Linear.Projection:perspective from linear-1.19.1.3, B

Percentage Accurate: 77.7% → 99.9%

Time: 5.8s

Alternatives: 8

Speedup: 0.7×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.7% 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: 99.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y - x} \cdot \left(y \cdot -2\right)\\ \mathbf{if}\;x \leq -1 \cdot 10^{-18}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 0.0001:\\ \;\;\;\;\frac{y}{y - x} \cdot \left(-2 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (/ x (- y x)) (* y -2.0))))
   (if (<= x -1e-18) t_0 (if (<= x 0.0001) (* (/ y (- y x)) (* -2.0 x)) t_0))))

C

double code(double x, double y) {
	double t_0 = (x / (y - x)) * (y * -2.0);
	double tmp;
	if (x <= -1e-18) {
		tmp = t_0;
	} else if (x <= 0.0001) {
		tmp = (y / (y - x)) * (-2.0 * x);
	} 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 = (x / (y - x)) * (y * (-2.0d0))
    if (x <= (-1d-18)) then
        tmp = t_0
    else if (x <= 0.0001d0) then
        tmp = (y / (y - x)) * ((-2.0d0) * x)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x / (y - x)) * (y * -2.0);
	double tmp;
	if (x <= -1e-18) {
		tmp = t_0;
	} else if (x <= 0.0001) {
		tmp = (y / (y - x)) * (-2.0 * x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x / (y - x)) * (y * -2.0)
	tmp = 0
	if x <= -1e-18:
		tmp = t_0
	elif x <= 0.0001:
		tmp = (y / (y - x)) * (-2.0 * x)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x / Float64(y - x)) * Float64(y * -2.0))
	tmp = 0.0
	if (x <= -1e-18)
		tmp = t_0;
	elseif (x <= 0.0001)
		tmp = Float64(Float64(y / Float64(y - x)) * Float64(-2.0 * x));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x / (y - x)) * (y * -2.0);
	tmp = 0.0;
	if (x <= -1e-18)
		tmp = t_0;
	elseif (x <= 0.0001)
		tmp = (y / (y - x)) * (-2.0 * x);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x / N[(y - x), $MachinePrecision]), $MachinePrecision] * N[(y * -2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1e-18], t$95$0, If[LessEqual[x, 0.0001], N[(N[(y / N[(y - x), $MachinePrecision]), $MachinePrecision] * N[(-2.0 * x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.0000000000000001e-18 or 1.00000000000000005e-4 < x

   1. Initial program 80.7%

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

   4. Applied rewrites 100.0%

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

   if -1.0000000000000001e-18 < x < 1.00000000000000005e-4

   1. Initial program 80.2%

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

   4. Applied rewrites 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-18}:\\ \;\;\;\;\frac{x}{y - x} \cdot \left(y \cdot -2\right)\\ \mathbf{elif}\;x \leq 0.0001:\\ \;\;\;\;\frac{y}{y - x} \cdot \left(-2 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y - x} \cdot \left(y \cdot -2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{y - x} \cdot \left(-2 \cdot x\right)\\ t_1 := \frac{\left(2 \cdot x\right) \cdot y}{x - y}\\ t_2 := \frac{\left(x + x\right) \cdot y}{x - y}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-41}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-299}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-46}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (/ y (- y x)) (* -2.0 x)))
        (t_1 (/ (* (* 2.0 x) y) (- x y)))
        (t_2 (/ (* (+ x x) y) (- x y))))
   (if (<= t_1 -1e-41)
     t_0
     (if (<= t_1 -5e-299)
       t_2
       (if (<= t_1 0.0) t_0 (if (<= t_1 5e-46) t_2 t_0))))))

