Linear.Projection:perspective from linear-1.19.1.3, A

Percentage Accurate: 100.0% → 100.0%

Time: 11.4s

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: 74.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -46000000000000 \lor \neg \left(y \leq 3.8 \cdot 10^{-68}\right):\\ \;\;\;\;\frac{y}{x - y}\\ \mathbf{else}:\\ \;\;\;\;1 + 2 \cdot \frac{y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -46000000000000.0) (not (<= y 3.8e-68)))
   (/ y (- x y))
   (+ 1.0 (* 2.0 (/ y x)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -46000000000000.0) || !(y <= 3.8e-68)) {
		tmp = y / (x - y);
	} 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 <= (-46000000000000.0d0)) .or. (.not. (y <= 3.8d-68))) then
        tmp = y / (x - y)
    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 <= -46000000000000.0) || !(y <= 3.8e-68)) {
		tmp = y / (x - y);
	} else {
		tmp = 1.0 + (2.0 * (y / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -46000000000000.0) or not (y <= 3.8e-68):
		tmp = y / (x - y)
	else:
		tmp = 1.0 + (2.0 * (y / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -46000000000000.0) || !(y <= 3.8e-68))
		tmp = Float64(y / Float64(x - y));
	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 <= -46000000000000.0) || ~((y <= 3.8e-68)))
		tmp = y / (x - y);
	else
		tmp = 1.0 + (2.0 * (y / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -46000000000000.0], N[Not[LessEqual[y, 3.8e-68]], $MachinePrecision]], N[(y / N[(x - y), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(2.0 * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.6e13 or 3.80000000000000038e-68 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 75.5%

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

   if -4.6e13 < y < 3.80000000000000038e-68

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 83.5%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -46000000000000 \lor \neg \left(y \leq 3.8 \cdot 10^{-68}\right):\\ \;\;\;\;\frac{y}{x - y}\\ \mathbf{else}:\\ \;\;\;\;1 + 2 \cdot \frac{y}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7000000000000:\\ \;\;\;\;\frac{y}{x - y}\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-66}:\\ \;\;\;\;1 + 2 \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{x}{y} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -7000000000000.0)
   (/ y (- x y))
   (if (<= y 2.2e-66) (+ 1.0 (* 2.0 (/ y x))) (+ (* -2.0 (/ x y)) -1.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -7000000000000.0) {
		tmp = y / (x - y);
	} else if (y <= 2.2e-66) {
		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 (y <= (-7000000000000.0d0)) then
        tmp = y / (x - y)
    else if (y <= 2.2d-66) 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 (y <= -7000000000000.0) {
		tmp = y / (x - y);
	} else if (y <= 2.2e-66) {
		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 y <= -7000000000000.0:
		tmp = y / (x - y)
	elif y <= 2.2e-66:
		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 (y <= -7000000000000.0)
		tmp = Float64(y / Float64(x - y));
	elseif (y <= 2.2e-66)
		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 (y <= -7000000000000.0)
		tmp = y / (x - y);
	elseif (y <= 2.2e-66)
		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[LessEqual[y, -7000000000000.0], N[(y / N[(x - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.2e-66], 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 3 regimes

2. if y < -7e12

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 79.2%

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

   if -7e12 < y < 2.2000000000000001e-66

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 83.0%

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

   if 2.2000000000000001e-66 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 73.6%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7000000000000:\\ \;\;\;\;\frac{y}{x - y}\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-66}:\\ \;\;\;\;1 + 2 \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{x}{y} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -64000000000000 \lor \neg \left(y \leq 2.25 \cdot 10^{-66}\right):\\ \;\;\;\;\frac{y}{x - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -64000000000000.0) (not (<= y 2.25e-66)))
   (/ y (- x y))
   (/ x (- x y))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -64000000000000.0) || !(y <= 2.25e-66)) {
		tmp = y / (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 <= (-64000000000000.0d0)) .or. (.not. (y <= 2.25d-66))) then
        tmp = y / (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 <= -64000000000000.0) || !(y <= 2.25e-66)) {
		tmp = y / (x - y);
	} else {
		tmp = x / (x - y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -64000000000000.0) or not (y <= 2.25e-66):
		tmp = y / (x - y)
	else:
		tmp = x / (x - y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -64000000000000.0) || !(y <= 2.25e-66))
		tmp = Float64(y / 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 <= -64000000000000.0) || ~((y <= 2.25e-66)))
		tmp = y / (x - y);
	else
		tmp = x / (x - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -64000000000000.0], N[Not[LessEqual[y, 2.25e-66]], $MachinePrecision]], N[(y / N[(x - y), $MachinePrecision]), $MachinePrecision], N[(x / N[(x - y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -6.4e13 or 2.2499999999999999e-66 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 75.9%

