Linear.Projection:perspective from linear-1.19.1.3, B

Percentage Accurate: 76.8% → 99.8%

Time: 5.6s

Alternatives: 7

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: 76.8% 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: 99.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2000 \lor \neg \left(x \leq 1.5 \cdot 10^{-57}\right):\\ \;\;\;\;\left(2 \cdot \frac{x}{x - y}\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{x - y} \cdot x\right) \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -2000.0) (not (<= x 1.5e-57)))
   (* (* 2.0 (/ x (- x y))) y)
   (* (* (/ y (- x y)) x) 2.0)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -2000.0) || !(x <= 1.5e-57)) {
		tmp = (2.0 * (x / (x - y))) * y;
	} else {
		tmp = ((y / (x - y)) * x) * 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-2000.0d0)) .or. (.not. (x <= 1.5d-57))) then
        tmp = (2.0d0 * (x / (x - y))) * y
    else
        tmp = ((y / (x - y)) * x) * 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -2000.0) || !(x <= 1.5e-57)) {
		tmp = (2.0 * (x / (x - y))) * y;
	} else {
		tmp = ((y / (x - y)) * x) * 2.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -2000.0) or not (x <= 1.5e-57):
		tmp = (2.0 * (x / (x - y))) * y
	else:
		tmp = ((y / (x - y)) * x) * 2.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -2000.0) || !(x <= 1.5e-57))
		tmp = Float64(Float64(2.0 * Float64(x / Float64(x - y))) * y);
	else
		tmp = Float64(Float64(Float64(y / Float64(x - y)) * x) * 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -2000.0) || ~((x <= 1.5e-57)))
		tmp = (2.0 * (x / (x - y))) * y;
	else
		tmp = ((y / (x - y)) * x) * 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -2000.0], N[Not[LessEqual[x, 1.5e-57]], $MachinePrecision]], N[(N[(2.0 * N[(x / N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], N[(N[(N[(y / N[(x - y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2e3 or 1.5e-57 < x

   1. Initial program 79.4%

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

      1. lift-/.f64 N/A

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

      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. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

   if -2e3 < x < 1.5e-57

   1. Initial program 75.9%

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

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2000 \lor \neg \left(x \leq 1.5 \cdot 10^{-57}\right):\\ \;\;\;\;\left(2 \cdot \frac{x}{x - y}\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{x - y} \cdot x\right) \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(x \cdot 2\right) \cdot y}{x - y}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+63} \lor \neg \left(t\_0 \leq -5 \cdot 10^{-292} \lor \neg \left(t\_0 \leq 0 \lor \neg \left(t\_0 \leq 4 \cdot 10^{-23}\right)\right)\right):\\ \;\;\;\;\left(2 \cdot \frac{x}{x - y}\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x + x\right) \cdot y}{x - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (* x 2.0) y) (- x y))))
   (if (or (<= t_0 -2e+63)
           (not
            (or (<= t_0 -5e-292)
                (not (or (<= t_0 0.0) (not (<= t_0 4e-23)))))))
     (* (* 2.0 (/ x (- x y))) y)
     (/ (* (+ x x) y) (- x y)))))

