Linear.Projection:perspective from linear-1.19.1.3, A

Percentage Accurate: 100.0% → 100.0%

Time: 9.3s

Alternatives: 9

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

   \[\frac{x + y}{x - y} \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 2: 75.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{-36} \lor \neg \left(y \leq 2.3 \cdot 10^{+27}\right):\\ \;\;\;\;-2 \cdot \frac{x}{y} + -1\\ \mathbf{else}:\\ \;\;\;\;1 + 2 \cdot \frac{y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -8e-36) (not (<= y 2.3e+27)))
   (+ (* -2.0 (/ x y)) -1.0)
   (+ 1.0 (* 2.0 (/ y x)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -8e-36) || !(y <= 2.3e+27)) {
		tmp = (-2.0 * (x / y)) + -1.0;
	} else {
		tmp = 1.0 + (2.0 * (y / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-8d-36)) .or. (.not. (y <= 2.3d+27))) then
        tmp = ((-2.0d0) * (x / y)) + (-1.0d0)
    else
        tmp = 1.0d0 + (2.0d0 * (y / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -8e-36) || !(y <= 2.3e+27)) {
		tmp = (-2.0 * (x / y)) + -1.0;
	} else {
		tmp = 1.0 + (2.0 * (y / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -8e-36) or not (y <= 2.3e+27):
		tmp = (-2.0 * (x / y)) + -1.0
	else:
		tmp = 1.0 + (2.0 * (y / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -8e-36) || !(y <= 2.3e+27))
		tmp = Float64(Float64(-2.0 * Float64(x / y)) + -1.0);
	else
		tmp = Float64(1.0 + Float64(2.0 * Float64(y / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -8e-36) || ~((y <= 2.3e+27)))
		tmp = (-2.0 * (x / y)) + -1.0;
	else
		tmp = 1.0 + (2.0 * (y / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -8e-36], N[Not[LessEqual[y, 2.3e+27]], $MachinePrecision]], N[(N[(-2.0 * N[(x / y), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(1.0 + N[(2.0 * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -7.9999999999999995e-36 or 2.3000000000000001e27 < y

   1. Initial program 99.9%

      \[\frac{x + y}{x - y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 74.6%

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

   if -7.9999999999999995e-36 < y < 2.3000000000000001e27

   1. Initial program 99.9%

      \[\frac{x + y}{x - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 78.6%

      \[\leadsto \color{blue}{1 + 2 \cdot \frac{y}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{-36} \lor \neg \left(y \leq 2.3 \cdot 10^{+27}\right):\\ \;\;\;\;-2 \cdot \frac{x}{y} + -1\\ \mathbf{else}:\\ \;\;\;\;1 + 2 \cdot \frac{y}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{-36}:\\ \;\;\;\;-1 - \frac{x}{y}\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{+26}:\\ \;\;\;\;1 + 2 \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.5e-36)
   (- -1.0 (/ x y))
   (if (<= y 2.1e+26) (+ 1.0 (* 2.0 (/ y x))) (/ y (- x y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.5e-36) {
		tmp = -1.0 - (x / y);
	} else if (y <= 2.1e+26) {
		tmp = 1.0 + (2.0 * (y / x));
	} else {
		tmp = y / (x - 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 <= (-2.5d-36)) then
        tmp = (-1.0d0) - (x / y)
    else if (y <= 2.1d+26) then
        tmp = 1.0d0 + (2.0d0 * (y / x))
    else
        tmp = y / (x - y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2.5e-36) {
		tmp = -1.0 - (x / y);
	} else if (y <= 2.1e+26) {
		tmp = 1.0 + (2.0 * (y / x));
	} else {
		tmp = y / (x - y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.5e-36:
		tmp = -1.0 - (x / y)
	elif y <= 2.1e+26:
		tmp = 1.0 + (2.0 * (y / x))
	else:
		tmp = y / (x - y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.5e-36)
		tmp = Float64(-1.0 - Float64(x / y));
	elseif (y <= 2.1e+26)
		tmp = Float64(1.0 + Float64(2.0 * Float64(y / x)));
	else
		tmp = Float64(y / Float64(x - y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.5e-36)
		tmp = -1.0 - (x / y);
	elseif (y <= 2.1e+26)
		tmp = 1.0 + (2.0 * (y / x));
	else
		tmp = y / (x - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.5e-36], N[(-1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.1e+26], N[(1.0 + N[(2.0 * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y / N[(x - y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.50000000000000002e-36

