Linear.Projection:perspective from linear-1.19.1.3, A

Percentage Accurate: 100.0% → 100.0%

Time: 5.6s

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 100.0%

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

Alternative 2: 73.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.65 \cdot 10^{-12} \lor \neg \left(y \leq 5 \cdot 10^{-103}\right):\\ \;\;\;\;-2 \cdot \frac{x}{y} + -1\\ \mathbf{else}:\\ \;\;\;\;1 + 2 \cdot \frac{y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.65e-12) (not (<= y 5e-103)))
   (+ (* -2.0 (/ x y)) -1.0)
   (+ 1.0 (* 2.0 (/ y x)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.65e-12) || !(y <= 5e-103)) {
		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 <= (-1.65d-12)) .or. (.not. (y <= 5d-103))) 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 <= -1.65e-12) || !(y <= 5e-103)) {
		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 <= -1.65e-12) or not (y <= 5e-103):
		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 <= -1.65e-12) || !(y <= 5e-103))
		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 <= -1.65e-12) || ~((y <= 5e-103)))
		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, -1.65e-12], N[Not[LessEqual[y, 5e-103]], $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 < -1.65e-12 or 4.99999999999999966e-103 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 79.5%

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

   if -1.65e-12 < y < 4.99999999999999966e-103

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 79.4%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.65 \cdot 10^{-12} \lor \neg \left(y \leq 5 \cdot 10^{-103}\right):\\ \;\;\;\;-2 \cdot \frac{x}{y} + -1\\ \mathbf{else}:\\ \;\;\;\;1 + 2 \cdot \frac{y}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{-12}:\\ \;\;\;\;\frac{x + y}{-y}\\ \mathbf{elif}\;y \leq 6.8 \cdot 10^{-106}:\\ \;\;\;\;1 + 2 \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;-1 - \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.2e-12)
   (/ (+ x y) (- y))
   (if (<= y 6.8e-106) (+ 1.0 (* 2.0 (/ y x))) (- -1.0 (/ x y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.2e-12) {
		tmp = (x + y) / -y;
	} else if (y <= 6.8e-106) {
		tmp = 1.0 + (2.0 * (y / x));
	} else {
		tmp = -1.0 - (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 <= (-2.2d-12)) then
        tmp = (x + y) / -y
    else if (y <= 6.8d-106) then
        tmp = 1.0d0 + (2.0d0 * (y / x))
    else
        tmp = (-1.0d0) - (x / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2.2e-12) {
		tmp = (x + y) / -y;
	} else if (y <= 6.8e-106) {
		tmp = 1.0 + (2.0 * (y / x));
	} else {
		tmp = -1.0 - (x / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.2e-12:
		tmp = (x + y) / -y
	elif y <= 6.8e-106:
		tmp = 1.0 + (2.0 * (y / x))
	else:
		tmp = -1.0 - (x / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.2e-12)
		tmp = Float64(Float64(x + y) / Float64(-y));
	elseif (y <= 6.8e-106)
		tmp = Float64(1.0 + Float64(2.0 * Float64(y / x)));
	else
		tmp = Float64(-1.0 - Float64(x / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.2e-12)
		tmp = (x + y) / -y;
	elseif (y <= 6.8e-106)
		tmp = 1.0 + (2.0 * (y / x));
	else
		tmp = -1.0 - (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.2e-12], N[(N[(x + y), $MachinePrecision] / (-y)), $MachinePrecision], If[LessEqual[y, 6.8e-106], N[(1.0 + N[(2.0 * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.19999999999999992e-12

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 85.0%

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

      1. neg-mul-1 85.0%

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

   5. Simplified 85.0%

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

   if -2.19999999999999992e-12 < y < 6.79999999999999965e-106

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 79.4%

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

   if 6.79999999999999965e-106 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 73.0%

