Linear.Projection:perspective from linear-1.19.1.3, B

Percentage Accurate: 76.9% → 99.0%

Time: 4.2s

Alternatives: 6

Speedup: 1.0×

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: 76.9% 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: 99.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{-106} \lor \neg \left(x \leq 6.8 \cdot 10^{-36}\right):\\ \;\;\;\;y \cdot \frac{2 \cdot x}{x - y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{2}{\frac{x - y}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.2e-106) (not (<= x 6.8e-36)))
   (* y (/ (* 2.0 x) (- x y)))
   (* x (/ 2.0 (/ (- x y) y)))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.2e-106) || !(x <= 6.8e-36)) {
		tmp = y * ((2.0 * x) / (x - y));
	} else {
		tmp = x * (2.0 / ((x - y) / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-1.2d-106)) .or. (.not. (x <= 6.8d-36))) then
        tmp = y * ((2.0d0 * x) / (x - y))
    else
        tmp = x * (2.0d0 / ((x - y) / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -1.2e-106) || !(x <= 6.8e-36)) {
		tmp = y * ((2.0 * x) / (x - y));
	} else {
		tmp = x * (2.0 / ((x - y) / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.2e-106) or not (x <= 6.8e-36):
		tmp = y * ((2.0 * x) / (x - y))
	else:
		tmp = x * (2.0 / ((x - y) / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.2e-106) || !(x <= 6.8e-36))
		tmp = Float64(y * Float64(Float64(2.0 * x) / Float64(x - y)));
	else
		tmp = Float64(x * Float64(2.0 / Float64(Float64(x - y) / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -1.2e-106) || ~((x <= 6.8e-36)))
		tmp = y * ((2.0 * x) / (x - y));
	else
		tmp = x * (2.0 / ((x - y) / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.2e-106], N[Not[LessEqual[x, 6.8e-36]], $MachinePrecision]], N[(y * N[(N[(2.0 * x), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(2.0 / N[(N[(x - y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.1999999999999999e-106 or 6.8000000000000005e-36 < x

   1. Initial program 79.7%

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

      1. associate-*l/ 99.9%

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

   3. Simplified 99.9%

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

   if -1.1999999999999999e-106 < x < 6.8000000000000005e-36

   1. Initial program 66.9%

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

      1. associate-/l* 99.9%

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

      2. associate-*r/ 99.9%

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

   3. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{-106} \lor \neg \left(x \leq 6.8 \cdot 10^{-36}\right):\\ \;\;\;\;y \cdot \frac{2 \cdot x}{x - y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{2}{\frac{x - y}{y}}\\ \end{array} \]

Alternative 2: 93.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.8 \cdot 10^{-207} \lor \neg \left(y \leq 5.6 \cdot 10^{-173}\right):\\ \;\;\;\;x \cdot \frac{2}{\frac{x - y}{y}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -5.8e-207) (not (<= y 5.6e-173)))
   (* x (/ 2.0 (/ (- x y) y)))
   (* 2.0 y)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -5.8e-207) || !(y <= 5.6e-173)) {
		tmp = x * (2.0 / ((x - y) / y));
	} else {
		tmp = 2.0 * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-5.8d-207)) .or. (.not. (y <= 5.6d-173))) then
        tmp = x * (2.0d0 / ((x - y) / y))
    else
        tmp = 2.0d0 * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -5.8e-207) || !(y <= 5.6e-173)) {
		tmp = x * (2.0 / ((x - y) / y));
	} else {
		tmp = 2.0 * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -5.8e-207) or not (y <= 5.6e-173):
		tmp = x * (2.0 / ((x - y) / y))
	else:
		tmp = 2.0 * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -5.8e-207) || !(y <= 5.6e-173))
		tmp = Float64(x * Float64(2.0 / Float64(Float64(x - y) / y)));
	else
		tmp = Float64(2.0 * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -5.8e-207) || ~((y <= 5.6e-173)))
		tmp = x * (2.0 / ((x - y) / y));
	else
		tmp = 2.0 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -5.8e-207], N[Not[LessEqual[y, 5.6e-173]], $MachinePrecision]], N[(x * N[(2.0 / N[(N[(x - y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5.80000000000000022e-207 or 5.5999999999999998e-173 < y

