Linear.Projection:perspective from linear-1.19.1.3, A

Percentage Accurate: 100.0% → 100.0%

Time: 7.0s

Alternatives: 8

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: 75.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.05 \cdot 10^{+76} \lor \neg \left(x \leq 210000000000\right):\\ \;\;\;\;1 + 2 \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{x}{y} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -2.05e+76) (not (<= x 210000000000.0)))
   (+ 1.0 (* 2.0 (/ y x)))
   (+ (* -2.0 (/ x y)) -1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -2.05e+76) || !(x <= 210000000000.0)) {
		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 <= (-2.05d+76)) .or. (.not. (x <= 210000000000.0d0))) 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 <= -2.05e+76) || !(x <= 210000000000.0)) {
		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 <= -2.05e+76) or not (x <= 210000000000.0):
		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 <= -2.05e+76) || !(x <= 210000000000.0))
		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 <= -2.05e+76) || ~((x <= 210000000000.0)))
		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, -2.05e+76], N[Not[LessEqual[x, 210000000000.0]], $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 < -2.0499999999999999e76 or 2.1e11 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 82.7%

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

   if -2.0499999999999999e76 < x < 2.1e11

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 78.4%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.05 \cdot 10^{+76} \lor \neg \left(x \leq 210000000000\right):\\ \;\;\;\;1 + 2 \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{x}{y} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{+76} \lor \neg \left(x \leq 1450000000000\right):\\ \;\;\;\;1 + 2 \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;-1 - \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -2.7e+76) (not (<= x 1450000000000.0)))
   (+ 1.0 (* 2.0 (/ y x)))
   (- -1.0 (/ x y))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -2.7e+76) || !(x <= 1450000000000.0)) {
		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 <= (-2.7d+76)) .or. (.not. (x <= 1450000000000.0d0))) 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 <= -2.7e+76) || !(x <= 1450000000000.0)) {
		tmp = 1.0 + (2.0 * (y / x));
	} else {
		tmp = -1.0 - (x / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -2.7e+76) or not (x <= 1450000000000.0):
		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 <= -2.7e+76) || !(x <= 1450000000000.0))
		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 <= -2.7e+76) || ~((x <= 1450000000000.0)))
		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, -2.7e+76], N[Not[LessEqual[x, 1450000000000.0]], $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 < -2.6999999999999999e76 or 1.45e12 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 82.7%

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

   if -2.6999999999999999e76 < x < 1.45e12

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 77.7%

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

   4. Taylor expanded in y around inf 77.9%

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

      1. sub-neg 77.9%

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

      2. metadata-eval 77.9%

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

      3. +-commutative 77.9%

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

      4. mul-1-neg 77.9%

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

      5. unsub-neg 77.9%

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

   6. Simplified 77.9%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{+76} \lor \neg \left(x \leq 1450000000000\right):\\ \;\;\;\;1 + 2 \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;-1 - \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{+76} \lor \neg \left(x \leq 4600000000000\right):\\ \;\;\;\;1 + \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;-1 - \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -2.6e+76) (not (<= x 4600000000000.0)))
   (+ 1.0 (/ y x))
   (- -1.0 (/ x y))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -2.6e+76) || !(x <= 4600000000000.0)) {
		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 <= (-2.6d+76)) .or. (.not. (x <= 4600000000000.0d0))) 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 <= -2.6e+76) || !(x <= 4600000000000.0)) {
		tmp = 1.0 + (y / x);
	} else {
		tmp = -1.0 - (x / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -2.6e+76) or not (x <= 4600000000000.0):
		tmp = 1.0 + (y / x)
	else:
		tmp = -1.0 - (x / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -2.6e+76) || !(x <= 4600000000000.0))
		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 <= -2.6e+76) || ~((x <= 4600000000000.0)))
		tmp = 1.0 + (y / x);
	else
		tmp = -1.0 - (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -2.6e+76], N[Not[LessEqual[x, 4600000000000.0]], $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 < -2.5999999999999999e76 or 4.6e12 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 82.1%

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

   4. Taylor expanded in x around inf 82.1%

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

   if -2.5999999999999999e76 < x < 4.6e12

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 77.7%

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

   4. Taylor expanded in y around inf 77.9%

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

      1. sub-neg 77.9%

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

      2. metadata-eval 77.9%

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

      3. +-commutative 77.9%

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

      4. mul-1-neg 77.9%

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

      5. unsub-neg 77.9%

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

   6. Simplified 77.9%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{+76} \lor \neg \left(x \leq 4600000000000\right):\\ \;\;\;\;1 + \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;-1 - \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.05 \cdot 10^{+76} \lor \neg \left(x \leq 1000000000000\right):\\ \;\;\;\;1 + \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -2.05e+76) (not (<= x 1000000000000.0))) (+ 1.0 (/ y x)) -1.0))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -2.05e+76) || !(x <= 1000000000000.0)) {
		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 <= (-2.05d+76)) .or. (.not. (x <= 1000000000000.0d0))) 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 <= -2.05e+76) || !(x <= 1000000000000.0)) {
		tmp = 1.0 + (y / x);
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -2.05e+76) or not (x <= 1000000000000.0):
		tmp = 1.0 + (y / x)
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -2.05e+76) || !(x <= 1000000000000.0))
		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 <= -2.05e+76) || ~((x <= 1000000000000.0)))
		tmp = 1.0 + (y / x);
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -2.05e+76], N[Not[LessEqual[x, 1000000000000.0]], $MachinePrecision]], N[(1.0 + N[(y / x), $MachinePrecision]), $MachinePrecision], -1.0]

