Linear.Projection:perspective from linear-1.19.1.3, A

Percentage Accurate: 100.0% → 100.0%

Time: 5.7s

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 100.0%

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

Alternative 2: 75.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.6 \cdot 10^{-75}:\\ \;\;\;\;\frac{y}{x - y}\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{+50}:\\ \;\;\;\;1 + \frac{y \cdot 2}{x}\\ \mathbf{else}:\\ \;\;\;\;-1 + -2 \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -6.6e-75)
   (/ y (- x y))
   (if (<= y 1.85e+50) (+ 1.0 (/ (* y 2.0) x)) (+ -1.0 (* -2.0 (/ x y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -6.6e-75) {
		tmp = y / (x - y);
	} else if (y <= 1.85e+50) {
		tmp = 1.0 + ((y * 2.0) / x);
	} else {
		tmp = -1.0 + (-2.0 * (x / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-6.6d-75)) then
        tmp = y / (x - y)
    else if (y <= 1.85d+50) then
        tmp = 1.0d0 + ((y * 2.0d0) / x)
    else
        tmp = (-1.0d0) + ((-2.0d0) * (x / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -6.6e-75) {
		tmp = y / (x - y);
	} else if (y <= 1.85e+50) {
		tmp = 1.0 + ((y * 2.0) / x);
	} else {
		tmp = -1.0 + (-2.0 * (x / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -6.6e-75:
		tmp = y / (x - y)
	elif y <= 1.85e+50:
		tmp = 1.0 + ((y * 2.0) / x)
	else:
		tmp = -1.0 + (-2.0 * (x / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -6.6e-75)
		tmp = Float64(y / Float64(x - y));
	elseif (y <= 1.85e+50)
		tmp = Float64(1.0 + Float64(Float64(y * 2.0) / x));
	else
		tmp = Float64(-1.0 + Float64(-2.0 * Float64(x / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -6.6e-75)
		tmp = y / (x - y);
	elseif (y <= 1.85e+50)
		tmp = 1.0 + ((y * 2.0) / x);
	else
		tmp = -1.0 + (-2.0 * (x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -6.6e-75], N[(y / N[(x - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.85e+50], N[(1.0 + N[(N[(y * 2.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(-2.0 * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.5999999999999999e-75

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{y}, \mathsf{\_.f64}\left(x, y\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 80.9%

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

   if -6.5999999999999999e-75 < y < 1.85e50

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. *-lft-identity N/A

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

      3. distribute-rgt-out-- N/A

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

      4. *-rgt-identity N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

         \[\leadsto 1 + \left(y \cdot \frac{1}{x}\right) \cdot 2 \]

      7. associate-*r* N/A

         \[\leadsto 1 + y \cdot \color{blue}{\left(\frac{1}{x} \cdot 2\right)} \]

      8. *-commutative N/A

         \[\leadsto 1 + y \cdot \left(2 \cdot \color{blue}{\frac{1}{x}}\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(y \cdot \left(2 \cdot \frac{1}{x}\right)\right)}\right) \]

      10. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(y \cdot \frac{2 \cdot 1}{\color{blue}{x}}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(y \cdot \frac{2}{x}\right)\right) \]

      12. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{y \cdot 2}{\color{blue}{x}}\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{y \cdot \left(1 - -1\right)}{x}\right)\right) \]

      14. distribute-rgt-out-- N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{1 \cdot y - -1 \cdot y}{x}\right)\right) \]

      15. *-lft-identity N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{y - -1 \cdot y}{x}\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(y - -1 \cdot y\right), \color{blue}{x}\right)\right) \]

      17. *-lft-identity N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(1 \cdot y - -1 \cdot y\right), x\right)\right) \]

      18. distribute-rgt-out-- N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(y \cdot \left(1 - -1\right)\right), x\right)\right) \]

      19. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(y \cdot 2\right), x\right)\right) \]

      20. *-lowering-*.f64 78.9%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, 2\right), x\right)\right) \]

   5. Simplified 78.9%

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

   if 1.85e50 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-2 \cdot \frac{x}{y} - 1} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto -2 \cdot \frac{x}{y} + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{x}{y} \cdot -2 + \left(\mathsf{neg}\left(\color{blue}{1}\right)\right) \]

