Linear.Projection:perspective from linear-1.19.1.3, B

Percentage Accurate: 76.9% → 99.8%

Time: 5.1s

Alternatives: 5

Speedup: 0.5×

Specification

Math

\[\begin{array}{l} \\ \frac{\left(x \cdot 2\right) \cdot y}{x - y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ (* (* x 2.0) y) (- x y)))

C

double code(double x, double y) {
	return ((x * 2.0) * y) / (x - y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((x * 2.0d0) * y) / (x - y)
end function

Java

public static double code(double x, double y) {
	return ((x * 2.0) * y) / (x - y);
}

Python

def code(x, y):
	return ((x * 2.0) * y) / (x - y)

Julia

function code(x, y)
	return Float64(Float64(Float64(x * 2.0) * y) / Float64(x - y))
end

MATLAB

function tmp = code(x, y)
	tmp = ((x * 2.0) * y) / (x - y);
end

Wolfram

code[x_, y_] := N[(N[(N[(x * 2.0), $MachinePrecision] * y), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision]

Initial Program: 76.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(x \cdot 2\right) \cdot y}{x - y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ (* (* x 2.0) y) (- x y)))

C

double code(double x, double y) {
	return ((x * 2.0) * y) / (x - y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((x * 2.0d0) * y) / (x - y)
end function

Java

public static double code(double x, double y) {
	return ((x * 2.0) * y) / (x - y);
}

Python

def code(x, y):
	return ((x * 2.0) * y) / (x - y)

Julia

function code(x, y)
	return Float64(Float64(Float64(x * 2.0) * y) / Float64(x - y))
end

MATLAB

function tmp = code(x, y)
	tmp = ((x * 2.0) * y) / (x - y);
end

Wolfram

code[x_, y_] := N[(N[(N[(x * 2.0), $MachinePrecision] * y), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(x \cdot \frac{2}{x - y}\right)\\ \mathbf{if}\;x \leq -5500000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 7000000000000:\\ \;\;\;\;\frac{\frac{x}{\frac{x - y}{y}}}{0.5}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* x (/ 2.0 (- x y))))))
   (if (<= x -5500000000000.0)
     t_0
     (if (<= x 7000000000000.0) (/ (/ x (/ (- x y) y)) 0.5) t_0))))

C

double code(double x, double y) {
	double t_0 = y * (x * (2.0 / (x - y)));
	double tmp;
	if (x <= -5500000000000.0) {
		tmp = t_0;
	} else if (x <= 7000000000000.0) {
		tmp = (x / ((x - y) / y)) / 0.5;
	} 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 * (2.0d0 / (x - y)))
    if (x <= (-5500000000000.0d0)) then
        tmp = t_0
    else if (x <= 7000000000000.0d0) then
        tmp = (x / ((x - y) / y)) / 0.5d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y * (x * (2.0 / (x - y)));
	double tmp;
	if (x <= -5500000000000.0) {
		tmp = t_0;
	} else if (x <= 7000000000000.0) {
		tmp = (x / ((x - y) / y)) / 0.5;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (x * (2.0 / (x - y)))
	tmp = 0
	if x <= -5500000000000.0:
		tmp = t_0
	elif x <= 7000000000000.0:
		tmp = (x / ((x - y) / y)) / 0.5
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(x * Float64(2.0 / Float64(x - y))))
	tmp = 0.0
	if (x <= -5500000000000.0)
		tmp = t_0;
	elseif (x <= 7000000000000.0)
		tmp = Float64(Float64(x / Float64(Float64(x - y) / y)) / 0.5);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * (x * (2.0 / (x - y)));
	tmp = 0.0;
	if (x <= -5500000000000.0)
		tmp = t_0;
	elseif (x <= 7000000000000.0)
		tmp = (x / ((x - y) / y)) / 0.5;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(x * N[(2.0 / N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5500000000000.0], t$95$0, If[LessEqual[x, 7000000000000.0], N[(N[(x / N[(N[(x - y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / 0.5), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -5.5e12 or 7e12 < x

   1. Initial program 80.3%

      \[\frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
   2. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

