Linear.Projection:perspective from linear-1.19.1.3, A

Percentage Accurate: 100.0% → 100.0%

Time: 7.3s

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: 75.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2300000000 \lor \neg \left(x \leq 4.9 \cdot 10^{+41}\right):\\ \;\;\;\;1 + 2 \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{x}{y} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -2300000000.0) (not (<= x 4.9e+41)))
   (+ 1.0 (* 2.0 (/ y x)))
   (+ (* -2.0 (/ x y)) -1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -2300000000.0) || !(x <= 4.9e+41)) {
		tmp = 1.0 + (2.0 * (y / x));
	} else {
		tmp = (-2.0 * (x / y)) + -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 ((x <= (-2300000000.0d0)) .or. (.not. (x <= 4.9d+41))) then
        tmp = 1.0d0 + (2.0d0 * (y / x))
    else
        tmp = ((-2.0d0) * (x / y)) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -2300000000.0) || !(x <= 4.9e+41)) {
		tmp = 1.0 + (2.0 * (y / x));
	} else {
		tmp = (-2.0 * (x / y)) + -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -2300000000.0) or not (x <= 4.9e+41):
		tmp = 1.0 + (2.0 * (y / x))
	else:
		tmp = (-2.0 * (x / y)) + -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -2300000000.0) || !(x <= 4.9e+41))
		tmp = Float64(1.0 + Float64(2.0 * Float64(y / x)));
	else
		tmp = Float64(Float64(-2.0 * Float64(x / y)) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -2300000000.0) || ~((x <= 4.9e+41)))
		tmp = 1.0 + (2.0 * (y / x));
	else
		tmp = (-2.0 * (x / y)) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -2300000000.0], N[Not[LessEqual[x, 4.9e+41]], $MachinePrecision]], N[(1.0 + N[(2.0 * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 * N[(x / y), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.3e9 or 4.8999999999999999e41 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 79.5%

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

   if -2.3e9 < x < 4.8999999999999999e41

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 78.9%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2300000000 \lor \neg \left(x \leq 4.9 \cdot 10^{+41}\right):\\ \;\;\;\;1 + 2 \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{x}{y} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -125000000000 \lor \neg \left(x \leq 1.66 \cdot 10^{+38}\right):\\ \;\;\;\;1 + 2 \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;-1 - \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -125000000000.0) (not (<= x 1.66e+38)))
   (+ 1.0 (* 2.0 (/ y x)))
   (- -1.0 (/ x y))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -125000000000.0) || !(x <= 1.66e+38)) {
		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 ((x <= (-125000000000.0d0)) .or. (.not. (x <= 1.66d+38))) 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 ((x <= -125000000000.0) || !(x <= 1.66e+38)) {
		tmp = 1.0 + (2.0 * (y / x));
	} else {
		tmp = -1.0 - (x / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -125000000000.0) or not (x <= 1.66e+38):
		tmp = 1.0 + (2.0 * (y / x))
	else:
		tmp = -1.0 - (x / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -125000000000.0) || !(x <= 1.66e+38))
		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 ((x <= -125000000000.0) || ~((x <= 1.66e+38)))
		tmp = 1.0 + (2.0 * (y / x));
	else
		tmp = -1.0 - (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -125000000000.0], N[Not[LessEqual[x, 1.66e+38]], $MachinePrecision]], 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 2 regimes

2. if x < -1.25e11 or 1.66e38 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 79.5%

