Linear.Projection:perspective from linear-1.19.1.3, A

Percentage Accurate: 100.0% → 100.0%

Time: 6.9s

Alternatives: 9

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 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. div-inv N/A

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

   2. flip3-- N/A

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

   3. clear-num N/A

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

   4. *-commutative N/A

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

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

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

   6. clear-num N/A

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

   7. flip3-- N/A

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

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

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

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

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

   10. +-lowering-+.f64 99.7

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

4. Applied egg-rr 99.7%

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

   1. distribute-rgt-in N/A

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

   2. *-commutative N/A

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

   3. div-inv N/A

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

   4. frac-2neg N/A

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

   5. distribute-frac-neg N/A

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

   6. unsub-neg N/A

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

   7. --lowering--.f64 N/A

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

   8. *-commutative N/A

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

   9. un-div-inv N/A

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

   10. /-lowering-/.f64 N/A

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

   11. --lowering--.f64 N/A

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

   12. /-lowering-/.f64 N/A

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

   13. neg-sub0 N/A

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

   14. associate-+l- N/A

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

   15. neg-sub0 N/A

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

   16. +-commutative N/A

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

   17. sub-neg N/A

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

   18. --lowering--.f64 100.0

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

6. Applied egg-rr 100.0%

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

7. Final simplification 100.0%

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

Alternative 2: 98.6% accurate, 0.4× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	double tmp;
	if (((x + y) / (x - y)) <= -0.5) {
		tmp = y / (x - y);
	} else {
		tmp = 1.0 + ((y + 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 (((x + y) / (x - y)) <= (-0.5d0)) then
        tmp = y / (x - y)
    else
        tmp = 1.0d0 + ((y + y) / x)
    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 = y / (x - y);
	} else {
		tmp = 1.0 + ((y + y) / x);
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 96.1%

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

   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 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

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

      2. associate-*r/ N/A

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

      3. div-sub N/A

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

      4. *-lft-identity N/A

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

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

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. associate-*r/ N/A

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

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

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. associate-*r/ N/A

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

      14. metadata-eval N/A

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

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

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

      16. *-lft-identity N/A

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

      17. /-lowering-/.f64 N/A

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

      18. cancel-sign-sub-inv N/A

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

      19. metadata-eval N/A

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

      20. *-lft-identity N/A

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

      21. +-lowering-+.f64 99.7

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

   5. Simplified 99.7%

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

Alternative 3: 98.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	double tmp;
	if (((x + y) / (x - y)) <= -0.5) {
		tmp = y / (x - y);
	} else {
		tmp = (x + 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 (((x + y) / (x - y)) <= (-0.5d0)) then
        tmp = y / (x - y)
    else
        tmp = (x + y) / x
    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 = y / (x - y);
	} else {
		tmp = (x + y) / x;
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 96.1%

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

   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 x around inf

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

   5. Simplified 98.5%

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

Alternative 4: 98.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 96.1%

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

   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 x around inf

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

   5. Simplified 98.5%

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

Alternative 5: 98.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 96.1%

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

   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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. sub-neg N/A

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

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

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

      7. /-lowering-/.f64 96.1

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

   8. Simplified 96.1%

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

   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 x around inf

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

   5. Simplified 98.5%

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

Alternative 6: 98.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	double tmp;
	if (((x + y) / (x - y)) <= -0.5) {
		tmp = -1.0 - (x / 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 = (-1.0d0) - (x / 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 = -1.0 - (x / y);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x + y) / (x - y)) <= -0.5:
		tmp = -1.0 - (x / 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(-1.0 - Float64(x / 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 = -1.0 - (x / 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[(-1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 96.1%

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

   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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. sub-neg N/A

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

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

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

      7. /-lowering-/.f64 96.1

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

   8. Simplified 96.1%

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

   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 x around inf

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

   5. Simplified 98.4%

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

Alternative 7: 97.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x + y}{x - y} \leq -1 \cdot 10^{-309}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (+ x y) (- x y)) -1e-309) -1.0 1.0))

C

double code(double x, double y) {
	double tmp;
	if (((x + y) / (x - y)) <= -1e-309) {
		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)) <= (-1d-309)) 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)) <= -1e-309) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x + y) / (x - y)) <= -1e-309:
		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)) <= -1e-309)
		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)) <= -1e-309)
		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], -1e-309], -1.0, 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x y) (-.f64 x y)) < -1.000000000000002e-309

   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 95.9%

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

   if -1.000000000000002e-309 < (/.f64 (+.f64 x y) (-.f64 x y))

   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 98.4%

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

Alternative 8: 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 9: 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 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, 0.4× 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 2024204 
(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)))