   1. Initial program 76.7%

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

      1. associate-/l* 95.1%

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

      2. associate-*r/ 94.9%

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

   3. Simplified 94.9%

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

   if -5.80000000000000022e-207 < y < 5.5999999999999998e-173

   1. Initial program 69.7%

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

      1. associate-/l* 50.4%

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

      2. associate-*r/ 50.3%

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

   3. Simplified 50.3%

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

   4. Taylor expanded in x around inf 92.5%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.8 \cdot 10^{-207} \lor \neg \left(y \leq 5.6 \cdot 10^{-173}\right):\\ \;\;\;\;x \cdot \frac{2}{\frac{x - y}{y}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot y\\ \end{array} \]

Alternative 3: 73.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{-109}:\\ \;\;\;\;2 \cdot y\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{+14}:\\ \;\;\;\;-2 \cdot \left(x + \frac{x \cdot x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -3.6e-109)
   (* 2.0 y)
   (if (<= x 2.1e+14) (* -2.0 (+ x (/ (* x x) y))) (* 2.0 y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -3.6e-109) {
		tmp = 2.0 * y;
	} else if (x <= 2.1e+14) {
		tmp = -2.0 * (x + ((x * x) / y));
	} else {
		tmp = 2.0 * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-3.6d-109)) then
        tmp = 2.0d0 * y
    else if (x <= 2.1d+14) then
        tmp = (-2.0d0) * (x + ((x * x) / y))
    else
        tmp = 2.0d0 * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -3.6e-109) {
		tmp = 2.0 * y;
	} else if (x <= 2.1e+14) {
		tmp = -2.0 * (x + ((x * x) / y));
	} else {
		tmp = 2.0 * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -3.6e-109:
		tmp = 2.0 * y
	elif x <= 2.1e+14:
		tmp = -2.0 * (x + ((x * x) / y))
	else:
		tmp = 2.0 * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -3.6e-109)
		tmp = Float64(2.0 * y);
	elseif (x <= 2.1e+14)
		tmp = Float64(-2.0 * Float64(x + Float64(Float64(x * x) / y)));
	else
		tmp = Float64(2.0 * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -3.6e-109)
		tmp = 2.0 * y;
	elseif (x <= 2.1e+14)
		tmp = -2.0 * (x + ((x * x) / y));
	else
		tmp = 2.0 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -3.6e-109], N[(2.0 * y), $MachinePrecision], If[LessEqual[x, 2.1e+14], N[(-2.0 * N[(x + N[(N[(x * x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.6000000000000001e-109 or 2.1e14 < x

   1. Initial program 78.3%

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

      1. associate-/l* 74.1%

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

      2. associate-*r/ 73.9%

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

   3. Simplified 73.9%

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

   4. Taylor expanded in x around inf 78.2%

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

   if -3.6000000000000001e-109 < x < 2.1e14

   1. Initial program 70.4%

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

      1. associate-/l* 99.9%

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

      2. associate-*r/ 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 80.2%

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

      1. distribute-lft-out 80.2%

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

      2. unpow2 80.2%

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

   6. Simplified 80.2%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{-109}:\\ \;\;\;\;2 \cdot y\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{+14}:\\ \;\;\;\;-2 \cdot \left(x + \frac{x \cdot x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot y\\ \end{array} \]

Alternative 4: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\frac{1}{y} - \frac{1}{x}} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ 2.0 (- (/ 1.0 y) (/ 1.0 x))))

C

double code(double x, double y) {
	return 2.0 / ((1.0 / y) - (1.0 / x));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 2.0d0 / ((1.0d0 / y) - (1.0d0 / x))
end function

Java

public static double code(double x, double y) {
	return 2.0 / ((1.0 / y) - (1.0 / x));
}

Python

def code(x, y):
	return 2.0 / ((1.0 / y) - (1.0 / x))

Julia

function code(x, y)
	return Float64(2.0 / Float64(Float64(1.0 / y) - Float64(1.0 / x)))
end

MATLAB

function tmp = code(x, y)
	tmp = 2.0 / ((1.0 / y) - (1.0 / x));
end

Wolfram

code[x_, y_] := N[(2.0 / N[(N[(1.0 / y), $MachinePrecision] - N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.1%

