Linear.Projection:perspective from linear-1.19.1.3, A

Percentage Accurate: 100.0% → 100.0%

Time: 7.1s

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: 75.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.2 \cdot 10^{+48} \lor \neg \left(y \leq 4.4 \cdot 10^{-36}\right):\\ \;\;\;\;-2 \cdot \frac{x}{y} + -1\\ \mathbf{else}:\\ \;\;\;\;1 + 2 \cdot \frac{y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -6.2e+48) (not (<= y 4.4e-36)))
   (+ (* -2.0 (/ x y)) -1.0)
   (+ 1.0 (* 2.0 (/ y x)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -6.2e+48) || !(y <= 4.4e-36)) {
		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 <= (-6.2d+48)) .or. (.not. (y <= 4.4d-36))) 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 <= -6.2e+48) || !(y <= 4.4e-36)) {
		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 <= -6.2e+48) or not (y <= 4.4e-36):
		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 <= -6.2e+48) || !(y <= 4.4e-36))
		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 <= -6.2e+48) || ~((y <= 4.4e-36)))
		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, -6.2e+48], N[Not[LessEqual[y, 4.4e-36]], $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 < -6.20000000000000011e48 or 4.3999999999999999e-36 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 77.2%

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

   if -6.20000000000000011e48 < y < 4.3999999999999999e-36

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 82.6%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.2 \cdot 10^{+48} \lor \neg \left(y \leq 4.4 \cdot 10^{-36}\right):\\ \;\;\;\;-2 \cdot \frac{x}{y} + -1\\ \mathbf{else}:\\ \;\;\;\;1 + 2 \cdot \frac{y}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{+49}:\\ \;\;\;\;-1 - \frac{x}{y}\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-36}:\\ \;\;\;\;1 + 2 \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.8e+49)
   (- -1.0 (/ x y))
   (if (<= y 4.2e-36) (+ 1.0 (* 2.0 (/ y x))) (/ y (- x y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.8e+49) {
		tmp = -1.0 - (x / y);
	} else if (y <= 4.2e-36) {
		tmp = 1.0 + (2.0 * (y / x));
	} else {
		tmp = y / (x - y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-2.8d+49)) then
        tmp = (-1.0d0) - (x / y)
    else if (y <= 4.2d-36) then
        tmp = 1.0d0 + (2.0d0 * (y / x))
    else
        tmp = y / (x - y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2.8e+49) {
		tmp = -1.0 - (x / y);
	} else if (y <= 4.2e-36) {
		tmp = 1.0 + (2.0 * (y / x));
	} else {
		tmp = y / (x - y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.8e+49:
		tmp = -1.0 - (x / y)
	elif y <= 4.2e-36:
		tmp = 1.0 + (2.0 * (y / x))
	else:
		tmp = y / (x - y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.8e+49)
		tmp = Float64(-1.0 - Float64(x / y));
	elseif (y <= 4.2e-36)
		tmp = Float64(1.0 + Float64(2.0 * Float64(y / x)));
	else
		tmp = Float64(y / Float64(x - y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.8e+49)
		tmp = -1.0 - (x / y);
	elseif (y <= 4.2e-36)
		tmp = 1.0 + (2.0 * (y / x));
	else
		tmp = y / (x - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.8e+49], N[(-1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.2e-36], N[(1.0 + N[(2.0 * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y / N[(x - y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.7999999999999998e49

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 82.6%

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

      1. neg-mul-1 82.6%

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

   5. Simplified 82.6%

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

   6. Taylor expanded in x around 0 82.6%

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

      1. sub-neg 82.6%

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

      2. *-commutative 82.6%

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

      3. metadata-eval 82.6%

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

      4. distribute-lft1-in 82.6%

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

      5. *-lft-identity 82.6%

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

      6. associate-*l/ 82.6%

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

      7. lft-mult-inverse 82.3%

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

      8. distribute-rgt-in 82.3%

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

      9. +-commutative 82.3%

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

      10. *-commutative 82.3%

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

      11. mul-1-neg 82.3%

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

      12. neg-sub0 82.3%

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

      13. distribute-rgt-in 82.3%

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

      14. lft-mult-inverse 82.6%

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

      15. associate--r+ 82.6%

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

      16. metadata-eval 82.6%

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

      17. associate-*l/ 82.6%

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

      18. *-lft-identity 82.6%

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

   8. Simplified 82.6%

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

   if -2.7999999999999998e49 < y < 4.19999999999999982e-36

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 82.6%

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

   if 4.19999999999999982e-36 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 72.4%

