Linear.Projection:perspective from linear-1.19.1.3, A

Percentage Accurate: 100.0% → 100.0%

Time: 7.4s

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, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{x - y}{x + y}} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ 1.0 (/ (- x y) (+ x y))))

C

double code(double x, double y) {
	return 1.0 / ((x - y) / (x + y));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 / ((x - y) / (x + y))
end function

Java

public static double code(double x, double y) {
	return 1.0 / ((x - y) / (x + y));
}

Python

def code(x, y):
	return 1.0 / ((x - y) / (x + y))

Julia

function code(x, y)
	return Float64(1.0 / Float64(Float64(x - y) / Float64(x + y)))
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 / ((x - y) / (x + y));
end

Wolfram

code[x_, y_] := N[(1.0 / N[(N[(x - y), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\frac{x + y}{x - y} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. clear-num N/A

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

   2. /-lowering-/.f64 N/A

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

   3. /-lowering-/.f64 N/A

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

   4. --lowering--.f64 N/A

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

   5. +-lowering-+.f64 99.9%

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

4. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{\frac{1}{\frac{x - y}{x + y}}} \]
5. Add Preprocessing

Alternative 2: 74.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -1 + -2 \cdot \frac{x}{y}\\ \mathbf{if}\;y \leq -4.1 \cdot 10^{+24}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{+43}:\\ \;\;\;\;1 + \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ -1.0 (* -2.0 (/ x y)))))
   (if (<= y -4.1e+24) t_0 (if (<= y 1.45e+43) (+ 1.0 (/ y x)) t_0))))

C

double code(double x, double y) {
	double t_0 = -1.0 + (-2.0 * (x / y));
	double tmp;
	if (y <= -4.1e+24) {
		tmp = t_0;
	} else if (y <= 1.45e+43) {
		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) + ((-2.0d0) * (x / y))
    if (y <= (-4.1d+24)) then
        tmp = t_0
    else if (y <= 1.45d+43) 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 + (-2.0 * (x / y));
	double tmp;
	if (y <= -4.1e+24) {
		tmp = t_0;
	} else if (y <= 1.45e+43) {
		tmp = 1.0 + (y / x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = -1.0 + (-2.0 * (x / y))
	tmp = 0
	if y <= -4.1e+24:
		tmp = t_0
	elif y <= 1.45e+43:
		tmp = 1.0 + (y / x)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(-1.0 + Float64(-2.0 * Float64(x / y)))
	tmp = 0.0
	if (y <= -4.1e+24)
		tmp = t_0;
	elseif (y <= 1.45e+43)
		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 + (-2.0 * (x / y));
	tmp = 0.0;
	if (y <= -4.1e+24)
		tmp = t_0;
	elseif (y <= 1.45e+43)
		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[(-2.0 * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -4.1e+24], t$95$0, If[LessEqual[y, 1.45e+43], N[(1.0 + N[(y / x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.1000000000000001e24 or 1.4500000000000001e43 < y

   1. Initial program 99.9%

      \[\frac{x + y}{x - y} \]
   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. associate-*l/ N/A

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

      4. metadata-eval N/A

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

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

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

      6. *-lft-identity N/A

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

      7. metadata-eval 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 81.0%

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

   5. Simplified 81.0%

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

   if -4.1000000000000001e24 < y < 1.4500000000000001e43

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 75.8%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. div-inv N/A

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

      6. div-inv N/A

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

      7. *-inverses N/A

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

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

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

      9. /-lowering-/.f64 75.8%

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

   7. Applied egg-rr 75.8%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.1 \cdot 10^{+24}:\\ \;\;\;\;-1 + -2 \cdot \frac{x}{y}\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{+43}:\\ \;\;\;\;1 + \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;-1 + -2 \cdot \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{x - y}\\ \mathbf{if}\;y \leq -3.75 \cdot 10^{+24}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{+38}:\\ \;\;\;\;1 + \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (- x y))))
   (if (<= y -3.75e+24) t_0 (if (<= y 1.3e+38) (+ 1.0 (/ y x)) t_0))))

C

double code(double x, double y) {
	double t_0 = y / (x - y);
	double tmp;
	if (y <= -3.75e+24) {
		tmp = t_0;
	} else if (y <= 1.3e+38) {
		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 <= (-3.75d+24)) then
        tmp = t_0
    else if (y <= 1.3d+38) 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 <= -3.75e+24) {
		tmp = t_0;
	} else if (y <= 1.3e+38) {
		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 <= -3.75e+24:
		tmp = t_0
	elif y <= 1.3e+38:
		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 <= -3.75e+24)
		tmp = t_0;
	elseif (y <= 1.3e+38)
		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 <= -3.75e+24)
		tmp = t_0;
	elseif (y <= 1.3e+38)
		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, -3.75e+24], t$95$0, If[LessEqual[y, 1.3e+38], N[(1.0 + N[(y / x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.75000000000000007e24 or 1.3e38 < y

   1. Initial program 99.9%

      \[\frac{x + y}{x - y} \]
   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 80.0%

