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 51.0% 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 79.2%

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

3. Taylor expanded in x around 0

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

   1. *-lowering-*.f64 50.3

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

5. Simplified 50.3%

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

6. Final simplification 50.3%

   \[\leadsto x \cdot -2 \]
7. 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 2024205 
(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)))