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

Alternative 4: 75.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.9 \cdot 10^{+49} \lor \neg \left(y \leq 4.4 \cdot 10^{-36}\right):\\ \;\;\;\;-1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -3.9e+49) (not (<= y 4.4e-36)))
   (- -1.0 (/ x y))
   (+ 1.0 (/ y x))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -3.9e+49) || !(y <= 4.4e-36)) {
		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 <= (-3.9d+49)) .or. (.not. (y <= 4.4d-36))) 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 <= -3.9e+49) || !(y <= 4.4e-36)) {
		tmp = -1.0 - (x / y);
	} else {
		tmp = 1.0 + (y / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -3.9e+49) or not (y <= 4.4e-36):
		tmp = -1.0 - (x / y)
	else:
		tmp = 1.0 + (y / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -3.9e+49) || !(y <= 4.4e-36))
		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 <= -3.9e+49) || ~((y <= 4.4e-36)))
		tmp = -1.0 - (x / y);
	else
		tmp = 1.0 + (y / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -3.9e+49], N[Not[LessEqual[y, 4.4e-36]], $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 < -3.9000000000000001e49 or 4.3999999999999999e-36 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 76.5%

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

      1. neg-mul-1 76.5%

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

   5. Simplified 76.5%

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

   6. Taylor expanded in x around 0 76.5%

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

      1. sub-neg 76.5%

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

      2. *-commutative 76.5%

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

      3. metadata-eval 76.5%

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

      4. distribute-lft1-in 76.5%

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

      5. *-lft-identity 76.5%

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

      6. associate-*l/ 76.5%

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

      7. lft-mult-inverse 76.3%

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

      8. distribute-rgt-in 76.3%

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

      9. +-commutative 76.3%

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

      10. *-commutative 76.3%

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

      11. mul-1-neg 76.3%

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

      12. neg-sub0 76.3%

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

      13. distribute-rgt-in 76.3%

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

      14. lft-mult-inverse 76.5%

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

      15. associate--r+ 76.5%

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

      16. metadata-eval 76.5%

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

      17. associate-*l/ 76.5%

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

      18. *-lft-identity 76.5%

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

   8. Simplified 76.5%

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

   if -3.9000000000000001e49 < y < 4.3999999999999999e-36

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 81.0%

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

   4. Taylor expanded in x around inf 81.0%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.9 \cdot 10^{+49} \lor \neg \left(y \leq 4.4 \cdot 10^{-36}\right):\\ \;\;\;\;-1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{y}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{+49}:\\ \;\;\;\;-1 - \frac{x}{y}\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{-36}:\\ \;\;\;\;1 + \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.4e+49)
   (- -1.0 (/ x y))
   (if (<= y 4.4e-36) (+ 1.0 (/ y x)) (/ y (- x y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.4e+49) {
		tmp = -1.0 - (x / y);
	} else if (y <= 4.4e-36) {
		tmp = 1.0 + (y / x);
	} else {
		tmp = y / (x - y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-2.4d+49)) then
        tmp = (-1.0d0) - (x / y)
    else if (y <= 4.4d-36) then
        tmp = 1.0d0 + (y / x)
    else
        tmp = y / (x - y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2.4e+49) {
		tmp = -1.0 - (x / y);
	} else if (y <= 4.4e-36) {
		tmp = 1.0 + (y / x);
	} else {
		tmp = y / (x - y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.4e+49:
		tmp = -1.0 - (x / y)
	elif y <= 4.4e-36:
		tmp = 1.0 + (y / x)
	else:
		tmp = y / (x - y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.4e+49)
		tmp = Float64(-1.0 - Float64(x / y));
	elseif (y <= 4.4e-36)
		tmp = Float64(1.0 + Float64(y / x));
	else
		tmp = Float64(y / Float64(x - y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.4e+49)
		tmp = -1.0 - (x / y);
	elseif (y <= 4.4e-36)
		tmp = 1.0 + (y / x);
	else
		tmp = y / (x - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.4e+49], N[(-1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.4e-36], N[(1.0 + N[(y / x), $MachinePrecision]), $MachinePrecision], N[(y / N[(x - y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.4e49