   1. Initial program 99.9%

      \[\frac{x + y}{x - y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 71.6%

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

   4. Taylor expanded in y around inf 71.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{y} - 1} \]
   5. Step-by-step derivation

      1. mul-1-neg 71.8%

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

      2. neg-sub0 71.8%

         \[\leadsto \color{blue}{\left(0 - \frac{x}{y}\right)} - 1 \]

      3. associate--r+ 71.8%

         \[\leadsto \color{blue}{0 - \left(\frac{x}{y} + 1\right)} \]

      4. +-commutative 71.8%

         \[\leadsto 0 - \color{blue}{\left(1 + \frac{x}{y}\right)} \]

      5. associate--r+ 71.8%

         \[\leadsto \color{blue}{\left(0 - 1\right) - \frac{x}{y}} \]

      6. metadata-eval 71.8%

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

   6. Simplified 71.8%

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

   if -2.50000000000000002e-36 < y < 2.1000000000000001e26

   1. Initial program 99.9%

      \[\frac{x + y}{x - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 78.6%

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

   if 2.1000000000000001e26 < y

   1. Initial program 99.8%

      \[\frac{x + y}{x - y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 75.5%

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

Alternative 4: 75.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.3 \cdot 10^{-36} \lor \neg \left(y \leq 3.7 \cdot 10^{+28}\right):\\ \;\;\;\;-1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -5.3e-36) (not (<= y 3.7e+28))) (- -1.0 (/ x y)) (/ x (- x y))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -5.3e-36) || !(y <= 3.7e+28)) {
		tmp = -1.0 - (x / y);
	} else {
		tmp = x / (x - 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 <= (-5.3d-36)) .or. (.not. (y <= 3.7d+28))) then
        tmp = (-1.0d0) - (x / y)
    else
        tmp = x / (x - y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -5.3e-36) || !(y <= 3.7e+28)) {
		tmp = -1.0 - (x / y);
	} else {
		tmp = x / (x - y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -5.3e-36) or not (y <= 3.7e+28):
		tmp = -1.0 - (x / y)
	else:
		tmp = x / (x - y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -5.3e-36) || !(y <= 3.7e+28))
		tmp = Float64(-1.0 - Float64(x / y));
	else
		tmp = Float64(x / Float64(x - y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -5.3e-36) || ~((y <= 3.7e+28)))
		tmp = -1.0 - (x / y);
	else
		tmp = x / (x - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -5.3e-36], N[Not[LessEqual[y, 3.7e+28]], $MachinePrecision]], N[(-1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision], N[(x / N[(x - y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5.2999999999999998e-36 or 3.6999999999999999e28 < y

   1. Initial program 99.9%

      \[\frac{x + y}{x - y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 73.5%

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

   4. Taylor expanded in y around inf 73.6%

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{y} - 1} \]
   5. Step-by-step derivation

      1. mul-1-neg 73.6%

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

      2. neg-sub0 73.6%

         \[\leadsto \color{blue}{\left(0 - \frac{x}{y}\right)} - 1 \]

      3. associate--r+ 73.6%

         \[\leadsto \color{blue}{0 - \left(\frac{x}{y} + 1\right)} \]

      4. +-commutative 73.6%

         \[\leadsto 0 - \color{blue}{\left(1 + \frac{x}{y}\right)} \]

      5. associate--r+ 73.6%

         \[\leadsto \color{blue}{\left(0 - 1\right) - \frac{x}{y}} \]

