Linear.Projection:perspective from linear-1.19.1.3, A

Percentage Accurate: 100.0% → 100.0%

Time: 6.4s

Alternatives: 7

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, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(N[(x / N[(x - y), $MachinePrecision]), $MachinePrecision] + N[(y / 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. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lift--.f64 N/A

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

   5. flip3-- N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{{x}^{3} - {y}^{3}}{x \cdot x + \left(y \cdot y + x \cdot y\right)}}} \cdot \left(x + y\right) \]

   6. clear-num N/A

      \[\leadsto \color{blue}{\frac{x \cdot x + \left(y \cdot y + x \cdot y\right)}{{x}^{3} - {y}^{3}}} \cdot \left(x + y\right) \]

   7. lift-+.f64 N/A

      \[\leadsto \frac{x \cdot x + \left(y \cdot y + x \cdot y\right)}{{x}^{3} - {y}^{3}} \cdot \color{blue}{\left(x + y\right)} \]

   8. distribute-lft-in N/A

      \[\leadsto \color{blue}{\frac{x \cdot x + \left(y \cdot y + x \cdot y\right)}{{x}^{3} - {y}^{3}} \cdot x + \frac{x \cdot x + \left(y \cdot y + x \cdot y\right)}{{x}^{3} - {y}^{3}} \cdot y} \]

   9. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x \cdot x + \left(y \cdot y + x \cdot y\right)}{{x}^{3} - {y}^{3}}, x, \frac{x \cdot x + \left(y \cdot y + x \cdot y\right)}{{x}^{3} - {y}^{3}} \cdot y\right)} \]

   10. clear-num N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{\frac{{x}^{3} - {y}^{3}}{x \cdot x + \left(y \cdot y + x \cdot y\right)}}}, x, \frac{x \cdot x + \left(y \cdot y + x \cdot y\right)}{{x}^{3} - {y}^{3}} \cdot y\right) \]

   11. flip3-- N/A

       \[\leadsto \mathsf{fma}\left(\frac{1}{\color{blue}{x - y}}, x, \frac{x \cdot x + \left(y \cdot y + x \cdot y\right)}{{x}^{3} - {y}^{3}} \cdot y\right) \]

   12. lift--.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{1}{\color{blue}{x - y}}, x, \frac{x \cdot x + \left(y \cdot y + x \cdot y\right)}{{x}^{3} - {y}^{3}} \cdot y\right) \]

   13. inv-pow N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{{\left(x - y\right)}^{-1}}, x, \frac{x \cdot x + \left(y \cdot y + x \cdot y\right)}{{x}^{3} - {y}^{3}} \cdot y\right) \]

   14. lower-pow.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{{\left(x - y\right)}^{-1}}, x, \frac{x \cdot x + \left(y \cdot y + x \cdot y\right)}{{x}^{3} - {y}^{3}} \cdot y\right) \]

   15. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left({\left(x - y\right)}^{-1}, x, \color{blue}{\frac{x \cdot x + \left(y \cdot y + x \cdot y\right)}{{x}^{3} - {y}^{3}} \cdot y}\right) \]

4. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left({\left(x - y\right)}^{-1}, x, {\left(x - y\right)}^{-1} \cdot y\right)} \]
5. Step-by-step derivation

   1. lift-fma.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-pow.f64 N/A

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

   7. unpow-1 N/A

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

   8. un-div-inv N/A

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

   9. lower-/.f64 N/A

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

   10. *-commutative N/A

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

   11. lift-pow.f64 N/A

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

   12. unpow-1 N/A

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

   13. un-div-inv N/A

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

   14. lower-/.f64 100.0

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

6. Applied rewrites 100.0%

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

7. Final simplification 100.0%

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

Alternative 2: 98.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x + y}{x - y} \leq -0.5:\\ \;\;\;\;\mathsf{fma}\left(\frac{-2}{y}, x, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{2}{x}, y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (+ x y) (- x y)) -0.5)
   (fma (/ -2.0 y) x -1.0)
   (fma (/ 2.0 x) y 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (((x + y) / (x - y)) <= -0.5) {
		tmp = fma((-2.0 / y), x, -1.0);
	} else {
		tmp = fma((2.0 / x), y, 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x + y) / Float64(x - y)) <= -0.5)
		tmp = fma(Float64(-2.0 / y), x, -1.0);
	else
		tmp = fma(Float64(2.0 / x), y, 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x + y), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision], -0.5], N[(N[(-2.0 / y), $MachinePrecision] * x + -1.0), $MachinePrecision], N[(N[(2.0 / x), $MachinePrecision] * y + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x y) (-.f64 x y)) < -0.5