C

double code(double x, double y) {
	double t_0 = (y / (y - x)) * (-2.0 * x);
	double t_1 = ((2.0 * x) * y) / (x - y);
	double t_2 = ((x + x) * y) / (x - y);
	double tmp;
	if (t_1 <= -1e-41) {
		tmp = t_0;
	} else if (t_1 <= -5e-299) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = t_0;
	} else if (t_1 <= 5e-46) {
		tmp = t_2;
	} 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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (y / (y - x)) * ((-2.0d0) * x)
    t_1 = ((2.0d0 * x) * y) / (x - y)
    t_2 = ((x + x) * y) / (x - y)
    if (t_1 <= (-1d-41)) then
        tmp = t_0
    else if (t_1 <= (-5d-299)) then
        tmp = t_2
    else if (t_1 <= 0.0d0) then
        tmp = t_0
    else if (t_1 <= 5d-46) then
        tmp = t_2
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (y / (y - x)) * (-2.0 * x);
	double t_1 = ((2.0 * x) * y) / (x - y);
	double t_2 = ((x + x) * y) / (x - y);
	double tmp;
	if (t_1 <= -1e-41) {
		tmp = t_0;
	} else if (t_1 <= -5e-299) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = t_0;
	} else if (t_1 <= 5e-46) {
		tmp = t_2;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y / (y - x)) * (-2.0 * x)
	t_1 = ((2.0 * x) * y) / (x - y)
	t_2 = ((x + x) * y) / (x - y)
	tmp = 0
	if t_1 <= -1e-41:
		tmp = t_0
	elif t_1 <= -5e-299:
		tmp = t_2
	elif t_1 <= 0.0:
		tmp = t_0
	elif t_1 <= 5e-46:
		tmp = t_2
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y / Float64(y - x)) * Float64(-2.0 * x))
	t_1 = Float64(Float64(Float64(2.0 * x) * y) / Float64(x - y))
	t_2 = Float64(Float64(Float64(x + x) * y) / Float64(x - y))
	tmp = 0.0
	if (t_1 <= -1e-41)
		tmp = t_0;
	elseif (t_1 <= -5e-299)
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = t_0;
	elseif (t_1 <= 5e-46)
		tmp = t_2;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y / (y - x)) * (-2.0 * x);
	t_1 = ((2.0 * x) * y) / (x - y);
	t_2 = ((x + x) * y) / (x - y);
	tmp = 0.0;
	if (t_1 <= -1e-41)
		tmp = t_0;
	elseif (t_1 <= -5e-299)
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = t_0;
	elseif (t_1 <= 5e-46)
		tmp = t_2;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y / N[(y - x), $MachinePrecision]), $MachinePrecision] * N[(-2.0 * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(2.0 * x), $MachinePrecision] * y), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(x + x), $MachinePrecision] * y), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-41], t$95$0, If[LessEqual[t$95$1, -5e-299], t$95$2, If[LessEqual[t$95$1, 0.0], t$95$0, If[LessEqual[t$95$1, 5e-46], t$95$2, t$95$0]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (*.f64 x #s(literal 2 binary64)) y) (-.f64 x y)) < -1.00000000000000001e-41 or -4.99999999999999956e-299 < (/.f64 (*.f64 (*.f64 x #s(literal 2 binary64)) y) (-.f64 x y)) < -0.0 or 4.99999999999999992e-46 < (/.f64 (*.f64 (*.f64 x #s(literal 2 binary64)) y) (-.f64 x y))

   1. Initial program 55.7%

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

   4. Applied rewrites 99.9%

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

   if -1.00000000000000001e-41 < (/.f64 (*.f64 (*.f64 x #s(literal 2 binary64)) y) (-.f64 x y)) < -4.99999999999999956e-299 or -0.0 < (/.f64 (*.f64 (*.f64 x #s(literal 2 binary64)) y) (-.f64 x y)) < 4.99999999999999992e-46

   1. Initial program 99.7%

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

      1. lift-*.f64 N/A

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

      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 99.7

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

   4. Applied rewrites 99.7%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(2 \cdot x\right) \cdot y}{x - y} \leq -1 \cdot 10^{-41}:\\ \;\;\;\;\frac{y}{y - x} \cdot \left(-2 \cdot x\right)\\ \mathbf{elif}\;\frac{\left(2 \cdot x\right) \cdot y}{x - y} \leq -5 \cdot 10^{-299}:\\ \;\;\;\;\frac{\left(x + x\right) \cdot y}{x - y}\\ \mathbf{elif}\;\frac{\left(2 \cdot x\right) \cdot y}{x - y} \leq 0:\\ \;\;\;\;\frac{y}{y - x} \cdot \left(-2 \cdot x\right)\\ \mathbf{elif}\;\frac{\left(2 \cdot x\right) \cdot y}{x - y} \leq 5 \cdot 10^{-46}:\\ \;\;\;\;\frac{\left(x + x\right) \cdot y}{x - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y - x} \cdot \left(-2 \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(x + x\right) \cdot y}{x - y}\\ t_1 := \frac{\left(2 \cdot x\right) \cdot y}{x - y}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-2 \cdot x\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-299}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, x, x\right) \cdot -2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+129}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (+ x x) y) (- x y))) (t_1 (/ (* (* 2.0 x) y) (- x y))))
   (if (<= t_1 (- INFINITY))
     (* -2.0 x)
     (if (<= t_1 -5e-299)
       t_0
       (if (<= t_1 0.0)
         (* (fma (/ x y) x x) -2.0)
         (if (<= t_1 2e+129) t_0 (* -2.0 x)))))))