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

   if -1.25e11 < x < 1.66e38

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 78.6%

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

      1. neg-mul-1 78.6%

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

   5. Simplified 78.6%

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

   6. Taylor expanded in x around 0 78.6%

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

      1. neg-mul-1 78.6%

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

      2. neg-sub0 78.6%

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

      3. associate--r+ 78.6%

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

      4. +-commutative 78.6%

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

      5. associate--r+ 78.6%

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

      6. metadata-eval 78.6%

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

   8. Simplified 78.6%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -125000000000 \lor \neg \left(x \leq 1.66 \cdot 10^{+38}\right):\\ \;\;\;\;1 + 2 \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;-1 - \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -17000000000 \lor \neg \left(x \leq 7.5 \cdot 10^{+40}\right):\\ \;\;\;\;1 + \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;-1 - \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -17000000000.0) (not (<= x 7.5e+40)))
   (+ 1.0 (/ y x))
   (- -1.0 (/ x y))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -17000000000.0) || !(x <= 7.5e+40)) {
		tmp = 1.0 + (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 ((x <= (-17000000000.0d0)) .or. (.not. (x <= 7.5d+40))) then
        tmp = 1.0d0 + (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 ((x <= -17000000000.0) || !(x <= 7.5e+40)) {
		tmp = 1.0 + (y / x);
	} else {
		tmp = -1.0 - (x / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -17000000000.0) or not (x <= 7.5e+40):
		tmp = 1.0 + (y / x)
	else:
		tmp = -1.0 - (x / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -17000000000.0) || !(x <= 7.5e+40))
		tmp = Float64(1.0 + 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 ((x <= -17000000000.0) || ~((x <= 7.5e+40)))
		tmp = 1.0 + (y / x);
	else
		tmp = -1.0 - (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -17000000000.0], N[Not[LessEqual[x, 7.5e+40]], $MachinePrecision]], N[(1.0 + N[(y / x), $MachinePrecision]), $MachinePrecision], N[(-1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.7e10 or 7.4999999999999996e40 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 78.9%

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

   4. Taylor expanded in x around inf 78.8%

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

   if -1.7e10 < x < 7.4999999999999996e40

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 78.6%

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

      1. neg-mul-1 78.6%

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

   5. Simplified 78.6%

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

   6. Taylor expanded in x around 0 78.6%

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

      1. neg-mul-1 78.6%

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

      2. neg-sub0 78.6%

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

      3. associate--r+ 78.6%

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

      4. +-commutative 78.6%

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

      5. associate--r+ 78.6%

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

      6. metadata-eval 78.6%

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

   8. Simplified 78.6%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -17000000000 \lor \neg \left(x \leq 7.5 \cdot 10^{+40}\right):\\ \;\;\;\;1 + \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;-1 - \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -95000000000 \lor \neg \left(x \leq 1.62 \cdot 10^{+38}\right):\\ \;\;\;\;1 + \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -95000000000.0) (not (<= x 1.62e+38))) (+ 1.0 (/ y x)) -1.0))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -95000000000.0) || !(x <= 1.62e+38)) {
		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 ((x <= (-95000000000.0d0)) .or. (.not. (x <= 1.62d+38))) 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 ((x <= -95000000000.0) || !(x <= 1.62e+38)) {
		tmp = 1.0 + (y / x);
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -95000000000.0) or not (x <= 1.62e+38):
		tmp = 1.0 + (y / x)
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -95000000000.0) || !(x <= 1.62e+38))
		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 ((x <= -95000000000.0) || ~((x <= 1.62e+38)))
		tmp = 1.0 + (y / x);
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -95000000000.0], N[Not[LessEqual[x, 1.62e+38]], $MachinePrecision]], N[(1.0 + N[(y / x), $MachinePrecision]), $MachinePrecision], -1.0]

Derivation

1. Split input into 2 regimes

2. if x < -9.5e10 or 1.62000000000000001e38 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 78.9%

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

   4. Taylor expanded in x around inf 78.8%

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

   if -9.5e10 < x < 1.62000000000000001e38

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 77.8%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -95000000000 \lor \neg \left(x \leq 1.62 \cdot 10^{+38}\right):\\ \;\;\;\;1 + \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{+16}:\\ \;\;\;\;\frac{x}{x - y}\\ \mathbf{elif}\;x \leq 1.52 \cdot 10^{+38}:\\ \;\;\;\;-1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -5.5e+16)
   (/ x (- x y))
   (if (<= x 1.52e+38) (- -1.0 (/ x y)) (+ 1.0 (/ y x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -5.5e+16) {
		tmp = x / (x - y);
	} else if (x <= 1.52e+38) {
		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 (x <= (-5.5d+16)) then
        tmp = x / (x - y)
    else if (x <= 1.52d+38) 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 (x <= -5.5e+16) {
		tmp = x / (x - y);
	} else if (x <= 1.52e+38) {
		tmp = -1.0 - (x / y);
	} else {
		tmp = 1.0 + (y / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -5.5e+16:
		tmp = x / (x - y)
	elif x <= 1.52e+38:
		tmp = -1.0 - (x / y)
	else:
		tmp = 1.0 + (y / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -5.5e+16)
		tmp = Float64(x / Float64(x - y));
	elseif (x <= 1.52e+38)
		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 (x <= -5.5e+16)
		tmp = x / (x - y);
	elseif (x <= 1.52e+38)
		tmp = -1.0 - (x / y);
	else
		tmp = 1.0 + (y / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -5.5e+16], N[(x / N[(x - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.52e+38], N[(-1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(y / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.5e16

