Linear.Projection:perspective from linear-1.19.1.3, B

Percentage Accurate: 77.5% → 99.7%

Time: 7.1s

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: 77.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.55 \cdot 10^{-62}:\\ \;\;\;\;x \cdot \frac{y \cdot 2}{x - y}\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+19}:\\ \;\;\;\;y \cdot \frac{2 \cdot x}{x - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot x}{\frac{x - y}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3.55e-62)
   (* x (/ (* y 2.0) (- x y)))
   (if (<= y 2e+19) (* y (/ (* 2.0 x) (- x y))) (/ (* 2.0 x) (/ (- x y) y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3.55e-62) {
		tmp = x * ((y * 2.0) / (x - y));
	} else if (y <= 2e+19) {
		tmp = y * ((2.0 * x) / (x - y));
	} else {
		tmp = (2.0 * x) / ((x - 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 <= (-3.55d-62)) then
        tmp = x * ((y * 2.0d0) / (x - y))
    else if (y <= 2d+19) then
        tmp = y * ((2.0d0 * x) / (x - y))
    else
        tmp = (2.0d0 * x) / ((x - y) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -3.55e-62) {
		tmp = x * ((y * 2.0) / (x - y));
	} else if (y <= 2e+19) {
		tmp = y * ((2.0 * x) / (x - y));
	} else {
		tmp = (2.0 * x) / ((x - y) / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -3.55e-62:
		tmp = x * ((y * 2.0) / (x - y))
	elif y <= 2e+19:
		tmp = y * ((2.0 * x) / (x - y))
	else:
		tmp = (2.0 * x) / ((x - y) / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3.55e-62)
		tmp = Float64(x * Float64(Float64(y * 2.0) / Float64(x - y)));
	elseif (y <= 2e+19)
		tmp = Float64(y * Float64(Float64(2.0 * x) / Float64(x - y)));
	else
		tmp = Float64(Float64(2.0 * x) / Float64(Float64(x - y) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -3.55e-62)
		tmp = x * ((y * 2.0) / (x - y));
	elseif (y <= 2e+19)
		tmp = y * ((2.0 * x) / (x - y));
	else
		tmp = (2.0 * x) / ((x - y) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3.55e-62], N[(x * N[(N[(y * 2.0), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2e+19], N[(y * N[(N[(2.0 * x), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * x), $MachinePrecision] / N[(N[(x - y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.5500000000000001e-62

   1. Initial program 77.6%

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

      1. associate-*l* N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

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

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

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

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

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

      7. --lowering--.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   if -3.5500000000000001e-62 < y < 2e19

   1. Initial program 78.7%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

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

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

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

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

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

      7. --lowering--.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   if 2e19 < y

   1. Initial program 76.8%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

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

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

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

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

      7. --lowering--.f64 99.9

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

   4. Applied egg-rr 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.55 \cdot 10^{-62}:\\ \;\;\;\;x \cdot \frac{y \cdot 2}{x - y}\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+19}:\\ \;\;\;\;y \cdot \frac{2 \cdot x}{x - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot x}{\frac{x - y}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{y \cdot 2}{x - y}\\ \mathbf{if}\;y \leq -1.5 \cdot 10^{-63}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 10000000000:\\ \;\;\;\;y \cdot \frac{2 \cdot x}{x - y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (/ (* y 2.0) (- x y)))))
   (if (<= y -1.5e-63)
     t_0
     (if (<= y 10000000000.0) (* y (/ (* 2.0 x) (- x y))) t_0))))