      3. metadata-eval N/A

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

      4. associate-*l/ N/A

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

      5. metadata-eval N/A

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

      6. distribute-rgt-out-- N/A

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

      7. *-lft-identity N/A

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

      8. +-commutative N/A

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

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \color{blue}{\left(\frac{-1 \cdot x - x}{y}\right)}\right) \]

      10. *-lft-identity N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left(\frac{-1 \cdot x - 1 \cdot x}{y}\right)\right) \]

      11. distribute-rgt-out-- N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left(\frac{x \cdot \left(-1 - 1\right)}{y}\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left(\frac{x \cdot -2}{y}\right)\right) \]

      13. associate-*l/ N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left(\frac{x}{y} \cdot \color{blue}{-2}\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left(-2 \cdot \color{blue}{\frac{x}{y}}\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(-2, \color{blue}{\left(\frac{x}{y}\right)}\right)\right) \]

      16. /-lowering-/.f64 78.2%

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(-2, \mathsf{/.f64}\left(x, \color{blue}{y}\right)\right)\right) \]

   5. Simplified 78.2%

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

Alternative 3: 74.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{-73}:\\ \;\;\;\;\frac{y}{x - y}\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{+49}:\\ \;\;\;\;1 + \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;-1 + -2 \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.3e-73)
   (/ y (- x y))
   (if (<= y 1.8e+49) (+ 1.0 (/ y x)) (+ -1.0 (* -2.0 (/ x y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.3e-73) {
		tmp = y / (x - y);
	} else if (y <= 1.8e+49) {
		tmp = 1.0 + (y / x);
	} else {
		tmp = -1.0 + (-2.0 * (x / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.3d-73)) then
        tmp = y / (x - y)
    else if (y <= 1.8d+49) then
        tmp = 1.0d0 + (y / x)
    else
        tmp = (-1.0d0) + ((-2.0d0) * (x / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.3e-73) {
		tmp = y / (x - y);
	} else if (y <= 1.8e+49) {
		tmp = 1.0 + (y / x);
	} else {
		tmp = -1.0 + (-2.0 * (x / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.3e-73:
		tmp = y / (x - y)
	elif y <= 1.8e+49:
		tmp = 1.0 + (y / x)
	else:
		tmp = -1.0 + (-2.0 * (x / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.3e-73)
		tmp = Float64(y / Float64(x - y));
	elseif (y <= 1.8e+49)
		tmp = Float64(1.0 + Float64(y / x));
	else
		tmp = Float64(-1.0 + Float64(-2.0 * Float64(x / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.3e-73)
		tmp = y / (x - y);
	elseif (y <= 1.8e+49)
		tmp = 1.0 + (y / x);
	else
		tmp = -1.0 + (-2.0 * (x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.3e-73], N[(y / N[(x - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.8e+49], N[(1.0 + N[(y / x), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(-2.0 * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.3e-73

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{y}, \mathsf{\_.f64}\left(x, y\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 80.9%

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

   if -1.3e-73 < y < 1.79999999999999998e49

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{\_.f64}\left(x, y\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 78.5%

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

   6. Taylor expanded in x around inf

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{y}{x}\right)}\right) \]

      2. /-lowering-/.f64 78.5%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(y, \color{blue}{x}\right)\right) \]

   8. Simplified 78.5%

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

   if 1.79999999999999998e49 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-2 \cdot \frac{x}{y} - 1} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto -2 \cdot \frac{x}{y} + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{x}{y} \cdot -2 + \left(\mathsf{neg}\left(\color{blue}{1}\right)\right) \]

      3. metadata-eval N/A

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

      4. associate-*l/ N/A

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

      5. metadata-eval N/A

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

      6. distribute-rgt-out-- N/A

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

      7. *-lft-identity N/A

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

      8. +-commutative N/A

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

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \color{blue}{\left(\frac{-1 \cdot x - x}{y}\right)}\right) \]

      10. *-lft-identity N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left(\frac{-1 \cdot x - 1 \cdot x}{y}\right)\right) \]