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

   1. associate-/l* 84.6%

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

   2. associate-*r/ 84.5%

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

3. Simplified 84.5%

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

   1. associate-*r/ 84.6%

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

   2. *-commutative 84.6%

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

   3. associate-/l* 84.4%

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

   4. div-sub 84.4%

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

   5. *-inverses 84.4%

      \[\leadsto \frac{2}{\frac{\frac{x}{y} - \color{blue}{1}}{x}} \]

   6. sub-neg 84.4%

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

   7. metadata-eval 84.4%

      \[\leadsto \frac{2}{\frac{\frac{x}{y} + \color{blue}{-1}}{x}} \]

5. Applied egg-rr 84.4%

   \[\leadsto \color{blue}{\frac{2}{\frac{\frac{x}{y} + -1}{x}}} \]

6. Taylor expanded in x around 0 99.7%

   \[\leadsto \frac{2}{\color{blue}{\frac{1}{y} - \frac{1}{x}}} \]

7. Final simplification 99.7%

   \[\leadsto \frac{2}{\frac{1}{y} - \frac{1}{x}} \]

Alternative 5: 73.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.55 \cdot 10^{-106}:\\ \;\;\;\;2 \cdot y\\ \mathbf{elif}\;x \leq 8 \cdot 10^{+21}:\\ \;\;\;\;x \cdot -2\\ \mathbf{else}:\\ \;\;\;\;2 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.55e-106) (* 2.0 y) (if (<= x 8e+21) (* x -2.0) (* 2.0 y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.55e-106) {
		tmp = 2.0 * y;
	} else if (x <= 8e+21) {
		tmp = x * -2.0;
	} else {
		tmp = 2.0 * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.55d-106)) then
        tmp = 2.0d0 * y
    else if (x <= 8d+21) then
        tmp = x * (-2.0d0)
    else
        tmp = 2.0d0 * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.55e-106) {
		tmp = 2.0 * y;
	} else if (x <= 8e+21) {
		tmp = x * -2.0;
	} else {
		tmp = 2.0 * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.55e-106:
		tmp = 2.0 * y
	elif x <= 8e+21:
		tmp = x * -2.0
	else:
		tmp = 2.0 * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.55e-106)
		tmp = Float64(2.0 * y);
	elseif (x <= 8e+21)
		tmp = Float64(x * -2.0);
	else
		tmp = Float64(2.0 * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.55e-106)
		tmp = 2.0 * y;
	elseif (x <= 8e+21)
		tmp = x * -2.0;
	else
		tmp = 2.0 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.55e-106], N[(2.0 * y), $MachinePrecision], If[LessEqual[x, 8e+21], N[(x * -2.0), $MachinePrecision], N[(2.0 * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.54999999999999993e-106 or 8e21 < x

   1. Initial program 78.3%

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

      1. associate-/l* 74.1%

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

      2. associate-*r/ 73.9%

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

   3. Simplified 73.9%

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

   4. Taylor expanded in x around inf 78.2%

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

   if -1.54999999999999993e-106 < x < 8e21

   1. Initial program 70.4%

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

      1. associate-/l* 99.9%

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

      2. associate-*r/ 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 80.2%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.55 \cdot 10^{-106}:\\ \;\;\;\;2 \cdot y\\ \mathbf{elif}\;x \leq 8 \cdot 10^{+21}:\\ \;\;\;\;x \cdot -2\\ \mathbf{else}:\\ \;\;\;\;2 \cdot y\\ \end{array} \]

Alternative 6: 50.9% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ 2 \cdot y \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.1%

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

   1. associate-/l* 84.6%

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

   2. associate-*r/ 84.5%

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

3. Simplified 84.5%

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

4. Taylor expanded in x around inf 54.9%

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

5. Final simplification 54.9%

   \[\leadsto 2 \cdot y \]

Developer target: 99.4% accurate, 0.7× 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 2023200 
(FPCore (x y)
  :name "Linear.Projection:perspective from linear-1.19.1.3, B"
  :precision binary64

  :herbie-target
  (if (< x -1.7210442634149447e+81) (* (/ (* 2.0 x) (- x y)) y) (if (< x 83645045635564430.0) (/ (* x 2.0) (/ (- x y) y)) (* (/ (* 2.0 x) (- x y)) y)))

  (/ (* (* x 2.0) y) (- x y)))