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 82.6%

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

      1. neg-mul-1 82.6%

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

   5. Simplified 82.6%

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

   6. Taylor expanded in x around 0 82.6%

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

      1. sub-neg 82.6%

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

      2. *-commutative 82.6%

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

      3. metadata-eval 82.6%

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

      4. distribute-lft1-in 82.6%

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

      5. *-lft-identity 82.6%

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

      6. associate-*l/ 82.6%

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

      7. lft-mult-inverse 82.3%

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

      8. distribute-rgt-in 82.3%

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

      9. +-commutative 82.3%

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

      10. *-commutative 82.3%

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

      11. mul-1-neg 82.3%

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

      12. neg-sub0 82.3%

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

      13. distribute-rgt-in 82.3%

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

      14. lft-mult-inverse 82.6%

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

      15. associate--r+ 82.6%

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

      16. metadata-eval 82.6%

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

      17. associate-*l/ 82.6%

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

      18. *-lft-identity 82.6%

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

   8. Simplified 82.6%

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

   if -2.4e49 < y < 4.3999999999999999e-36

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 81.0%

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

   4. Taylor expanded in x around inf 81.0%

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

   if 4.3999999999999999e-36 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 72.4%

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

Alternative 6: 74.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.2 \cdot 10^{+48}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{-36}:\\ \;\;\;\;1 + \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -8.2e+48) -1.0 (if (<= y 4.4e-36) (+ 1.0 (/ y x)) -1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -8.2e+48) {
		tmp = -1.0;
	} else if (y <= 4.4e-36) {
		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 <= (-8.2d+48)) then
        tmp = -1.0d0
    else if (y <= 4.4d-36) 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 <= -8.2e+48) {
		tmp = -1.0;
	} else if (y <= 4.4e-36) {
		tmp = 1.0 + (y / x);
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -8.2e+48:
		tmp = -1.0
	elif y <= 4.4e-36:
		tmp = 1.0 + (y / x)
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -8.2e+48)
		tmp = -1.0;
	elseif (y <= 4.4e-36)
		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 <= -8.2e+48)
		tmp = -1.0;
	elseif (y <= 4.4e-36)
		tmp = 1.0 + (y / x);
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -8.2e+48], -1.0, If[LessEqual[y, 4.4e-36], N[(1.0 + N[(y / x), $MachinePrecision]), $MachinePrecision], -1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -8.2000000000000005e48 or 4.3999999999999999e-36 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 75.9%

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

   if -8.2000000000000005e48 < y < 4.3999999999999999e-36

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 81.0%

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

   4. Taylor expanded in x around inf 81.0%

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

Alternative 7: 74.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{+52}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{-36}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -4e+52) -1.0 (if (<= y 4.4e-36) 1.0 -1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -4e+52) {
		tmp = -1.0;
	} else if (y <= 4.4e-36) {
		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 <= (-4d+52)) then
        tmp = -1.0d0
    else if (y <= 4.4d-36) 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 <= -4e+52) {
		tmp = -1.0;
	} else if (y <= 4.4e-36) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -4e+52:
		tmp = -1.0
	elif y <= 4.4e-36:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -4e+52)
		tmp = -1.0;
	elseif (y <= 4.4e-36)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -4e+52)
		tmp = -1.0;
	elseif (y <= 4.4e-36)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -4e+52], -1.0, If[LessEqual[y, 4.4e-36], 1.0, -1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -4e52 or 4.3999999999999999e-36 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 76.3%

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

   if -4e52 < y < 4.3999999999999999e-36

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 80.1%

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

Alternative 8: 48.8% 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 47.3%

   \[\leadsto \color{blue}{-1} \]
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 2024181 
(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)))