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

   4. Taylor expanded in y around inf 73.4%

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

      1. sub-neg 73.4%

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

      2. metadata-eval 73.4%

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

      3. +-commutative 73.4%

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

      4. mul-1-neg 73.4%

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

      5. unsub-neg 73.4%

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

   6. Simplified 73.4%

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

Alternative 4: 73.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{-12} \lor \neg \left(y \leq 8 \cdot 10^{-104}\right):\\ \;\;\;\;-1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.45e-12) (not (<= y 8e-104))) (- -1.0 (/ x y)) (/ x (- x y))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.45e-12) || !(y <= 8e-104)) {
		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 <= (-1.45d-12)) .or. (.not. (y <= 8d-104))) 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 <= -1.45e-12) || !(y <= 8e-104)) {
		tmp = -1.0 - (x / y);
	} else {
		tmp = x / (x - y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.45e-12) or not (y <= 8e-104):
		tmp = -1.0 - (x / y)
	else:
		tmp = x / (x - y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.45e-12) || !(y <= 8e-104))
		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 <= -1.45e-12) || ~((y <= 8e-104)))
		tmp = -1.0 - (x / y);
	else
		tmp = x / (x - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.45e-12], N[Not[LessEqual[y, 8e-104]], $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 < -1.4500000000000001e-12 or 7.99999999999999941e-104 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 78.4%

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

   4. Taylor expanded in y around inf 78.7%

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

      1. sub-neg 78.7%

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

      2. metadata-eval 78.7%

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

      3. +-commutative 78.7%

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

      4. mul-1-neg 78.7%

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

      5. unsub-neg 78.7%

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

   6. Simplified 78.7%

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

   if -1.4500000000000001e-12 < y < 7.99999999999999941e-104

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 79.1%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{-12} \lor \neg \left(y \leq 8 \cdot 10^{-104}\right):\\ \;\;\;\;-1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{-12} \lor \neg \left(y \leq 5.4 \cdot 10^{-102}\right):\\ \;\;\;\;-1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.4e-12) (not (<= y 5.4e-102)))
   (- -1.0 (/ x y))
   (+ 1.0 (/ y x))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.4e-12) || !(y <= 5.4e-102)) {
		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 <= (-1.4d-12)) .or. (.not. (y <= 5.4d-102))) 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 <= -1.4e-12) || !(y <= 5.4e-102)) {
		tmp = -1.0 - (x / y);
	} else {
		tmp = 1.0 + (y / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.4e-12) or not (y <= 5.4e-102):
		tmp = -1.0 - (x / y)
	else:
		tmp = 1.0 + (y / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.4e-12) || !(y <= 5.4e-102))
		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 <= -1.4e-12) || ~((y <= 5.4e-102)))
		tmp = -1.0 - (x / y);
	else
		tmp = 1.0 + (y / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.4e-12], N[Not[LessEqual[y, 5.4e-102]], $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 < -1.4000000000000001e-12 or 5.4e-102 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 78.4%

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

   4. Taylor expanded in y around inf 78.7%

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

      1. sub-neg 78.7%

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

      2. metadata-eval 78.7%

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

      3. +-commutative 78.7%

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

      4. mul-1-neg 78.7%

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

      5. unsub-neg 78.7%

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

   6. Simplified 78.7%

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

   if -1.4000000000000001e-12 < y < 5.4e-102

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 79.1%

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

   4. Taylor expanded in x around inf 79.1%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{-12} \lor \neg \left(y \leq 5.4 \cdot 10^{-102}\right):\\ \;\;\;\;-1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{y}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{-12}:\\ \;\;\;\;\frac{x + y}{-y}\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{-101}:\\ \;\;\;\;\frac{x}{x - y}\\ \mathbf{else}:\\ \;\;\;\;-1 - \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.4e-12)
   (/ (+ x y) (- y))
   (if (<= y 4.6e-101) (/ x (- x y)) (- -1.0 (/ x y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.4e-12) {
		tmp = (x + y) / -y;
	} else if (y <= 4.6e-101) {
		tmp = x / (x - y);
	} else {
		tmp = -1.0 - (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 <= (-2.4d-12)) then
        tmp = (x + y) / -y
    else if (y <= 4.6d-101) then
        tmp = x / (x - y)
    else
        tmp = (-1.0d0) - (x / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2.4e-12) {
		tmp = (x + y) / -y;
	} else if (y <= 4.6e-101) {
		tmp = x / (x - y);
	} else {
		tmp = -1.0 - (x / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.4e-12:
		tmp = (x + y) / -y
	elif y <= 4.6e-101:
		tmp = x / (x - y)
	else:
		tmp = -1.0 - (x / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.4e-12)
		tmp = Float64(Float64(x + y) / Float64(-y));
	elseif (y <= 4.6e-101)
		tmp = Float64(x / Float64(x - y));
	else
		tmp = Float64(-1.0 - Float64(x / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.4e-12)
		tmp = (x + y) / -y;
	elseif (y <= 4.6e-101)
		tmp = x / (x - y);
	else
		tmp = -1.0 - (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.4e-12], N[(N[(x + y), $MachinePrecision] / (-y)), $MachinePrecision], If[LessEqual[y, 4.6e-101], N[(x / N[(x - y), $MachinePrecision]), $MachinePrecision], N[(-1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.39999999999999987e-12