      6. metadata-eval 73.6%

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

   6. Simplified 73.6%

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

   if -5.2999999999999998e-36 < y < 3.6999999999999999e28

   1. Initial program 99.9%

      \[\frac{x + y}{x - y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 78.3%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.3 \cdot 10^{-36} \lor \neg \left(y \leq 3.7 \cdot 10^{+28}\right):\\ \;\;\;\;-1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{-36} \lor \neg \left(y \leq 7.8 \cdot 10^{+28}\right):\\ \;\;\;\;-1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -2.1e-36) (not (<= y 7.8e+28)))
   (- -1.0 (/ x y))
   (+ 1.0 (/ y x))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -2.1e-36) || !(y <= 7.8e+28)) {
		tmp = -1.0 - (x / y);
	} else {
		tmp = 1.0 + (y / x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-2.1d-36)) .or. (.not. (y <= 7.8d+28))) then
        tmp = (-1.0d0) - (x / y)
    else
        tmp = 1.0d0 + (y / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -2.1e-36) || !(y <= 7.8e+28)) {
		tmp = -1.0 - (x / y);
	} else {
		tmp = 1.0 + (y / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -2.1e-36) or not (y <= 7.8e+28):
		tmp = -1.0 - (x / y)
	else:
		tmp = 1.0 + (y / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -2.1e-36) || !(y <= 7.8e+28))
		tmp = Float64(-1.0 - Float64(x / y));
	else
		tmp = Float64(1.0 + Float64(y / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -2.1e-36) || ~((y <= 7.8e+28)))
		tmp = -1.0 - (x / y);
	else
		tmp = 1.0 + (y / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -2.1e-36], N[Not[LessEqual[y, 7.8e+28]], $MachinePrecision]], N[(-1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(y / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.09999999999999991e-36 or 7.7999999999999997e28 < y

   1. Initial program 99.9%

      \[\frac{x + y}{x - y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 73.5%

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

   4. Taylor expanded in y around inf 73.6%

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{y} - 1} \]
   5. Step-by-step derivation

      1. mul-1-neg 73.6%

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

      2. neg-sub0 73.6%

         \[\leadsto \color{blue}{\left(0 - \frac{x}{y}\right)} - 1 \]

      3. associate--r+ 73.6%

         \[\leadsto \color{blue}{0 - \left(\frac{x}{y} + 1\right)} \]

      4. +-commutative 73.6%

         \[\leadsto 0 - \color{blue}{\left(1 + \frac{x}{y}\right)} \]

      5. associate--r+ 73.6%

         \[\leadsto \color{blue}{\left(0 - 1\right) - \frac{x}{y}} \]

      6. metadata-eval 73.6%

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

   6. Simplified 73.6%

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

   if -2.09999999999999991e-36 < y < 7.7999999999999997e28

   1. Initial program 99.9%

      \[\frac{x + y}{x - y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 78.3%

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

   4. Taylor expanded in x around inf 78.1%

      \[\leadsto \color{blue}{1 + \frac{y}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{-36} \lor \neg \left(y \leq 7.8 \cdot 10^{+28}\right):\\ \;\;\;\;-1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{y}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9.2 \cdot 10^{-39}:\\ \;\;\;\;-1 - \frac{x}{y}\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{+26}:\\ \;\;\;\;\frac{x}{x - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -9.2e-39)
   (- -1.0 (/ x y))
   (if (<= y 1.4e+26) (/ x (- x y)) (/ y (- x y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -9.2e-39) {
		tmp = -1.0 - (x / y);
	} else if (y <= 1.4e+26) {
		tmp = x / (x - y);
	} else {
		tmp = y / (x - 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 <= (-9.2d-39)) then
        tmp = (-1.0d0) - (x / y)
    else if (y <= 1.4d+26) then
        tmp = x / (x - y)
    else
        tmp = y / (x - y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -9.2e-39) {
		tmp = -1.0 - (x / y);
	} else if (y <= 1.4e+26) {
		tmp = x / (x - y);
	} else {
		tmp = y / (x - y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -9.2e-39:
		tmp = -1.0 - (x / y)
	elif y <= 1.4e+26:
		tmp = x / (x - y)
	else:
		tmp = y / (x - y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -9.2e-39)
		tmp = Float64(-1.0 - Float64(x / y));
	elseif (y <= 1.4e+26)
		tmp = Float64(x / Float64(x - y));
	else
		tmp = Float64(y / Float64(x - y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -9.2e-39)
		tmp = -1.0 - (x / y);
	elseif (y <= 1.4e+26)
		tmp = x / (x - y);
	else
		tmp = y / (x - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -9.2e-39], N[(-1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.4e+26], N[(x / N[(x - y), $MachinePrecision]), $MachinePrecision], N[(y / N[(x - y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -9.20000000000000033e-39

   1. Initial program 99.9%

      \[\frac{x + y}{x - y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 71.6%

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

   4. Taylor expanded in y around inf 71.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{y} - 1} \]
   5. Step-by-step derivation

      1. mul-1-neg 71.8%

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

      2. neg-sub0 71.8%

         \[\leadsto \color{blue}{\left(0 - \frac{x}{y}\right)} - 1 \]