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

   if -6.4e13 < y < 2.2499999999999999e-66

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 82.8%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -64000000000000 \lor \neg \left(y \leq 2.25 \cdot 10^{-66}\right):\\ \;\;\;\;\frac{y}{x - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{+14} \lor \neg \left(y \leq 2.25 \cdot 10^{-66}\right):\\ \;\;\;\;-1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.7e+14) (not (<= y 2.25e-66)))
   (- -1.0 (/ x y))
   (/ x (- x y))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.7e+14) || !(y <= 2.25e-66)) {
		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 <= (-1.7d+14)) .or. (.not. (y <= 2.25d-66))) 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 <= -1.7e+14) || !(y <= 2.25e-66)) {
		tmp = -1.0 - (x / y);
	} else {
		tmp = x / (x - y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.7e+14) or not (y <= 2.25e-66):
		tmp = -1.0 - (x / y)
	else:
		tmp = x / (x - y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.7e+14) || !(y <= 2.25e-66))
		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 <= -1.7e+14) || ~((y <= 2.25e-66)))
		tmp = -1.0 - (x / y);
	else
		tmp = x / (x - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.7e+14], N[Not[LessEqual[y, 2.25e-66]], $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 < -1.7e14 or 2.2499999999999999e-66 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 75.4%

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

      1. neg-mul-1 75.4%

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

   5. Simplified 75.4%

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

   6. Taylor expanded in x around 0 75.4%

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

      1. sub-neg 75.4%

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

      2. *-commutative 75.4%

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

      3. metadata-eval 75.4%

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

      4. distribute-lft1-in 75.4%

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

      5. *-rgt-identity 75.4%

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

      6. associate-*r/ 75.4%

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

      7. rgt-mult-inverse 75.2%

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

      8. distribute-lft-in 75.2%

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

      9. +-commutative 75.2%

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

      10. *-commutative 75.2%

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

      11. mul-1-neg 75.2%

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

      12. neg-sub0 75.2%

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

      13. distribute-lft-in 75.2%

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

      14. rgt-mult-inverse 75.4%

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

      15. associate--r+ 75.4%

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

      16. metadata-eval 75.4%

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

      17. associate-*r/ 75.4%

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

      18. *-rgt-identity 75.4%

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

   8. Simplified 75.4%

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

   if -1.7e14 < y < 2.2499999999999999e-66

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 82.8%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{+14} \lor \neg \left(y \leq 2.25 \cdot 10^{-66}\right):\\ \;\;\;\;-1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{+14} \lor \neg \left(y \leq 2.25 \cdot 10^{-66}\right):\\ \;\;\;\;-1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.7e+14) (not (<= y 2.25e-66)))
   (- -1.0 (/ x y))
   (+ 1.0 (/ y x))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.7e+14) || !(y <= 2.25e-66)) {
		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 <= (-1.7d+14)) .or. (.not. (y <= 2.25d-66))) 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 <= -1.7e+14) || !(y <= 2.25e-66)) {
		tmp = -1.0 - (x / y);
	} else {
		tmp = 1.0 + (y / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.7e+14) or not (y <= 2.25e-66):
		tmp = -1.0 - (x / y)
	else:
		tmp = 1.0 + (y / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.7e+14) || !(y <= 2.25e-66))
		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 <= -1.7e+14) || ~((y <= 2.25e-66)))
		tmp = -1.0 - (x / y);
	else
		tmp = 1.0 + (y / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.7e+14], N[Not[LessEqual[y, 2.25e-66]], $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 < -1.7e14 or 2.2499999999999999e-66 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 75.4%