C

double code(double x, double y) {
	double t_0 = x * ((y * 2.0) / (x - y));
	double tmp;
	if (y <= -1.5e-63) {
		tmp = t_0;
	} else if (y <= 10000000000.0) {
		tmp = y * ((2.0 * x) / (x - y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * ((y * 2.0d0) / (x - y))
    if (y <= (-1.5d-63)) then
        tmp = t_0
    else if (y <= 10000000000.0d0) then
        tmp = y * ((2.0d0 * x) / (x - y))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x * ((y * 2.0) / (x - y));
	double tmp;
	if (y <= -1.5e-63) {
		tmp = t_0;
	} else if (y <= 10000000000.0) {
		tmp = y * ((2.0 * x) / (x - y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x * ((y * 2.0) / (x - y))
	tmp = 0
	if y <= -1.5e-63:
		tmp = t_0
	elif y <= 10000000000.0:
		tmp = y * ((2.0 * x) / (x - y))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x * Float64(Float64(y * 2.0) / Float64(x - y)))
	tmp = 0.0
	if (y <= -1.5e-63)
		tmp = t_0;
	elseif (y <= 10000000000.0)
		tmp = Float64(y * Float64(Float64(2.0 * x) / Float64(x - y)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x * ((y * 2.0) / (x - y));
	tmp = 0.0;
	if (y <= -1.5e-63)
		tmp = t_0;
	elseif (y <= 10000000000.0)
		tmp = y * ((2.0 * x) / (x - y));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x * N[(N[(y * 2.0), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.5e-63], t$95$0, If[LessEqual[y, 10000000000.0], N[(y * N[(N[(2.0 * x), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.4999999999999999e-63 or 1e10 < y

   1. Initial program 77.6%

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

      1. associate-*l* N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

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

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

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

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

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

      7. --lowering--.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   if -1.4999999999999999e-63 < y < 1e10

   1. Initial program 78.3%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

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

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

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

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

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

      7. --lowering--.f64 100.0

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

   4. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.5 \cdot 10^{-63}:\\ \;\;\;\;x \cdot \frac{y \cdot 2}{x - y}\\ \mathbf{elif}\;y \leq 10000000000:\\ \;\;\;\;y \cdot \frac{2 \cdot x}{x - y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y \cdot 2}{x - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{y \cdot 2}{x - y}\\ \mathbf{if}\;y \leq -1.8 \cdot 10^{-184}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 5.9 \cdot 10^{-195}:\\ \;\;\;\;y \cdot 2\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (/ (* y 2.0) (- x y)))))
   (if (<= y -1.8e-184) t_0 (if (<= y 5.9e-195) (* y 2.0) t_0))))

C

double code(double x, double y) {
	double t_0 = x * ((y * 2.0) / (x - y));
	double tmp;
	if (y <= -1.8e-184) {
		tmp = t_0;
	} else if (y <= 5.9e-195) {
		tmp = y * 2.0;
	} 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 * 2.0d0) / (x - y))
    if (y <= (-1.8d-184)) then
        tmp = t_0
    else if (y <= 5.9d-195) then
        tmp = y * 2.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x * ((y * 2.0) / (x - y));
	double tmp;
	if (y <= -1.8e-184) {
		tmp = t_0;
	} else if (y <= 5.9e-195) {
		tmp = y * 2.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x * ((y * 2.0) / (x - y))
	tmp = 0
	if y <= -1.8e-184:
		tmp = t_0
	elif y <= 5.9e-195:
		tmp = y * 2.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x * Float64(Float64(y * 2.0) / Float64(x - y)))
	tmp = 0.0
	if (y <= -1.8e-184)
		tmp = t_0;
	elseif (y <= 5.9e-195)
		tmp = Float64(y * 2.0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x * ((y * 2.0) / (x - y));
	tmp = 0.0;
	if (y <= -1.8e-184)
		tmp = t_0;
	elseif (y <= 5.9e-195)
		tmp = y * 2.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x * N[(N[(y * 2.0), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.8e-184], t$95$0, If[LessEqual[y, 5.9e-195], N[(y * 2.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.8000000000000001e-184 or 5.90000000000000006e-195 < y