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. associate-*r/ N/A

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

      4. div-sub N/A

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

      5. sub-neg N/A

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

      6. remove-double-neg N/A

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

      7. mul-1-neg N/A

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

      8. neg-mul-1 N/A

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

      9. distribute-lft-out-- N/A

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

      10. mul-1-neg N/A

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

      11. distribute-neg-frac N/A

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

   5. Applied rewrites 99.6%

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

   if -0.5 < (/.f64 (+.f64 x y) (-.f64 x y))

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-lft-identity N/A

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

      3. associate-*l/ N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{2}}{x}, y, 1\right) \]

      8. lower-/.f64 98.6

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{2}{x}}, y, 1\right) \]

   5. Applied rewrites 98.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{2}{x}, y, 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 98.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x + y}{x - y} \leq -0.5:\\ \;\;\;\;\mathsf{fma}\left(\frac{-2}{y}, x, -1\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (+ x y) (- x y)) -0.5) (fma (/ -2.0 y) x -1.0) 1.0))

C

double code(double x, double y) {
	double tmp;
	if (((x + y) / (x - y)) <= -0.5) {
		tmp = fma((-2.0 / y), x, -1.0);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x + y) / Float64(x - y)) <= -0.5)
		tmp = fma(Float64(-2.0 / y), x, -1.0);
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x + y), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision], -0.5], N[(N[(-2.0 / y), $MachinePrecision] * x + -1.0), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x y) (-.f64 x y)) < -0.5

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. associate-*r/ N/A

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

      4. div-sub N/A

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

      5. sub-neg N/A

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

      6. remove-double-neg N/A

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

      7. mul-1-neg N/A

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

      8. neg-mul-1 N/A

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

      9. distribute-lft-out-- N/A

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

      10. mul-1-neg N/A

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

      11. distribute-neg-frac N/A

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

   5. Applied rewrites 99.6%

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

   if -0.5 < (/.f64 (+.f64 x y) (-.f64 x y))

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 97.5%

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

Alternative 4: 97.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x + y}{x - y} \leq -0.5:\\ \;\;\;\;\frac{x + y}{-y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (+ x y) (- x y)) -0.5) (/ (+ x y) (- y)) 1.0))

C

double code(double x, double y) {
	double tmp;
	if (((x + y) / (x - y)) <= -0.5) {
		tmp = (x + y) / -y;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (((x + y) / (x - y)) <= (-0.5d0)) then
        tmp = (x + y) / -y
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((x + y) / (x - y)) <= -0.5) {
		tmp = (x + y) / -y;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x + y) / (x - y)) <= -0.5:
		tmp = (x + y) / -y
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x + y) / Float64(x - y)) <= -0.5)
		tmp = Float64(Float64(x + y) / Float64(-y));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((x + y) / (x - y)) <= -0.5)
		tmp = (x + y) / -y;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x + y), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision], -0.5], N[(N[(x + y), $MachinePrecision] / (-y)), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x y) (-.f64 x y)) < -0.5

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 98.5

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

   5. Applied rewrites 98.5%

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

   if -0.5 < (/.f64 (+.f64 x y) (-.f64 x y))

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 97.5%

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

Alternative 5: 97.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x + y}{x - y} \leq -4 \cdot 10^{-311}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (+ x y) (- x y)) -4e-311) -1.0 1.0))

C

double code(double x, double y) {
	double tmp;
	if (((x + y) / (x - y)) <= -4e-311) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (((x + y) / (x - y)) <= (-4d-311)) then
        tmp = -1.0d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((x + y) / (x - y)) <= -4e-311) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x + y) / (x - y)) <= -4e-311:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x + y) / Float64(x - y)) <= -4e-311)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((x + y) / (x - y)) <= -4e-311)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x + y), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision], -4e-311], -1.0, 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x y) (-.f64 x y)) < -3.99999999999979e-311

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 98.5%

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

   if -3.99999999999979e-311 < (/.f64 (+.f64 x y) (-.f64 x y))

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 97.5%

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

Alternative 6: 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 7: 49.6% accurate, 18.0× 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 y around inf

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

5. Applied rewrites 53.1%

   \[\leadsto \color{blue}{-1} \]
6. Add Preprocessing

Developer Target 1: 100.0% accurate, 0.4× 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 2024276 
(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)))