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

      1. neg-mul-1 75.4%

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

   5. Simplified 75.4%

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

   6. Taylor expanded in x around 0 75.4%

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

      1. sub-neg 75.4%

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

      2. *-commutative 75.4%

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

      3. metadata-eval 75.4%

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

      4. distribute-lft1-in 75.4%

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

      5. *-rgt-identity 75.4%

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

      6. associate-*r/ 75.4%

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

      7. rgt-mult-inverse 75.2%

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

      8. distribute-lft-in 75.2%

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

      9. +-commutative 75.2%

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

      10. *-commutative 75.2%

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

      11. mul-1-neg 75.2%

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

      12. neg-sub0 75.2%

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

      13. distribute-lft-in 75.2%

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

      14. rgt-mult-inverse 75.4%

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

      15. associate--r+ 75.4%

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

      16. metadata-eval 75.4%

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

      17. associate-*r/ 75.4%

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

      18. *-rgt-identity 75.4%

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

   8. Simplified 75.4%

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

   if -1.7e14 < y < 2.2499999999999999e-66

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 82.8%

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

   4. Taylor expanded in x around inf 82.5%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{+14} \lor \neg \left(y \leq 2.25 \cdot 10^{-66}\right):\\ \;\;\;\;-1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{y}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.5 \cdot 10^{+14}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 6 \cdot 10^{-67}:\\ \;\;\;\;1 + \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.5e+14) -1.0 (if (<= y 6e-67) (+ 1.0 (/ y x)) -1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.5e+14) {
		tmp = -1.0;
	} else if (y <= 6e-67) {
		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 <= (-1.5d+14)) then
        tmp = -1.0d0
    else if (y <= 6d-67) 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 <= -1.5e+14) {
		tmp = -1.0;
	} else if (y <= 6e-67) {
		tmp = 1.0 + (y / x);
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.5e+14:
		tmp = -1.0
	elif y <= 6e-67:
		tmp = 1.0 + (y / x)
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.5e+14)
		tmp = -1.0;
	elseif (y <= 6e-67)
		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 <= -1.5e+14)
		tmp = -1.0;
	elseif (y <= 6e-67)
		tmp = 1.0 + (y / x);
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.5e+14], -1.0, If[LessEqual[y, 6e-67], N[(1.0 + N[(y / x), $MachinePrecision]), $MachinePrecision], -1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -1.5e14 or 6.00000000000000065e-67 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 74.6%

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

   if -1.5e14 < y < 6.00000000000000065e-67

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 83.3%

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

   4. Taylor expanded in x around inf 83.0%

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

Alternative 8: 73.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6500000000000:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 6 \cdot 10^{-67}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -6500000000000.0) -1.0 (if (<= y 6e-67) 1.0 -1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -6500000000000.0) {
		tmp = -1.0;
	} else if (y <= 6e-67) {
		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 <= (-6500000000000.0d0)) then
        tmp = -1.0d0
    else if (y <= 6d-67) 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 <= -6500000000000.0) {
		tmp = -1.0;
	} else if (y <= 6e-67) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -6500000000000.0:
		tmp = -1.0
	elif y <= 6e-67:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -6500000000000.0)
		tmp = -1.0;
	elseif (y <= 6e-67)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -6500000000000.0)
		tmp = -1.0;
	elseif (y <= 6e-67)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -6500000000000.0], -1.0, If[LessEqual[y, 6e-67], 1.0, -1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -6.5e12 or 6.00000000000000065e-67 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 74.6%

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

   if -6.5e12 < y < 6.00000000000000065e-67

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 82.7%

      \[\leadsto \color{blue}{1} \]
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 100.0%

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

3. Taylor expanded in x around 0 46.1%

   \[\leadsto \color{blue}{-1} \]
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 2024165 
(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)))