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 85.0%

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

      1. neg-mul-1 85.0%

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

   5. Simplified 85.0%

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

   if -2.39999999999999987e-12 < y < 4.5999999999999999e-101

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 79.1%

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

   if 4.5999999999999999e-101 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 73.0%

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

   4. Taylor expanded in y around inf 73.4%

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

      1. sub-neg 73.4%

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

      2. metadata-eval 73.4%

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

      3. +-commutative 73.4%

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

      4. mul-1-neg 73.4%

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

      5. unsub-neg 73.4%

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

   6. Simplified 73.4%

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

Alternative 7: 72.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{-12}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{-101}:\\ \;\;\;\;1 + \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.2e-12) -1.0 (if (<= y 4.6e-101) (+ 1.0 (/ y x)) -1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.2e-12) {
		tmp = -1.0;
	} else if (y <= 4.6e-101) {
		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 <= (-2.2d-12)) then
        tmp = -1.0d0
    else if (y <= 4.6d-101) 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 <= -2.2e-12) {
		tmp = -1.0;
	} else if (y <= 4.6e-101) {
		tmp = 1.0 + (y / x);
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.2e-12:
		tmp = -1.0
	elif y <= 4.6e-101:
		tmp = 1.0 + (y / x)
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.2e-12)
		tmp = -1.0;
	elseif (y <= 4.6e-101)
		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 <= -2.2e-12)
		tmp = -1.0;
	elseif (y <= 4.6e-101)
		tmp = 1.0 + (y / x);
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.2e-12], -1.0, If[LessEqual[y, 4.6e-101], N[(1.0 + N[(y / x), $MachinePrecision]), $MachinePrecision], -1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -2.19999999999999992e-12 or 4.5999999999999999e-101 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 78.0%

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

   if -2.19999999999999992e-12 < y < 4.5999999999999999e-101

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 79.1%

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

   4. Taylor expanded in x around inf 79.1%

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

Alternative 8: 72.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{-12}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-109}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.2e-12) -1.0 (if (<= y 5e-109) 1.0 -1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.2e-12) {
		tmp = -1.0;
	} else if (y <= 5e-109) {
		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 <= (-2.2d-12)) then
        tmp = -1.0d0
    else if (y <= 5d-109) 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 <= -2.2e-12) {
		tmp = -1.0;
	} else if (y <= 5e-109) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.2e-12:
		tmp = -1.0
	elif y <= 5e-109:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.2e-12)
		tmp = -1.0;
	elseif (y <= 5e-109)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.2e-12)
		tmp = -1.0;
	elseif (y <= 5e-109)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.2e-12], -1.0, If[LessEqual[y, 5e-109], 1.0, -1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -2.19999999999999992e-12 or 5.0000000000000002e-109 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 78.0%

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

   if -2.19999999999999992e-12 < y < 5.0000000000000002e-109

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 78.5%

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

Alternative 9: 50.3% 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 100.0%

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

3. Taylor expanded in x around 0 55.4%

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

Developer Target 1: 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 2024186 
(FPCore (x y)
  :name "Linear.Projection:perspective from linear-1.19.1.3, A"
  :precision binary64

  :alt
  (! :herbie-platform default (/ 1 (- (/ x (+ x y)) (/ y (+ x y)))))

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