   1. Initial program 78.4%

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

      1. associate-*l* N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

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

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

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

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

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

      7. --lowering--.f64 97.3

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

   4. Applied egg-rr 97.3%

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

   if -1.8000000000000001e-184 < y < 5.90000000000000006e-195

   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. *-commutative N/A

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

      2. *-lowering-*.f64 91.9

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

   5. Simplified 91.9%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{-184}:\\ \;\;\;\;x \cdot \frac{y \cdot 2}{x - y}\\ \mathbf{elif}\;y \leq 5.9 \cdot 10^{-195}:\\ \;\;\;\;y \cdot 2\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y \cdot 2}{x - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.9 \cdot 10^{-24}:\\ \;\;\;\;y \cdot 2\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{+93}:\\ \;\;\;\;-2 \cdot \mathsf{fma}\left(x, \frac{x}{y}, x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(y, \frac{2}{x}, 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -4.9e-24)
   (* y 2.0)
   (if (<= x 1.15e+93)
     (* -2.0 (fma x (/ x y) x))
     (* y (fma y (/ 2.0 x) 2.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -4.9e-24) {
		tmp = y * 2.0;
	} else if (x <= 1.15e+93) {
		tmp = -2.0 * fma(x, (x / y), x);
	} else {
		tmp = y * fma(y, (2.0 / x), 2.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -4.9e-24)
		tmp = Float64(y * 2.0);
	elseif (x <= 1.15e+93)
		tmp = Float64(-2.0 * fma(x, Float64(x / y), x));
	else
		tmp = Float64(y * fma(y, Float64(2.0 / x), 2.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -4.9e-24], N[(y * 2.0), $MachinePrecision], If[LessEqual[x, 1.15e+93], N[(-2.0 * N[(x * N[(x / y), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[(y * N[(y * N[(2.0 / x), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.9000000000000001e-24

   1. Initial program 79.7%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 72.2

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

   5. Simplified 72.2%

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

   if -4.9000000000000001e-24 < x < 1.1500000000000001e93

   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

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-/l* N/A

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

      7. unpow2 N/A

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

      8. distribute-rgt-out N/A

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

      9. +-commutative N/A

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

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

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. associate-/l* N/A

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

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

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

      15. /-lowering-/.f64 78.4

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

   5. Simplified 78.4%

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

   if 1.1500000000000001e93 < x

   1. Initial program 70.7%

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

   3. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. associate-*l/ N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. associate-*r/ N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

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

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

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

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

      13. associate-*r/ N/A

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

      14. metadata-eval N/A

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

      15. /-lowering-/.f64 83.7

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

   5. Simplified 83.7%

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

Alternative 5: 74.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{-29}:\\ \;\;\;\;y \cdot 2\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{+93}:\\ \;\;\;\;-2 \cdot \mathsf{fma}\left(x, \frac{x}{y}, x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -7.5e-29)
   (* y 2.0)
   (if (<= x 1.15e+93) (* -2.0 (fma x (/ x y) x)) (* y 2.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -7.5e-29) {
		tmp = y * 2.0;
	} else if (x <= 1.15e+93) {
		tmp = -2.0 * fma(x, (x / y), x);
	} else {
		tmp = y * 2.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -7.5e-29)
		tmp = Float64(y * 2.0);
	elseif (x <= 1.15e+93)
		tmp = Float64(-2.0 * fma(x, Float64(x / y), x));
	else
		tmp = Float64(y * 2.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -7.5e-29], N[(y * 2.0), $MachinePrecision], If[LessEqual[x, 1.15e+93], N[(-2.0 * N[(x * N[(x / y), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[(y * 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -7.50000000000000006e-29 or 1.1500000000000001e93 < x