Derivation

1. Split input into 2 regimes

2. if x < -2.0499999999999999e76 or 1e12 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 82.1%

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

   4. Taylor expanded in x around inf 82.1%

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

   if -2.0499999999999999e76 < x < 1e12

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 77.2%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.05 \cdot 10^{+76} \lor \neg \left(x \leq 1000000000000\right):\\ \;\;\;\;1 + \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.05 \cdot 10^{+76}:\\ \;\;\;\;1 + \frac{y}{x}\\ \mathbf{elif}\;x \leq 1650000000000:\\ \;\;\;\;-1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.05e+76)
   (+ 1.0 (/ y x))
   (if (<= x 1650000000000.0) (- -1.0 (/ x y)) (/ (+ x y) x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.05e+76) {
		tmp = 1.0 + (y / x);
	} else if (x <= 1650000000000.0) {
		tmp = -1.0 - (x / y);
	} else {
		tmp = (x + 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 <= (-2.05d+76)) then
        tmp = 1.0d0 + (y / x)
    else if (x <= 1650000000000.0d0) then
        tmp = (-1.0d0) - (x / y)
    else
        tmp = (x + y) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -2.05e+76) {
		tmp = 1.0 + (y / x);
	} else if (x <= 1650000000000.0) {
		tmp = -1.0 - (x / y);
	} else {
		tmp = (x + y) / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.05e+76:
		tmp = 1.0 + (y / x)
	elif x <= 1650000000000.0:
		tmp = -1.0 - (x / y)
	else:
		tmp = (x + y) / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.05e+76)
		tmp = Float64(1.0 + Float64(y / x));
	elseif (x <= 1650000000000.0)
		tmp = Float64(-1.0 - Float64(x / y));
	else
		tmp = Float64(Float64(x + y) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.05e+76)
		tmp = 1.0 + (y / x);
	elseif (x <= 1650000000000.0)
		tmp = -1.0 - (x / y);
	else
		tmp = (x + y) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.05e+76], N[(1.0 + N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1650000000000.0], N[(-1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision], N[(N[(x + y), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.0499999999999999e76

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 88.1%

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

   4. Taylor expanded in x around inf 88.1%

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

   if -2.0499999999999999e76 < x < 1.65e12

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 77.7%

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

   4. Taylor expanded in y around inf 77.9%

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

      1. sub-neg 77.9%

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

      2. metadata-eval 77.9%

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

      3. +-commutative 77.9%

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

      4. mul-1-neg 77.9%

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

      5. unsub-neg 77.9%

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

   6. Simplified 77.9%

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

   if 1.65e12 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 77.3%

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

   4. Taylor expanded in x around inf 77.3%

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

   5. Taylor expanded in x around 0 77.3%

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

      1. +-commutative 77.3%

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

   7. Simplified 77.3%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.05 \cdot 10^{+76}:\\ \;\;\;\;1 + \frac{y}{x}\\ \mathbf{elif}\;x \leq 1650000000000:\\ \;\;\;\;-1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{+76}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1020000000000:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.1e+76) 1.0 (if (<= x 1020000000000.0) -1.0 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.1e+76) {
		tmp = 1.0;
	} else if (x <= 1020000000000.0) {
		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 <= (-2.1d+76)) then
        tmp = 1.0d0
    else if (x <= 1020000000000.0d0) 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 <= -2.1e+76) {
		tmp = 1.0;
	} else if (x <= 1020000000000.0) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.1e+76:
		tmp = 1.0
	elif x <= 1020000000000.0:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.1e+76)
		tmp = 1.0;
	elseif (x <= 1020000000000.0)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.1e+76)
		tmp = 1.0;
	elseif (x <= 1020000000000.0)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.1e+76], 1.0, If[LessEqual[x, 1020000000000.0], -1.0, 1.0]]

Derivation

1. Split input into 2 regimes

2. if x < -2.10000000000000007e76 or 1.02e12 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 81.6%

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

   if -2.10000000000000007e76 < x < 1.02e12

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 77.2%

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

Alternative 8: 50.0% 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 53.9%

   \[\leadsto \color{blue}{-1} \]
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 2024157 
(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)))