Linear.Projection:perspective from linear-1.19.1.3, A

Percentage Accurate: 100.0% → 100.0%

Time: 7.2s

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, 0.8× 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 100.0%

   \[\frac{x + y}{x - y} \]
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 100.0%

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

4. Applied egg-rr 100.0%

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

Alternative 2: 73.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{x - y}\\ \mathbf{if}\;y \leq -6.8 \cdot 10^{-56}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-71}:\\ \;\;\;\;1 + \frac{y \cdot 2}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (- x y))))
   (if (<= y -6.8e-56) t_0 (if (<= y 2e-71) (+ 1.0 (/ (* y 2.0) x)) t_0))))

C

double code(double x, double y) {
	double t_0 = y / (x - y);
	double tmp;
	if (y <= -6.8e-56) {
		tmp = t_0;
	} else if (y <= 2e-71) {
		tmp = 1.0 + ((y * 2.0) / 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 <= (-6.8d-56)) then
        tmp = t_0
    else if (y <= 2d-71) then
        tmp = 1.0d0 + ((y * 2.0d0) / 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 <= -6.8e-56) {
		tmp = t_0;
	} else if (y <= 2e-71) {
		tmp = 1.0 + ((y * 2.0) / x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y / (x - y)
	tmp = 0
	if y <= -6.8e-56:
		tmp = t_0
	elif y <= 2e-71:
		tmp = 1.0 + ((y * 2.0) / x)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y / Float64(x - y))
	tmp = 0.0
	if (y <= -6.8e-56)
		tmp = t_0;
	elseif (y <= 2e-71)
		tmp = Float64(1.0 + Float64(Float64(y * 2.0) / 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 <= -6.8e-56)
		tmp = t_0;
	elseif (y <= 2e-71)
		tmp = 1.0 + ((y * 2.0) / 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, -6.8e-56], t$95$0, If[LessEqual[y, 2e-71], N[(1.0 + N[(N[(y * 2.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -6.79999999999999964e-56 or 1.9999999999999998e-71 < 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 72.8%

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

   if -6.79999999999999964e-56 < y < 1.9999999999999998e-71

   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 85.9%

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

   5. Simplified 85.9%

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

Alternative 3: 73.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{x - y}\\ \mathbf{if}\;y \leq -4.5 \cdot 10^{-56}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{-73}:\\ \;\;\;\;\frac{x + y}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (- x y))))
   (if (<= y -4.5e-56) t_0 (if (<= y 3.6e-73) (/ (+ x y) x) t_0))))

C

double code(double x, double y) {
	double t_0 = y / (x - y);
	double tmp;
	if (y <= -4.5e-56) {
		tmp = t_0;
	} else if (y <= 3.6e-73) {
		tmp = (x + 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 <= (-4.5d-56)) then
        tmp = t_0
    else if (y <= 3.6d-73) then
        tmp = (x + 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 <= -4.5e-56) {
		tmp = t_0;
	} else if (y <= 3.6e-73) {
		tmp = (x + y) / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y / (x - y)
	tmp = 0
	if y <= -4.5e-56:
		tmp = t_0
	elif y <= 3.6e-73:
		tmp = (x + 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 <= -4.5e-56)
		tmp = t_0;
	elseif (y <= 3.6e-73)
		tmp = Float64(Float64(x + 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 <= -4.5e-56)
		tmp = t_0;
	elseif (y <= 3.6e-73)
		tmp = (x + 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, -4.5e-56], t$95$0, If[LessEqual[y, 3.6e-73], N[(N[(x + y), $MachinePrecision] / x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.5000000000000001e-56 or 3.5999999999999999e-73 < 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 72.8%

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

   if -4.5000000000000001e-56 < y < 3.5999999999999999e-73

   1. Initial program 100.0%

      \[\frac{x + y}{x - y} \]
   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 85.7%

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

Alternative 4: 73.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{x - y}\\ \mathbf{if}\;y \leq -2.3 \cdot 10^{-55}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{-71}:\\ \;\;\;\;\frac{x}{x - y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (- x y))))
   (if (<= y -2.3e-55) t_0 (if (<= y 1.55e-71) (/ x (- x y)) t_0))))

C

double code(double x, double y) {
	double t_0 = y / (x - y);
	double tmp;
	if (y <= -2.3e-55) {
		tmp = t_0;
	} else if (y <= 1.55e-71) {
		tmp = x / (x - y);
	} 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 <= (-2.3d-55)) then
        tmp = t_0
    else if (y <= 1.55d-71) then
        tmp = x / (x - y)
    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 <= -2.3e-55) {
		tmp = t_0;
	} else if (y <= 1.55e-71) {
		tmp = x / (x - y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y / (x - y)
	tmp = 0
	if y <= -2.3e-55:
		tmp = t_0
	elif y <= 1.55e-71:
		tmp = x / (x - y)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y / Float64(x - y))
	tmp = 0.0
	if (y <= -2.3e-55)
		tmp = t_0;
	elseif (y <= 1.55e-71)
		tmp = Float64(x / Float64(x - y));
	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 <= -2.3e-55)
		tmp = t_0;
	elseif (y <= 1.55e-71)
		tmp = x / (x - y);
	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, -2.3e-55], t$95$0, If[LessEqual[y, 1.55e-71], N[(x / N[(x - y), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.30000000000000011e-55 or 1.55000000000000001e-71 < 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 72.8%

