Linear.Projection:perspective from linear-1.19.1.3, B

Percentage Accurate: 76.4% → 98.6%

Time: 7.9s

Alternatives: 6

Speedup: 0.5×

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: 76.4% 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: 98.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(x \cdot 2\right) \cdot y}{x - y}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+51} \lor \neg \left(t\_0 \leq -1 \cdot 10^{-303} \lor \neg \left(t\_0 \leq 5 \cdot 10^{-305} \lor \neg \left(t\_0 \leq 4 \cdot 10^{-86}\right)\right)\right):\\ \;\;\;\;\left(\frac{y}{x - y} \cdot x\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x + x\right) \cdot y}{x - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (* x 2.0) y) (- x y))))
   (if (or (<= t_0 -1e+51)
           (not
            (or (<= t_0 -1e-303)
                (not (or (<= t_0 5e-305) (not (<= t_0 4e-86)))))))
     (* (* (/ y (- x y)) x) 2.0)
     (/ (* (+ 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 <= -1e+51) || !((t_0 <= -1e-303) || !((t_0 <= 5e-305) || !(t_0 <= 4e-86)))) {
		tmp = ((y / (x - y)) * x) * 2.0;
	} 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 <= (-1d+51)) .or. (.not. (t_0 <= (-1d-303)) .or. (.not. (t_0 <= 5d-305) .or. (.not. (t_0 <= 4d-86))))) then
        tmp = ((y / (x - y)) * x) * 2.0d0
    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 <= -1e+51) || !((t_0 <= -1e-303) || !((t_0 <= 5e-305) || !(t_0 <= 4e-86)))) {
		tmp = ((y / (x - y)) * x) * 2.0;
	} 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 <= -1e+51) or not ((t_0 <= -1e-303) or not ((t_0 <= 5e-305) or not (t_0 <= 4e-86))):
		tmp = ((y / (x - y)) * x) * 2.0
	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 <= -1e+51) || !((t_0 <= -1e-303) || !((t_0 <= 5e-305) || !(t_0 <= 4e-86))))
		tmp = Float64(Float64(Float64(y / Float64(x - y)) * x) * 2.0);
	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 <= -1e+51) || ~(((t_0 <= -1e-303) || ~(((t_0 <= 5e-305) || ~((t_0 <= 4e-86)))))))
		tmp = ((y / (x - y)) * x) * 2.0;
	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, -1e+51], N[Not[Or[LessEqual[t$95$0, -1e-303], N[Not[Or[LessEqual[t$95$0, 5e-305], N[Not[LessEqual[t$95$0, 4e-86]], $MachinePrecision]]], $MachinePrecision]]], $MachinePrecision]], N[(N[(N[(y / N[(x - y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * 2.0), $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)) < -1e51 or -9.99999999999999931e-304 < (/.f64 (*.f64 (*.f64 x #s(literal 2 binary64)) y) (-.f64 x y)) < 4.99999999999999985e-305 or 4.00000000000000034e-86 < (/.f64 (*.f64 (*.f64 x #s(literal 2 binary64)) y) (-.f64 x y))

   1. Initial program 50.8%

      \[\frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
   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 99.9

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

   4. Applied rewrites 99.9%

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

   if -1e51 < (/.f64 (*.f64 (*.f64 x #s(literal 2 binary64)) y) (-.f64 x y)) < -9.99999999999999931e-304 or 4.99999999999999985e-305 < (/.f64 (*.f64 (*.f64 x #s(literal 2 binary64)) y) (-.f64 x y)) < 4.00000000000000034e-86