      11. distribute-rgt-out-- N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left(\frac{x \cdot \left(-1 - 1\right)}{y}\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left(\frac{x \cdot -2}{y}\right)\right) \]

      13. associate-*l/ N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left(\frac{x}{y} \cdot \color{blue}{-2}\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left(-2 \cdot \color{blue}{\frac{x}{y}}\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(-2, \color{blue}{\left(\frac{x}{y}\right)}\right)\right) \]

      16. /-lowering-/.f64 78.2%

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(-2, \mathsf{/.f64}\left(x, \color{blue}{y}\right)\right)\right) \]

   5. Simplified 78.2%

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

Alternative 4: 74.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{x - y}\\ \mathbf{if}\;y \leq -1.8 \cdot 10^{-73}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{-6}:\\ \;\;\;\;1 + \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (- x y))))
   (if (<= y -1.8e-73) t_0 (if (<= y 7.2e-6) (+ 1.0 (/ y x)) t_0))))

C

double code(double x, double y) {
	double t_0 = y / (x - y);
	double tmp;
	if (y <= -1.8e-73) {
		tmp = t_0;
	} else if (y <= 7.2e-6) {
		tmp = 1.0 + (y / x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y / (x - y)
    if (y <= (-1.8d-73)) then
        tmp = t_0
    else if (y <= 7.2d-6) then
        tmp = 1.0d0 + (y / x)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y / (x - y);
	double tmp;
	if (y <= -1.8e-73) {
		tmp = t_0;
	} else if (y <= 7.2e-6) {
		tmp = 1.0 + (y / x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y / (x - y)
	tmp = 0
	if y <= -1.8e-73:
		tmp = t_0
	elif y <= 7.2e-6:
		tmp = 1.0 + (y / x)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y / Float64(x - y))
	tmp = 0.0
	if (y <= -1.8e-73)
		tmp = t_0;
	elseif (y <= 7.2e-6)
		tmp = Float64(1.0 + Float64(y / x));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y / (x - y);
	tmp = 0.0;
	if (y <= -1.8e-73)
		tmp = t_0;
	elseif (y <= 7.2e-6)
		tmp = 1.0 + (y / x);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y / N[(x - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.8e-73], t$95$0, If[LessEqual[y, 7.2e-6], N[(1.0 + N[(y / x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.8e-73 or 7.19999999999999967e-6 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{y}, \mathsf{\_.f64}\left(x, y\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 78.0%

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

   if -1.8e-73 < y < 7.19999999999999967e-6

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{\_.f64}\left(x, y\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 80.4%

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

   6. Taylor expanded in x around inf

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{y}{x}\right)}\right) \]

      2. /-lowering-/.f64 80.4%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(y, \color{blue}{x}\right)\right) \]

   8. Simplified 80.4%

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

Alternative 5: 74.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -1 - \frac{x}{y}\\ \mathbf{if}\;y \leq -8 \cdot 10^{-74}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{+48}:\\ \;\;\;\;1 + \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- -1.0 (/ x y))))
   (if (<= y -8e-74) t_0 (if (<= y 1.6e+48) (+ 1.0 (/ y x)) t_0))))

C

double code(double x, double y) {
	double t_0 = -1.0 - (x / y);
	double tmp;
	if (y <= -8e-74) {
		tmp = t_0;
	} else if (y <= 1.6e+48) {
		tmp = 1.0 + (y / x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (-1.0d0) - (x / y)
    if (y <= (-8d-74)) then
        tmp = t_0
    else if (y <= 1.6d+48) then
        tmp = 1.0d0 + (y / x)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = -1.0 - (x / y);
	double tmp;
	if (y <= -8e-74) {
		tmp = t_0;
	} else if (y <= 1.6e+48) {
		tmp = 1.0 + (y / x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = -1.0 - (x / y)
	tmp = 0
	if y <= -8e-74:
		tmp = t_0
	elif y <= 1.6e+48:
		tmp = 1.0 + (y / x)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(-1.0 - Float64(x / y))
	tmp = 0.0
	if (y <= -8e-74)
		tmp = t_0;
	elseif (y <= 1.6e+48)
		tmp = Float64(1.0 + Float64(y / x));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = -1.0 - (x / y);
	tmp = 0.0;
	if (y <= -8e-74)
		tmp = t_0;
	elseif (y <= 1.6e+48)
		tmp = 1.0 + (y / x);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(-1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -8e-74], t$95$0, If[LessEqual[y, 1.6e+48], N[(1.0 + N[(y / x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -7.99999999999999966e-74 or 1.6000000000000001e48 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{y}, \mathsf{\_.f64}\left(x, y\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 79.5%