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

   if -2.30000000000000011e-55 < y < 1.55000000000000001e-71

   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 85.5%

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

Alternative 5: 73.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -1 - \frac{x}{y}\\ \mathbf{if}\;y \leq -1.35 \cdot 10^{-55}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{-71}:\\ \;\;\;\;\frac{x}{x - y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- -1.0 (/ x y))))
   (if (<= y -1.35e-55) t_0 (if (<= y 3.3e-71) (/ x (- x y)) t_0))))

C

double code(double x, double y) {
	double t_0 = -1.0 - (x / y);
	double tmp;
	if (y <= -1.35e-55) {
		tmp = t_0;
	} else if (y <= 3.3e-71) {
		tmp = x / (x - y);
	} 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 <= (-1.35d-55)) then
        tmp = t_0
    else if (y <= 3.3d-71) then
        tmp = x / (x - y)
    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 <= -1.35e-55) {
		tmp = t_0;
	} else if (y <= 3.3e-71) {
		tmp = x / (x - y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = -1.0 - (x / y)
	tmp = 0
	if y <= -1.35e-55:
		tmp = t_0
	elif y <= 3.3e-71:
		tmp = x / (x - y)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(-1.0 - Float64(x / y))
	tmp = 0.0
	if (y <= -1.35e-55)
		tmp = t_0;
	elseif (y <= 3.3e-71)
		tmp = Float64(x / Float64(x - y));
	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 <= -1.35e-55)
		tmp = t_0;
	elseif (y <= 3.3e-71)
		tmp = x / (x - y);
	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, -1.35e-55], t$95$0, If[LessEqual[y, 3.3e-71], N[(x / N[(x - y), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.35000000000000002e-55 or 3.3000000000000002e-71 < 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 72.8%

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

   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 72.5%

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

   8. Simplified 72.5%

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

   if -1.35000000000000002e-55 < y < 3.3000000000000002e-71

   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 85.5%

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

Alternative 6: 72.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -1 - \frac{x}{y}\\ \mathbf{if}\;y \leq -7.4 \cdot 10^{-56}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{-71}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- -1.0 (/ x y))))
   (if (<= y -7.4e-56) t_0 (if (<= y 1.15e-71) 1.0 t_0))))

C

double code(double x, double y) {
	double t_0 = -1.0 - (x / y);
	double tmp;
	if (y <= -7.4e-56) {
		tmp = t_0;
	} else if (y <= 1.15e-71) {
		tmp = 1.0;
	} 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 <= (-7.4d-56)) then
        tmp = t_0
    else if (y <= 1.15d-71) then
        tmp = 1.0d0
    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 <= -7.4e-56) {
		tmp = t_0;
	} else if (y <= 1.15e-71) {
		tmp = 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = -1.0 - (x / y)
	tmp = 0
	if y <= -7.4e-56:
		tmp = t_0
	elif y <= 1.15e-71:
		tmp = 1.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(-1.0 - Float64(x / y))
	tmp = 0.0
	if (y <= -7.4e-56)
		tmp = t_0;
	elseif (y <= 1.15e-71)
		tmp = 1.0;
	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 <= -7.4e-56)
		tmp = t_0;
	elseif (y <= 1.15e-71)
		tmp = 1.0;
	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, -7.4e-56], t$95$0, If[LessEqual[y, 1.15e-71], 1.0, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -7.4000000000000004e-56 or 1.1499999999999999e-71 < 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 72.8%

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

   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 72.5%

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

   8. Simplified 72.5%

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

   if -7.4000000000000004e-56 < y < 1.1499999999999999e-71

   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 85.2%

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

Alternative 7: 72.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.8 \cdot 10^{-56}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-71}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -7.8e-56) -1.0 (if (<= y 2e-71) 1.0 -1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -7.8e-56) {
		tmp = -1.0;
	} else if (y <= 2e-71) {
		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 <= (-7.8d-56)) then
        tmp = -1.0d0
    else if (y <= 2d-71) 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 <= -7.8e-56) {
		tmp = -1.0;
	} else if (y <= 2e-71) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -7.8e-56:
		tmp = -1.0
	elif y <= 2e-71:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -7.8e-56)
		tmp = -1.0;
	elseif (y <= 2e-71)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -7.8e-56)
		tmp = -1.0;
	elseif (y <= 2e-71)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -7.8e-56], -1.0, If[LessEqual[y, 2e-71], 1.0, -1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -7.8e-56 or 1.9999999999999998e-71 < 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 72.0%

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

   if -7.8e-56 < y < 1.9999999999999998e-71

   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 85.2%

      \[\leadsto \color{blue}{1} \]
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 100.0%

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

Alternative 9: 49.6% 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 50.1%

   \[\leadsto \color{blue}{-1} \]
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 2024158 
(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)))