C

double code(double x, double y) {
	double t_0 = ((x + x) * y) / (x - y);
	double t_1 = ((2.0 * x) * y) / (x - y);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -2.0 * x;
	} else if (t_1 <= -5e-299) {
		tmp = t_0;
	} else if (t_1 <= 0.0) {
		tmp = fma((x / y), x, x) * -2.0;
	} else if (t_1 <= 2e+129) {
		tmp = t_0;
	} else {
		tmp = -2.0 * x;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(x + x) * y) / Float64(x - y))
	t_1 = Float64(Float64(Float64(2.0 * x) * y) / Float64(x - y))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-2.0 * x);
	elseif (t_1 <= -5e-299)
		tmp = t_0;
	elseif (t_1 <= 0.0)
		tmp = Float64(fma(Float64(x / y), x, x) * -2.0);
	elseif (t_1 <= 2e+129)
		tmp = t_0;
	else
		tmp = Float64(-2.0 * x);
	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]}, Block[{t$95$1 = N[(N[(N[(2.0 * x), $MachinePrecision] * y), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(-2.0 * x), $MachinePrecision], If[LessEqual[t$95$1, -5e-299], t$95$0, If[LessEqual[t$95$1, 0.0], N[(N[(N[(x / y), $MachinePrecision] * x + x), $MachinePrecision] * -2.0), $MachinePrecision], If[LessEqual[t$95$1, 2e+129], t$95$0, N[(-2.0 * x), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (*.f64 x #s(literal 2 binary64)) y) (-.f64 x y)) < -inf.0 or 2e129 < (/.f64 (*.f64 (*.f64 x #s(literal 2 binary64)) y) (-.f64 x y))

   1. Initial program 4.8%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 67.5

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

   5. Applied rewrites 67.5%

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

   if -inf.0 < (/.f64 (*.f64 (*.f64 x #s(literal 2 binary64)) y) (-.f64 x y)) < -4.99999999999999956e-299 or -0.0 < (/.f64 (*.f64 (*.f64 x #s(literal 2 binary64)) y) (-.f64 x y)) < 2e129

   1. Initial program 99.7%

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

      1. lift-*.f64 N/A

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

      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 99.7

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

   4. Applied rewrites 99.7%

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

   if -4.99999999999999956e-299 < (/.f64 (*.f64 (*.f64 x #s(literal 2 binary64)) y) (-.f64 x y)) < -0.0

   1. Initial program 11.9%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 56.9

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

   5. Applied rewrites 56.9%

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

   7. Applied rewrites 56.9%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(2 \cdot x\right) \cdot y}{x - y} \leq -\infty:\\ \;\;\;\;-2 \cdot x\\ \mathbf{elif}\;\frac{\left(2 \cdot x\right) \cdot y}{x - y} \leq -5 \cdot 10^{-299}:\\ \;\;\;\;\frac{\left(x + x\right) \cdot y}{x - y}\\ \mathbf{elif}\;\frac{\left(2 \cdot x\right) \cdot y}{x - y} \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, x, x\right) \cdot -2\\ \mathbf{elif}\;\frac{\left(2 \cdot x\right) \cdot y}{x - y} \leq 2 \cdot 10^{+129}:\\ \;\;\;\;\frac{\left(x + x\right) \cdot y}{x - y}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.43 \cdot 10^{+45}:\\ \;\;\;\;y + y\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-46}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, x, x\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;y + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -3.43e+45)
   (+ y y)
   (if (<= x 2.2e-46) (* (fma (/ x y) x x) -2.0) (+ y y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -3.43e+45) {
		tmp = y + y;
	} else if (x <= 2.2e-46) {
		tmp = fma((x / y), x, x) * -2.0;
	} else {
		tmp = y + y;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -3.43e+45)
		tmp = Float64(y + y);
	elseif (x <= 2.2e-46)
		tmp = Float64(fma(Float64(x / y), x, x) * -2.0);
	else
		tmp = Float64(y + y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -3.43e+45], N[(y + y), $MachinePrecision], If[LessEqual[x, 2.2e-46], N[(N[(N[(x / y), $MachinePrecision] * x + x), $MachinePrecision] * -2.0), $MachinePrecision], N[(y + y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.43000000000000001e45 or 2.2000000000000001e-46 < x