   1. Initial program 98.6%

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

      1. lift-*.f64 N/A

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

      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 98.6

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

   4. Applied rewrites 98.6%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot 2\right) \cdot y}{x - y} \leq -1 \cdot 10^{+51} \lor \neg \left(\frac{\left(x \cdot 2\right) \cdot y}{x - y} \leq -1 \cdot 10^{-303} \lor \neg \left(\frac{\left(x \cdot 2\right) \cdot y}{x - y} \leq 5 \cdot 10^{-305} \lor \neg \left(\frac{\left(x \cdot 2\right) \cdot y}{x - y} \leq 4 \cdot 10^{-86}\right)\right)\right):\\ \;\;\;\;\left(\frac{y}{x - y} \cdot x\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x + x\right) \cdot y}{x - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(x \cdot 2\right) \cdot y}{x - y}\\ \mathbf{if}\;t\_0 \leq -\infty \lor \neg \left(t\_0 \leq -1 \cdot 10^{-303} \lor \neg \left(t\_0 \leq 5 \cdot 10^{-305} \lor \neg \left(t\_0 \leq 4 \cdot 10^{+80}\right)\right)\right):\\ \;\;\;\;-2 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x + x\right) \cdot y}{x - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (* x 2.0) y) (- x y))))
   (if (or (<= t_0 (- INFINITY))
           (not
            (or (<= t_0 -1e-303)
                (not (or (<= t_0 5e-305) (not (<= t_0 4e+80)))))))
     (* -2.0 x)
     (/ (* (+ 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 <= -((double) INFINITY)) || !((t_0 <= -1e-303) || !((t_0 <= 5e-305) || !(t_0 <= 4e+80)))) {
		tmp = -2.0 * x;
	} else {
		tmp = ((x + x) * y) / (x - y);
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = ((x * 2.0) * y) / (x - y);
	double tmp;
	if ((t_0 <= -Double.POSITIVE_INFINITY) || !((t_0 <= -1e-303) || !((t_0 <= 5e-305) || !(t_0 <= 4e+80)))) {
		tmp = -2.0 * x;
	} 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 <= -math.inf) or not ((t_0 <= -1e-303) or not ((t_0 <= 5e-305) or not (t_0 <= 4e+80))):
		tmp = -2.0 * x
	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 <= Float64(-Inf)) || !((t_0 <= -1e-303) || !((t_0 <= 5e-305) || !(t_0 <= 4e+80))))
		tmp = Float64(-2.0 * x);
	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 <= -Inf) || ~(((t_0 <= -1e-303) || ~(((t_0 <= 5e-305) || ~((t_0 <= 4e+80)))))))
		tmp = -2.0 * x;
	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, (-Infinity)], N[Not[Or[LessEqual[t$95$0, -1e-303], N[Not[Or[LessEqual[t$95$0, 5e-305], N[Not[LessEqual[t$95$0, 4e+80]], $MachinePrecision]]], $MachinePrecision]]], $MachinePrecision]], N[(-2.0 * x), $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)) < -inf.0 or -9.99999999999999931e-304 < (/.f64 (*.f64 (*.f64 x #s(literal 2 binary64)) y) (-.f64 x y)) < 4.99999999999999985e-305 or 4e80 < (/.f64 (*.f64 (*.f64 x #s(literal 2 binary64)) y) (-.f64 x y))

   1. Initial program 9.7%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 64.4

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

   5. Applied rewrites 64.4%

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

   if -inf.0 < (/.f64 (*.f64 (*.f64 x #s(literal 2 binary64)) y) (-.f64 x y)) < -9.99999999999999931e-304 or 4.99999999999999985e-305 < (/.f64 (*.f64 (*.f64 x #s(literal 2 binary64)) y) (-.f64 x y)) < 4e80

   1. Initial program 98.8%

      \[\frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
   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. *-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 98.8

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

   4. Applied rewrites 98.8%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot 2\right) \cdot y}{x - y} \leq -\infty \lor \neg \left(\frac{\left(x \cdot 2\right) \cdot y}{x - y} \leq -1 \cdot 10^{-303} \lor \neg \left(\frac{\left(x \cdot 2\right) \cdot y}{x - y} \leq 5 \cdot 10^{-305} \lor \neg \left(\frac{\left(x \cdot 2\right) \cdot y}{x - y} \leq 4 \cdot 10^{+80}\right)\right)\right):\\ \;\;\;\;-2 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x + x\right) \cdot y}{x - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.9 \cdot 10^{-109} \lor \neg \left(y \leq 3.2 \cdot 10^{-147}\right):\\ \;\;\;\;\left(\frac{y}{x - y} \cdot x\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(\frac{y}{x}, \mathsf{fma}\left(\frac{y}{x}, y, y\right), y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -2.9e-109) (not (<= y 3.2e-147)))
   (* (* (/ y (- x y)) x) 2.0)
   (* 2.0 (fma (/ y x) (fma (/ y x) y y) y))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -2.9e-109) || !(y <= 3.2e-147)) {
		tmp = ((y / (x - y)) * x) * 2.0;
	} else {
		tmp = 2.0 * fma((y / x), fma((y / x), y, y), y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -2.9e-109) || !(y <= 3.2e-147))
		tmp = Float64(Float64(Float64(y / Float64(x - y)) * x) * 2.0);
	else
		tmp = Float64(2.0 * fma(Float64(y / x), fma(Float64(y / x), y, y), y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -2.9e-109], N[Not[LessEqual[y, 3.2e-147]], $MachinePrecision]], N[(N[(N[(y / N[(x - y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * 2.0), $MachinePrecision], N[(2.0 * N[(N[(y / x), $MachinePrecision] * N[(N[(y / x), $MachinePrecision] * y + y), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.9e-109 or 3.19999999999999979e-147 < y

   1. Initial program 82.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. 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 99.2

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

   4. Applied rewrites 99.2%

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

   if -2.9e-109 < y < 3.19999999999999979e-147

   1. Initial program 79.3%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{2 \cdot y + \left(2 \cdot \frac{{y}^{2}}{x} + 2 \cdot \frac{{y}^{3}}{{x}^{2}}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(2 \cdot \frac{{y}^{2}}{x} + 2 \cdot \frac{{y}^{3}}{{x}^{2}}\right) + 2 \cdot y} \]