C

double code(double x, double y) {
	double t_0 = ((x * 2.0) * y) / (x - y);
	double tmp;
	if ((t_0 <= -2e+63) || !((t_0 <= -5e-292) || !((t_0 <= 0.0) || !(t_0 <= 4e-23)))) {
		tmp = (2.0 * (x / (x - y))) * y;
	} else {
		tmp = ((x + x) * y) / (x - y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((x * 2.0d0) * y) / (x - y)
    if ((t_0 <= (-2d+63)) .or. (.not. (t_0 <= (-5d-292)) .or. (.not. (t_0 <= 0.0d0) .or. (.not. (t_0 <= 4d-23))))) then
        tmp = (2.0d0 * (x / (x - y))) * y
    else
        tmp = ((x + x) * y) / (x - 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 <= -2e+63) || !((t_0 <= -5e-292) || !((t_0 <= 0.0) || !(t_0 <= 4e-23)))) {
		tmp = (2.0 * (x / (x - y))) * y;
	} else {
		tmp = ((x + x) * y) / (x - y);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = ((x * 2.0) * y) / (x - y)
	tmp = 0
	if (t_0 <= -2e+63) or not ((t_0 <= -5e-292) or not ((t_0 <= 0.0) or not (t_0 <= 4e-23))):
		tmp = (2.0 * (x / (x - y))) * y
	else:
		tmp = ((x + x) * y) / (x - 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 <= -2e+63) || !((t_0 <= -5e-292) || !((t_0 <= 0.0) || !(t_0 <= 4e-23))))
		tmp = Float64(Float64(2.0 * Float64(x / Float64(x - y))) * y);
	else
		tmp = Float64(Float64(Float64(x + x) * y) / Float64(x - 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 <= -2e+63) || ~(((t_0 <= -5e-292) || ~(((t_0 <= 0.0) || ~((t_0 <= 4e-23)))))))
		tmp = (2.0 * (x / (x - y))) * y;
	else
		tmp = ((x + x) * y) / (x - 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[Or[LessEqual[t$95$0, -2e+63], N[Not[Or[LessEqual[t$95$0, -5e-292], N[Not[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, 4e-23]], $MachinePrecision]]], $MachinePrecision]]], $MachinePrecision]], N[(N[(2.0 * N[(x / N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], N[(N[(N[(x + x), $MachinePrecision] * y), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (*.f64 x #s(literal 2 binary64)) y) (-.f64 x y)) < -2.00000000000000012e63 or -4.99999999999999981e-292 < (/.f64 (*.f64 (*.f64 x #s(literal 2 binary64)) y) (-.f64 x y)) < 0.0 or 3.99999999999999984e-23 < (/.f64 (*.f64 (*.f64 x #s(literal 2 binary64)) y) (-.f64 x y))

   1. Initial program 39.4%

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

      1. lift-/.f64 N/A

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

      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. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   if -2.00000000000000012e63 < (/.f64 (*.f64 (*.f64 x #s(literal 2 binary64)) y) (-.f64 x y)) < -4.99999999999999981e-292 or 0.0 < (/.f64 (*.f64 (*.f64 x #s(literal 2 binary64)) y) (-.f64 x y)) < 3.99999999999999984e-23

   1. Initial program 99.2%

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

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

      3. count-2-rev N/A

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

      4. lower-+.f64 99.2

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

   4. Applied rewrites 99.2%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot 2\right) \cdot y}{x - y} \leq -2 \cdot 10^{+63} \lor \neg \left(\frac{\left(x \cdot 2\right) \cdot y}{x - y} \leq -5 \cdot 10^{-292} \lor \neg \left(\frac{\left(x \cdot 2\right) \cdot y}{x - y} \leq 0 \lor \neg \left(\frac{\left(x \cdot 2\right) \cdot y}{x - y} \leq 4 \cdot 10^{-23}\right)\right)\right):\\ \;\;\;\;\left(2 \cdot \frac{x}{x - y}\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x + x\right) \cdot y}{x - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(x + x\right) \cdot y}{x - y}\\ \mathbf{if}\;x \leq -8.2 \cdot 10^{+128}:\\ \;\;\;\;y + y\\ \mathbf{elif}\;x \leq -1.8 \cdot 10^{-176}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-179}:\\ \;\;\;\;-2 \cdot x\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+164}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{x}, y, y\right) \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (+ x x) y) (- x y))))
   (if (<= x -8.2e+128)
     (+ y y)
     (if (<= x -1.8e-176)
       t_0
       (if (<= x 1.9e-179)
         (* -2.0 x)
         (if (<= x 1.1e+164) t_0 (* (fma (/ y x) y y) 2.0)))))))