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

   if -3.75000000000000007e24 < y < 1.3e38

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 76.2%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. div-inv N/A

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

      6. div-inv N/A

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

      7. *-inverses N/A

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

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

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

      9. /-lowering-/.f64 76.2%

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

   7. Applied egg-rr 76.2%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.75 \cdot 10^{+24}:\\ \;\;\;\;\frac{y}{x - y}\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{+38}:\\ \;\;\;\;1 + \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -1 - \frac{x}{y}\\ \mathbf{if}\;y \leq -6.4 \cdot 10^{+23}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{+41}:\\ \;\;\;\;1 + \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- -1.0 (/ x y))))
   (if (<= y -6.4e+23) t_0 (if (<= y 1.1e+41) (+ 1.0 (/ y x)) t_0))))

C

double code(double x, double y) {
	double t_0 = -1.0 - (x / y);
	double tmp;
	if (y <= -6.4e+23) {
		tmp = t_0;
	} else if (y <= 1.1e+41) {
		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 <= (-6.4d+23)) then
        tmp = t_0
    else if (y <= 1.1d+41) 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 <= -6.4e+23) {
		tmp = t_0;
	} else if (y <= 1.1e+41) {
		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 <= -6.4e+23:
		tmp = t_0
	elif y <= 1.1e+41:
		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 <= -6.4e+23)
		tmp = t_0;
	elseif (y <= 1.1e+41)
		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 <= -6.4e+23)
		tmp = t_0;
	elseif (y <= 1.1e+41)
		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, -6.4e+23], t$95$0, If[LessEqual[y, 1.1e+41], N[(1.0 + N[(y / x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -6.4e23 or 1.09999999999999995e41 < y

   1. Initial program 99.9%

      \[\frac{x + y}{x - y} \]
   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 80.4%

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

   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 80.1%

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

   8. Simplified 80.1%

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

   if -6.4e23 < y < 1.09999999999999995e41

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 75.8%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. div-inv N/A

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

      6. div-inv N/A

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

      7. *-inverses N/A

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

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

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

      9. /-lowering-/.f64 75.8%

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

   7. Applied egg-rr 75.8%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.4 \cdot 10^{+23}:\\ \;\;\;\;-1 - \frac{x}{y}\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{+41}:\\ \;\;\;\;1 + \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;-1 - \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.8 \cdot 10^{+23}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{+39}:\\ \;\;\;\;1 + \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -7.8e+23) -1.0 (if (<= y 2.8e+39) (+ 1.0 (/ y x)) -1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -7.8e+23) {
		tmp = -1.0;
	} else if (y <= 2.8e+39) {
		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 <= (-7.8d+23)) then
        tmp = -1.0d0
    else if (y <= 2.8d+39) 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 <= -7.8e+23) {
		tmp = -1.0;
	} else if (y <= 2.8e+39) {
		tmp = 1.0 + (y / x);
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -7.8e+23:
		tmp = -1.0
	elif y <= 2.8e+39:
		tmp = 1.0 + (y / x)
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -7.8e+23)
		tmp = -1.0;
	elseif (y <= 2.8e+39)
		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 <= -7.8e+23)
		tmp = -1.0;
	elseif (y <= 2.8e+39)
		tmp = 1.0 + (y / x);
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -7.8e+23], -1.0, If[LessEqual[y, 2.8e+39], N[(1.0 + N[(y / x), $MachinePrecision]), $MachinePrecision], -1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -7.8000000000000001e23 or 2.80000000000000001e39 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 79.2%

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

   if -7.8000000000000001e23 < y < 2.80000000000000001e39

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 76.2%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. div-inv N/A

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

      6. div-inv N/A

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

      7. *-inverses N/A

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

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

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

      9. /-lowering-/.f64 76.2%

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

   7. Applied egg-rr 76.2%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.8 \cdot 10^{+23}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{+39}:\\ \;\;\;\;1 + \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{+24}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+28}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1e+24) -1.0 (if (<= y 5e+28) 1.0 -1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1e+24) {
		tmp = -1.0;
	} else if (y <= 5e+28) {
		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 <= (-1d+24)) then
        tmp = -1.0d0
    else if (y <= 5d+28) 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 <= -1e+24) {
		tmp = -1.0;
	} else if (y <= 5e+28) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1e+24:
		tmp = -1.0
	elif y <= 5e+28:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1e+24)
		tmp = -1.0;
	elseif (y <= 5e+28)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1e+24)
		tmp = -1.0;
	elseif (y <= 5e+28)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1e+24], -1.0, If[LessEqual[y, 5e+28], 1.0, -1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -9.9999999999999998e23 or 4.99999999999999957e28 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 78.8%

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

   if -9.9999999999999998e23 < y < 4.99999999999999957e28

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 75.8%

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

Alternative 7: 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 8: 49.8% 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

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

5. Simplified 49.5%

   \[\leadsto \color{blue}{-1} \]
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 2024138 
(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)))