      3. associate--r+ 71.8%

         \[\leadsto \color{blue}{0 - \left(\frac{x}{y} + 1\right)} \]

      4. +-commutative 71.8%

         \[\leadsto 0 - \color{blue}{\left(1 + \frac{x}{y}\right)} \]

      5. associate--r+ 71.8%

         \[\leadsto \color{blue}{\left(0 - 1\right) - \frac{x}{y}} \]

      6. metadata-eval 71.8%

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

   6. Simplified 71.8%

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

   if -9.20000000000000033e-39 < y < 1.4e26

   1. Initial program 99.9%

      \[\frac{x + y}{x - y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 78.3%

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

   if 1.4e26 < y

   1. Initial program 99.8%

      \[\frac{x + y}{x - y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 75.5%

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

Alternative 7: 74.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{-35}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{+27}:\\ \;\;\;\;1 + \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.8e-35) -1.0 (if (<= y 8.5e+27) (+ 1.0 (/ y x)) -1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.8e-35) {
		tmp = -1.0;
	} else if (y <= 8.5e+27) {
		tmp = 1.0 + (y / x);
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.8d-35)) then
        tmp = -1.0d0
    else if (y <= 8.5d+27) then
        tmp = 1.0d0 + (y / x)
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.8e-35) {
		tmp = -1.0;
	} else if (y <= 8.5e+27) {
		tmp = 1.0 + (y / x);
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.8e-35:
		tmp = -1.0
	elif y <= 8.5e+27:
		tmp = 1.0 + (y / x)
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.8e-35)
		tmp = -1.0;
	elseif (y <= 8.5e+27)
		tmp = Float64(1.0 + Float64(y / x));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.8e-35)
		tmp = -1.0;
	elseif (y <= 8.5e+27)
		tmp = 1.0 + (y / x);
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.8e-35], -1.0, If[LessEqual[y, 8.5e+27], N[(1.0 + N[(y / x), $MachinePrecision]), $MachinePrecision], -1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -1.80000000000000009e-35 or 8.5e27 < y

   1. Initial program 99.9%

      \[\frac{x + y}{x - y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 72.8%

      \[\leadsto \color{blue}{-1} \]

   if -1.80000000000000009e-35 < y < 8.5e27

   1. Initial program 99.9%

      \[\frac{x + y}{x - y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 78.3%

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

   4. Taylor expanded in x around inf 78.1%

      \[\leadsto \color{blue}{1 + \frac{y}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 74.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{-38}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{+28}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -6.5e-38) -1.0 (if (<= y 3.5e+28) 1.0 -1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -6.5e-38) {
		tmp = -1.0;
	} else if (y <= 3.5e+28) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-6.5d-38)) then
        tmp = -1.0d0
    else if (y <= 3.5d+28) then
        tmp = 1.0d0
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -6.5e-38) {
		tmp = -1.0;
	} else if (y <= 3.5e+28) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -6.5e-38:
		tmp = -1.0
	elif y <= 3.5e+28:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -6.5e-38)
		tmp = -1.0;
	elseif (y <= 3.5e+28)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -6.5e-38)
		tmp = -1.0;
	elseif (y <= 3.5e+28)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -6.5e-38], -1.0, If[LessEqual[y, 3.5e+28], 1.0, -1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -6.49999999999999949e-38 or 3.5e28 < y

   1. Initial program 99.9%

      \[\frac{x + y}{x - y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 72.8%

      \[\leadsto \color{blue}{-1} \]

   if -6.49999999999999949e-38 < y < 3.5e28

   1. Initial program 99.9%

      \[\frac{x + y}{x - y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 77.5%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 49.9% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return -1.0

Julia

function code(x, y)
	return -1.0
end

MATLAB

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

Wolfram

code[x_, y_] := -1.0

Derivation

1. Initial program 99.9%

   \[\frac{x + y}{x - y} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 45.9%

   \[\leadsto \color{blue}{-1} \]
4. Add Preprocessing

Developer target: 100.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024110 
(FPCore (x y)
  :name "Linear.Projection:perspective from linear-1.19.1.3, A"
  :precision binary64

  :alt
  (/ 1.0 (- (/ x (+ x y)) (/ y (+ x y))))

  (/ (+ x y) (- x y)))