C

double code(double x, double y) {
	double t_0 = ((x + x) * y) / (x - y);
	double tmp;
	if (x <= -8.2e+128) {
		tmp = y + y;
	} else if (x <= -1.8e-176) {
		tmp = t_0;
	} else if (x <= 1.9e-179) {
		tmp = -2.0 * x;
	} else if (x <= 1.1e+164) {
		tmp = t_0;
	} else {
		tmp = fma((y / x), y, y) * 2.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(x + x) * y) / Float64(x - y))
	tmp = 0.0
	if (x <= -8.2e+128)
		tmp = Float64(y + y);
	elseif (x <= -1.8e-176)
		tmp = t_0;
	elseif (x <= 1.9e-179)
		tmp = Float64(-2.0 * x);
	elseif (x <= 1.1e+164)
		tmp = t_0;
	else
		tmp = Float64(fma(Float64(y / x), y, y) * 2.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(x + x), $MachinePrecision] * y), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -8.2e+128], N[(y + y), $MachinePrecision], If[LessEqual[x, -1.8e-176], t$95$0, If[LessEqual[x, 1.9e-179], N[(-2.0 * x), $MachinePrecision], If[LessEqual[x, 1.1e+164], t$95$0, N[(N[(N[(y / x), $MachinePrecision] * y + y), $MachinePrecision] * 2.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -8.20000000000000023e128

   1. Initial program 74.1%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 92.4

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

   5. Applied rewrites 92.4%

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

   7. Applied rewrites 92.4%

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

   if -8.20000000000000023e128 < x < -1.8000000000000001e-176 or 1.89999999999999987e-179 < x < 1.10000000000000003e164

   1. Initial program 89.4%

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

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

      3. count-2-rev N/A

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

      4. lower-+.f64 89.4

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

   4. Applied rewrites 89.4%

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

   if -1.8000000000000001e-176 < x < 1.89999999999999987e-179

   1. Initial program 59.2%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 87.0

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

   5. Applied rewrites 87.0%

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

   if 1.10000000000000003e164 < x

   1. Initial program 54.6%

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

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 66.7

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

   4. Applied rewrites 66.7%

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

   5. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l/ N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 96.6

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

   7. Applied rewrites 96.6%

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

Alternative 4: 73.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.55 \cdot 10^{+63} \lor \neg \left(y \leq 1.25 \cdot 10^{-49}\right):\\ \;\;\;\;-2 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{x}, y, y\right) \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.55e+63) (not (<= y 1.25e-49)))
   (* -2.0 x)
   (* (fma (/ y x) y y) 2.0)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.55e+63) || !(y <= 1.25e-49)) {
		tmp = -2.0 * x;
	} else {
		tmp = fma((y / x), y, y) * 2.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.55e+63) || !(y <= 1.25e-49))
		tmp = Float64(-2.0 * x);
	else
		tmp = Float64(fma(Float64(y / x), y, y) * 2.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.55e+63], N[Not[LessEqual[y, 1.25e-49]], $MachinePrecision]], N[(-2.0 * x), $MachinePrecision], N[(N[(N[(y / x), $MachinePrecision] * y + y), $MachinePrecision] * 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.55e63 or 1.25e-49 < y

   1. Initial program 79.6%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 74.3

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

   5. Applied rewrites 74.3%

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

   if -1.55e63 < y < 1.25e-49

   1. Initial program 75.8%

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

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 81.2

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

   4. Applied rewrites 81.2%

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

   5. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l/ N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 79.7