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

   6. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. sub-neg N/A

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

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(-1, \color{blue}{\left(\frac{x}{y}\right)}\right) \]

      7. /-lowering-/.f64 79.3%

         \[\leadsto \mathsf{\_.f64}\left(-1, \mathsf{/.f64}\left(x, \color{blue}{y}\right)\right) \]

   8. Simplified 79.3%

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

   if -7.99999999999999966e-74 < y < 1.6000000000000001e48

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{\_.f64}\left(x, y\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 78.5%

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

   6. Taylor expanded in x around inf

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{y}{x}\right)}\right) \]

      2. /-lowering-/.f64 78.5%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(y, \color{blue}{x}\right)\right) \]

   8. Simplified 78.5%

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

Alternative 6: 74.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-73}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{+53}:\\ \;\;\;\;1 + \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2e-73) -1.0 (if (<= y 2.3e+53) (+ 1.0 (/ y x)) -1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2e-73) {
		tmp = -1.0;
	} else if (y <= 2.3e+53) {
		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 <= (-2d-73)) then
        tmp = -1.0d0
    else if (y <= 2.3d+53) 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 <= -2e-73) {
		tmp = -1.0;
	} else if (y <= 2.3e+53) {
		tmp = 1.0 + (y / x);
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2e-73:
		tmp = -1.0
	elif y <= 2.3e+53:
		tmp = 1.0 + (y / x)
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2e-73)
		tmp = -1.0;
	elseif (y <= 2.3e+53)
		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 <= -2e-73)
		tmp = -1.0;
	elseif (y <= 2.3e+53)
		tmp = 1.0 + (y / x);
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2e-73], -1.0, If[LessEqual[y, 2.3e+53], N[(1.0 + N[(y / x), $MachinePrecision]), $MachinePrecision], -1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -1.99999999999999999e-73 or 2.3000000000000002e53 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-1} \]
   4. Step-by-step derivation

   5. Simplified 78.9%

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

   if -1.99999999999999999e-73 < y < 2.3000000000000002e53

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{\_.f64}\left(x, y\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 78.5%

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

   6. Taylor expanded in x around inf

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{y}{x}\right)}\right) \]

      2. /-lowering-/.f64 78.5%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(y, \color{blue}{x}\right)\right) \]

   8. Simplified 78.5%

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

Alternative 7: 74.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.05 \cdot 10^{-73}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{-6}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.05e-73) -1.0 (if (<= y 9.5e-6) 1.0 -1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.05e-73) {
		tmp = -1.0;
	} else if (y <= 9.5e-6) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-2.05d-73)) then
        tmp = -1.0d0
    else if (y <= 9.5d-6) then
        tmp = 1.0d0
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2.05e-73) {
		tmp = -1.0;
	} else if (y <= 9.5e-6) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.05e-73:
		tmp = -1.0
	elif y <= 9.5e-6:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.05e-73)
		tmp = -1.0;
	elseif (y <= 9.5e-6)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.05e-73)
		tmp = -1.0;
	elseif (y <= 9.5e-6)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.05e-73], -1.0, If[LessEqual[y, 9.5e-6], 1.0, -1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -2.05000000000000008e-73 or 9.5000000000000005e-6 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-1} \]
   4. Step-by-step derivation

   5. Simplified 77.3%

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

   if -2.05000000000000008e-73 < y < 9.5000000000000005e-6

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 79.9%

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

Alternative 8: 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

   \[\leadsto \color{blue}{-1} \]
4. Step-by-step derivation

5. Simplified 51.7%

   \[\leadsto \color{blue}{-1} \]
6. 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 2024161 
(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)))