      2. distribute-lft-out N/A

         \[\leadsto \color{blue}{2 \cdot \left(\frac{{y}^{2}}{x} + \frac{{y}^{3}}{{x}^{2}}\right)} + 2 \cdot y \]

      3. distribute-lft-out N/A

         \[\leadsto \color{blue}{2 \cdot \left(\left(\frac{{y}^{2}}{x} + \frac{{y}^{3}}{{x}^{2}}\right) + y\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{2 \cdot \left(\left(\frac{{y}^{2}}{x} + \frac{{y}^{3}}{{x}^{2}}\right) + y\right)} \]

      5. unpow2 N/A

         \[\leadsto 2 \cdot \left(\left(\frac{\color{blue}{y \cdot y}}{x} + \frac{{y}^{3}}{{x}^{2}}\right) + y\right) \]

      6. associate-/l* N/A

         \[\leadsto 2 \cdot \left(\left(\color{blue}{y \cdot \frac{y}{x}} + \frac{{y}^{3}}{{x}^{2}}\right) + y\right) \]

      7. unpow3 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. times-frac N/A

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

      11. distribute-rgt-out N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. unpow2 N/A

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

      16. associate-*l/ N/A

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

      17. lower-fma.f64 N/A

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

      18. lower-/.f64 89.4

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

   5. Applied rewrites 89.4%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.9 \cdot 10^{-109} \lor \neg \left(y \leq 3.2 \cdot 10^{-147}\right):\\ \;\;\;\;\left(\frac{y}{x - y} \cdot x\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(\frac{y}{x}, \mathsf{fma}\left(\frac{y}{x}, y, y\right), y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{-21}:\\ \;\;\;\;-2 \cdot x\\ \mathbf{elif}\;y \leq 3 \cdot 10^{-79}:\\ \;\;\;\;y + y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{x}{y}, x\right) \cdot -2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -7e-21)
   (* -2.0 x)
   (if (<= y 3e-79) (+ y y) (* (fma x (/ x y) x) -2.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -7e-21) {
		tmp = -2.0 * x;
	} else if (y <= 3e-79) {
		tmp = y + y;
	} else {
		tmp = fma(x, (x / y), x) * -2.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -7e-21)
		tmp = Float64(-2.0 * x);
	elseif (y <= 3e-79)
		tmp = Float64(y + y);
	else
		tmp = Float64(fma(x, Float64(x / y), x) * -2.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -7e-21], N[(-2.0 * x), $MachinePrecision], If[LessEqual[y, 3e-79], N[(y + y), $MachinePrecision], N[(N[(x * N[(x / y), $MachinePrecision] + x), $MachinePrecision] * -2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.0000000000000007e-21

   1. Initial program 82.9%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 73.1

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

   5. Applied rewrites 73.1%

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

   if -7.0000000000000007e-21 < y < 3e-79

   1. Initial program 81.7%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 83.5

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

   5. Applied rewrites 83.5%

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

   7. Applied rewrites 83.5%

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

   if 3e-79 < y

   1. Initial program 79.6%

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

   3. Taylor expanded in y around inf

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

      1. distribute-lft-out N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 80.6

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

   5. Applied rewrites 80.6%

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

Alternative 5: 74.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{-21} \lor \neg \left(y \leq 3 \cdot 10^{-79}\right):\\ \;\;\;\;-2 \cdot x\\ \mathbf{else}:\\ \;\;\;\;y + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -7e-21) (not (<= y 3e-79))) (* -2.0 x) (+ y y)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -7e-21) || !(y <= 3e-79)) {
		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 <= (-7d-21)) .or. (.not. (y <= 3d-79))) 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 <= -7e-21) || !(y <= 3e-79)) {
		tmp = -2.0 * x;
	} else {
		tmp = y + y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -7e-21) or not (y <= 3e-79):
		tmp = -2.0 * x
	else:
		tmp = y + y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -7e-21) || !(y <= 3e-79))
		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 <= -7e-21) || ~((y <= 3e-79)))
		tmp = -2.0 * x;
	else
		tmp = y + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -7e-21], N[Not[LessEqual[y, 3e-79]], $MachinePrecision]], N[(-2.0 * x), $MachinePrecision], N[(y + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -7.0000000000000007e-21 or 3e-79 < y

   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. lower-*.f64 76.8

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

   5. Applied rewrites 76.8%

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

   if -7.0000000000000007e-21 < y < 3e-79

   1. Initial program 81.7%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 83.5

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

   5. Applied rewrites 83.5%

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

   7. Applied rewrites 83.5%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{-21} \lor \neg \left(y \leq 3 \cdot 10^{-79}\right):\\ \;\;\;\;-2 \cdot x\\ \mathbf{else}:\\ \;\;\;\;y + y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 50.8% 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 81.4%

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

3. Taylor expanded in x around inf

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

   1. lower-*.f64 49.4

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

5. Applied rewrites 49.4%

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

7. Applied rewrites 49.4%

   \[\leadsto y + \color{blue}{y} \]
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 2024329 
(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)))