   1. Initial program 80.8%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 75.2

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

   5. Applied rewrites 75.2%

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

   7. Applied rewrites 75.2%

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

   if -3.43000000000000001e45 < x < 2.2000000000000001e-46

   1. Initial program 80.1%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 81.9

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

   5. Applied rewrites 81.9%

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

   7. Applied rewrites 81.9%

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

Alternative 5: 74.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.43 \cdot 10^{+45}:\\ \;\;\;\;y + y\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-46}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, -2, -2\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;y + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -3.43e+45)
   (+ y y)
   (if (<= x 2.2e-46) (* (fma (/ x y) -2.0 -2.0) x) (+ y y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -3.43e+45) {
		tmp = y + y;
	} else if (x <= 2.2e-46) {
		tmp = fma((x / y), -2.0, -2.0) * x;
	} else {
		tmp = y + y;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -3.43e+45)
		tmp = Float64(y + y);
	elseif (x <= 2.2e-46)
		tmp = Float64(fma(Float64(x / y), -2.0, -2.0) * x);
	else
		tmp = Float64(y + y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -3.43e+45], N[(y + y), $MachinePrecision], If[LessEqual[x, 2.2e-46], N[(N[(N[(x / y), $MachinePrecision] * -2.0 + -2.0), $MachinePrecision] * x), $MachinePrecision], N[(y + y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.43000000000000001e45 or 2.2000000000000001e-46 < x

   1. Initial program 80.8%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 75.2

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

   5. Applied rewrites 75.2%

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

   7. Applied rewrites 75.2%

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

   if -3.43000000000000001e45 < x < 2.2000000000000001e-46

   1. Initial program 80.1%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 81.9

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

   5. Applied rewrites 81.9%

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

Alternative 6: 74.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.43 \cdot 10^{+45}:\\ \;\;\;\;y + y\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-46}:\\ \;\;\;\;-2 \cdot x\\ \mathbf{else}:\\ \;\;\;\;y + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -3.43e+45) (+ y y) (if (<= x 2.5e-46) (* -2.0 x) (+ y y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -3.43e+45) {
		tmp = y + y;
	} else if (x <= 2.5e-46) {
		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 (x <= (-3.43d+45)) then
        tmp = y + y
    else if (x <= 2.5d-46) 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 (x <= -3.43e+45) {
		tmp = y + y;
	} else if (x <= 2.5e-46) {
		tmp = -2.0 * x;
	} else {
		tmp = y + y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -3.43e+45:
		tmp = y + y
	elif x <= 2.5e-46:
		tmp = -2.0 * x
	else:
		tmp = y + y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -3.43e+45)
		tmp = Float64(y + y);
	elseif (x <= 2.5e-46)
		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 (x <= -3.43e+45)
		tmp = y + y;
	elseif (x <= 2.5e-46)
		tmp = -2.0 * x;
	else
		tmp = y + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -3.43e+45], N[(y + y), $MachinePrecision], If[LessEqual[x, 2.5e-46], N[(-2.0 * x), $MachinePrecision], N[(y + y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.43000000000000001e45 or 2.49999999999999996e-46 < x

   1. Initial program 80.8%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 75.2

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

   5. Applied rewrites 75.2%

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

   7. Applied rewrites 75.2%

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

   if -3.43000000000000001e45 < x < 2.49999999999999996e-46

   1. Initial program 80.1%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 81.0

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

   5. Applied rewrites 81.0%

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

Alternative 7: 50.4% 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 80.5%

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

3. Taylor expanded in x around inf

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

   1. lower-*.f64 48.5

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

5. Applied rewrites 48.5%

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

7. Applied rewrites 48.5%

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

Alternative 8: 3.3% accurate, 6.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.5%

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

3. Taylor expanded in x around inf

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

   1. lower-*.f64 48.5

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

5. Applied rewrites 48.5%

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

7. Applied rewrites 48.5%

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

9. Applied rewrites 3.3%

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

Developer Target 1: 99.6% 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 2024298 
(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)))