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

   7. Applied rewrites 79.7%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.55 \cdot 10^{+63} \lor \neg \left(y \leq 1.25 \cdot 10^{-49}\right):\\ \;\;\;\;-2 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{x}, y, y\right) \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.55 \cdot 10^{+63} \lor \neg \left(y \leq 1.25 \cdot 10^{-49}\right):\\ \;\;\;\;-2 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{2}{x}, y, 2\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.55e+63) (not (<= y 1.25e-49)))
   (* -2.0 x)
   (* (fma (/ 2.0 x) y 2.0) y)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.55e+63) || !(y <= 1.25e-49)) {
		tmp = -2.0 * x;
	} else {
		tmp = fma((2.0 / x), y, 2.0) * y;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.55e+63) || !(y <= 1.25e-49))
		tmp = Float64(-2.0 * x);
	else
		tmp = Float64(fma(Float64(2.0 / x), y, 2.0) * y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.55e+63], N[Not[LessEqual[y, 1.25e-49]], $MachinePrecision]], N[(-2.0 * x), $MachinePrecision], N[(N[(N[(2.0 / x), $MachinePrecision] * y + 2.0), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.55e63 or 1.25e-49 < y

   1. Initial program 79.6%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 74.3

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

   5. Applied rewrites 74.3%

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

   if -1.55e63 < y < 1.25e-49

   1. Initial program 75.8%

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

   3. Taylor expanded in x around inf

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

      1. distribute-lft-out N/A

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

      2. unpow2 N/A

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

      3. associate-*l/ N/A

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

      4. distribute-lft-out N/A

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

      5. associate-*l* N/A

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

      6. distribute-rgt-in N/A

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

      7. *-commutative N/A

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

      8. *-lft-identity N/A

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

      9. associate-*l/ N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. +-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. associate-*r/ N/A

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

      17. metadata-eval N/A

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

      18. lower-/.f64 79.7

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

   5. Applied rewrites 79.7%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.55 \cdot 10^{+63} \lor \neg \left(y \leq 1.25 \cdot 10^{-49}\right):\\ \;\;\;\;-2 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{2}{x}, y, 2\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.55 \cdot 10^{+63} \lor \neg \left(y \leq 1.28 \cdot 10^{-48}\right):\\ \;\;\;\;-2 \cdot x\\ \mathbf{else}:\\ \;\;\;\;y + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.55e+63) (not (<= y 1.28e-48))) (* -2.0 x) (+ y y)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.55e+63) || !(y <= 1.28e-48)) {
		tmp = -2.0 * x;
	} 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 ((y <= (-1.55d+63)) .or. (.not. (y <= 1.28d-48))) then
        tmp = (-2.0d0) * x
    else
        tmp = y + y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.55e+63) || !(y <= 1.28e-48)) {
		tmp = -2.0 * x;
	} else {
		tmp = y + y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.55e+63) or not (y <= 1.28e-48):
		tmp = -2.0 * x
	else:
		tmp = y + y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.55e+63) || !(y <= 1.28e-48))
		tmp = Float64(-2.0 * x);
	else
		tmp = Float64(y + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.55e+63) || ~((y <= 1.28e-48)))
		tmp = -2.0 * x;
	else
		tmp = y + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.55e+63], N[Not[LessEqual[y, 1.28e-48]], $MachinePrecision]], N[(-2.0 * x), $MachinePrecision], N[(y + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.55e63 or 1.28000000000000001e-48 < y

   1. Initial program 79.6%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 74.3

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

   5. Applied rewrites 74.3%

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

   if -1.55e63 < y < 1.28000000000000001e-48

   1. Initial program 75.8%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 79.5

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

   5. Applied rewrites 79.5%

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

   7. Applied rewrites 79.5%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.55 \cdot 10^{+63} \lor \neg \left(y \leq 1.28 \cdot 10^{-48}\right):\\ \;\;\;\;-2 \cdot x\\ \mathbf{else}:\\ \;\;\;\;y + y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 51.3% 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 77.7%

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

3. Taylor expanded in x around inf

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

   1. lower-*.f64 52.0

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

5. Applied rewrites 52.0%

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

7. Applied rewrites 52.0%

   \[\leadsto y + \color{blue}{y} \]
8. 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 2024320 
(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)))