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 76.4%

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

   if -5.5e16 < x < 1.51999999999999996e38

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 78.2%

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

      1. neg-mul-1 78.2%

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

   5. Simplified 78.2%

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

   6. Taylor expanded in x around 0 78.2%

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

      1. neg-mul-1 78.2%

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

      2. neg-sub0 78.2%

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

      3. associate--r+ 78.2%

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

      4. +-commutative 78.2%

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

      5. associate--r+ 78.2%

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

      6. metadata-eval 78.2%

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

   8. Simplified 78.2%

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

   if 1.51999999999999996e38 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 82.5%

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

   4. Taylor expanded in x around inf 82.5%

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

Alternative 7: 75.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -24000000000:\\ \;\;\;\;\frac{x}{x + y}\\ \mathbf{elif}\;x \leq 2.55 \cdot 10^{+39}:\\ \;\;\;\;-1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -24000000000.0)
   (/ x (+ x y))
   (if (<= x 2.55e+39) (- -1.0 (/ x y)) (+ 1.0 (/ y x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -24000000000.0) {
		tmp = x / (x + y);
	} else if (x <= 2.55e+39) {
		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 (x <= (-24000000000.0d0)) then
        tmp = x / (x + y)
    else if (x <= 2.55d+39) 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 (x <= -24000000000.0) {
		tmp = x / (x + y);
	} else if (x <= 2.55e+39) {
		tmp = -1.0 - (x / y);
	} else {
		tmp = 1.0 + (y / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -24000000000.0:
		tmp = x / (x + y)
	elif x <= 2.55e+39:
		tmp = -1.0 - (x / y)
	else:
		tmp = 1.0 + (y / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -24000000000.0)
		tmp = Float64(x / Float64(x + y));
	elseif (x <= 2.55e+39)
		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 (x <= -24000000000.0)
		tmp = x / (x + y);
	elseif (x <= 2.55e+39)
		tmp = -1.0 - (x / y);
	else
		tmp = 1.0 + (y / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -24000000000.0], N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.55e+39], N[(-1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(y / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.4e10

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 75.6%

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

      1. div-inv 75.4%

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

      2. sub-neg 75.4%

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

      3. add-sqr-sqrt 37.5%

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

      4. sqrt-unprod 70.9%

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

      5. sqr-neg 70.9%

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

      6. sqrt-unprod 38.5%

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

      7. add-sqr-sqrt 75.3%

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

   5. Applied egg-rr 75.3%

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

      1. associate-*r/ 75.6%

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

      2. *-rgt-identity 75.6%

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

   7. Simplified 75.6%

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

   if -2.4e10 < x < 2.5499999999999999e39

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 78.6%

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

      1. neg-mul-1 78.6%

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

   5. Simplified 78.6%

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

   6. Taylor expanded in x around 0 78.6%

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

      1. neg-mul-1 78.6%

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

      2. neg-sub0 78.6%

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

      3. associate--r+ 78.6%

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

      4. +-commutative 78.6%

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

      5. associate--r+ 78.6%

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

      6. metadata-eval 78.6%

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

   8. Simplified 78.6%

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

   if 2.5499999999999999e39 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 82.5%

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

   4. Taylor expanded in x around inf 82.5%

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

Alternative 8: 74.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+14}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{+40}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1e+14) 1.0 (if (<= x 1.7e+40) -1.0 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1e+14) {
		tmp = 1.0;
	} else if (x <= 1.7e+40) {
		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 (x <= (-1d+14)) then
        tmp = 1.0d0
    else if (x <= 1.7d+40) 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 (x <= -1e+14) {
		tmp = 1.0;
	} else if (x <= 1.7e+40) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1e+14:
		tmp = 1.0
	elif x <= 1.7e+40:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1e+14)
		tmp = 1.0;
	elseif (x <= 1.7e+40)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1e+14)
		tmp = 1.0;
	elseif (x <= 1.7e+40)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1e+14], 1.0, If[LessEqual[x, 1.7e+40], -1.0, 1.0]]

Derivation

1. Split input into 2 regimes

2. if x < -1e14 or 1.69999999999999994e40 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 78.7%

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

   if -1e14 < x < 1.69999999999999994e40

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 77.4%

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

Alternative 9: 49.5% 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 51.9%

   \[\leadsto \color{blue}{-1} \]
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 2024170 
(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)))