   1. Initial program 76.0%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 76.4

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

   5. Simplified 76.4%

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

   if -7.50000000000000006e-29 < x < 1.1500000000000001e93

   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

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-/l* N/A

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

      7. unpow2 N/A

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

      8. distribute-rgt-out N/A

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

      9. +-commutative N/A

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

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

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. associate-/l* N/A

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

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

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

      15. /-lowering-/.f64 78.4

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

   5. Simplified 78.4%

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

Alternative 6: 74.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{-26}:\\ \;\;\;\;y \cdot 2\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{+92}:\\ \;\;\;\;x \cdot -2\\ \mathbf{else}:\\ \;\;\;\;y \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -6.2e-26) (* y 2.0) (if (<= x 5.8e+92) (* x -2.0) (* y 2.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -6.2e-26) {
		tmp = y * 2.0;
	} else if (x <= 5.8e+92) {
		tmp = x * -2.0;
	} else {
		tmp = y * 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 <= (-6.2d-26)) then
        tmp = y * 2.0d0
    else if (x <= 5.8d+92) then
        tmp = x * (-2.0d0)
    else
        tmp = y * 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -6.2e-26) {
		tmp = y * 2.0;
	} else if (x <= 5.8e+92) {
		tmp = x * -2.0;
	} else {
		tmp = y * 2.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -6.2e-26:
		tmp = y * 2.0
	elif x <= 5.8e+92:
		tmp = x * -2.0
	else:
		tmp = y * 2.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -6.2e-26)
		tmp = Float64(y * 2.0);
	elseif (x <= 5.8e+92)
		tmp = Float64(x * -2.0);
	else
		tmp = Float64(y * 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -6.2e-26)
		tmp = y * 2.0;
	elseif (x <= 5.8e+92)
		tmp = x * -2.0;
	else
		tmp = y * 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -6.2e-26], N[(y * 2.0), $MachinePrecision], If[LessEqual[x, 5.8e+92], N[(x * -2.0), $MachinePrecision], N[(y * 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -6.19999999999999966e-26 or 5.8000000000000001e92 < x

   1. Initial program 76.0%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 76.4

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

   5. Simplified 76.4%

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

   if -6.19999999999999966e-26 < x < 5.8000000000000001e92

   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. *-lowering-*.f64 77.3

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

   5. Simplified 77.3%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{-26}:\\ \;\;\;\;y \cdot 2\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{+92}:\\ \;\;\;\;x \cdot -2\\ \mathbf{else}:\\ \;\;\;\;y \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 50.5% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.9%

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

3. Taylor expanded in x around 0

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

   1. *-lowering-*.f64 52.5

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

5. Simplified 52.5%

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

6. Final simplification 52.5%

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

Developer Target 1: 99.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2 \cdot x}{x - y} \cdot y\\ \mathbf{if}\;x < -1.7210442634149447 \cdot 10^{+81}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x < 83645045635564430:\\ \;\;\;\;\frac{x \cdot 2}{\frac{x - y}{y}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (/ (* 2.0 x) (- x y)) y)))
   (if (< x -1.7210442634149447e+81)
     t_0
     (if (< x 83645045635564430.0) (/ (* x 2.0) (/ (- x y) y)) t_0))))

C

double code(double x, double y) {
	double t_0 = ((2.0 * x) / (x - y)) * y;
	double tmp;
	if (x < -1.7210442634149447e+81) {
		tmp = t_0;
	} else if (x < 83645045635564430.0) {
		tmp = (x * 2.0) / ((x - y) / y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((2.0d0 * x) / (x - y)) * y
    if (x < (-1.7210442634149447d+81)) then
        tmp = t_0
    else if (x < 83645045635564430.0d0) then
        tmp = (x * 2.0d0) / ((x - y) / y)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = ((2.0 * x) / (x - y)) * y;
	double tmp;
	if (x < -1.7210442634149447e+81) {
		tmp = t_0;
	} else if (x < 83645045635564430.0) {
		tmp = (x * 2.0) / ((x - y) / y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = ((2.0 * x) / (x - y)) * y
	tmp = 0
	if x < -1.7210442634149447e+81:
		tmp = t_0
	elif x < 83645045635564430.0:
		tmp = (x * 2.0) / ((x - y) / y)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(2.0 * x) / Float64(x - y)) * y)
	tmp = 0.0
	if (x < -1.7210442634149447e+81)
		tmp = t_0;
	elseif (x < 83645045635564430.0)
		tmp = Float64(Float64(x * 2.0) / Float64(Float64(x - y) / y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = ((2.0 * x) / (x - y)) * y;
	tmp = 0.0;
	if (x < -1.7210442634149447e+81)
		tmp = t_0;
	elseif (x < 83645045635564430.0)
		tmp = (x * 2.0) / ((x - y) / y);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(2.0 * x), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]}, If[Less[x, -1.7210442634149447e+81], t$95$0, If[Less[x, 83645045635564430.0], N[(N[(x * 2.0), $MachinePrecision] / N[(N[(x - y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Reproduce

herbie shell --seed 2024204 
(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)))