      4. associate-/l* N/A

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

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

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

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

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

      7. --lowering--.f64 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{y \cdot \left(x \cdot \frac{2}{x - y}\right)} \]
   4. Add Preprocessing

   if -5.5e12 < x < 7e12

   1. Initial program 79.0%

      \[\frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
   2. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

      4. associate-/l* N/A

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

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

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

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

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

      7. --lowering--.f64 78.5%

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

   3. Simplified 78.5%

      \[\leadsto \color{blue}{y \cdot \left(x \cdot \frac{2}{x - y}\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. *-commutative N/A

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

      5. div-inv N/A

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

      6. associate-/r* N/A

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

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

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

      8. associate-*r/ N/A

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

      9. clear-num N/A

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

      10. un-div-inv N/A

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

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

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

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

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

      13. --lowering--.f64 N/A

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

      14. metadata-eval 100.0%

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

   6. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{\frac{x}{\frac{x - y}{y}}}{0.5}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 99.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y \cdot \left(x \cdot 2\right)}{x - y}\\ t_1 := y \cdot \left(x \cdot \frac{2}{x - y}\right)\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+59}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq -2 \cdot 10^{-307}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+59}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* y (* x 2.0)) (- x y))) (t_1 (* y (* x (/ 2.0 (- x y))))))
   (if (<= t_0 -1e+59)
     t_1
     (if (<= t_0 -2e-307)
       t_0
       (if (<= t_0 0.0) t_1 (if (<= t_0 5e+59) t_0 t_1))))))

C

double code(double x, double y) {
	double t_0 = (y * (x * 2.0)) / (x - y);
	double t_1 = y * (x * (2.0 / (x - y)));
	double tmp;
	if (t_0 <= -1e+59) {
		tmp = t_1;
	} else if (t_0 <= -2e-307) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 5e+59) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = (y * (x * 2.0d0)) / (x - y)
    t_1 = y * (x * (2.0d0 / (x - y)))
    if (t_0 <= (-1d+59)) then
        tmp = t_1
    else if (t_0 <= (-2d-307)) then
        tmp = t_0
    else if (t_0 <= 0.0d0) then
        tmp = t_1
    else if (t_0 <= 5d+59) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (y * (x * 2.0)) / (x - y);
	double t_1 = y * (x * (2.0 / (x - y)));
	double tmp;
	if (t_0 <= -1e+59) {
		tmp = t_1;
	} else if (t_0 <= -2e-307) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 5e+59) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y * (x * 2.0)) / (x - y)
	t_1 = y * (x * (2.0 / (x - y)))
	tmp = 0
	if t_0 <= -1e+59:
		tmp = t_1
	elif t_0 <= -2e-307:
		tmp = t_0
	elif t_0 <= 0.0:
		tmp = t_1
	elif t_0 <= 5e+59:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y * Float64(x * 2.0)) / Float64(x - y))
	t_1 = Float64(y * Float64(x * Float64(2.0 / Float64(x - y))))
	tmp = 0.0
	if (t_0 <= -1e+59)
		tmp = t_1;
	elseif (t_0 <= -2e-307)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 5e+59)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y * (x * 2.0)) / (x - y);
	t_1 = y * (x * (2.0 / (x - y)));
	tmp = 0.0;
	if (t_0 <= -1e+59)
		tmp = t_1;
	elseif (t_0 <= -2e-307)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 5e+59)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * N[(x * 2.0), $MachinePrecision]), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y * N[(x * N[(2.0 / N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+59], t$95$1, If[LessEqual[t$95$0, -2e-307], t$95$0, If[LessEqual[t$95$0, 0.0], t$95$1, If[LessEqual[t$95$0, 5e+59], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (*.f64 x #s(literal 2 binary64)) y) (-.f64 x y)) < -9.99999999999999972e58 or -1.99999999999999982e-307 < (/.f64 (*.f64 (*.f64 x #s(literal 2 binary64)) y) (-.f64 x y)) < 0.0 or 4.9999999999999997e59 < (/.f64 (*.f64 (*.f64 x #s(literal 2 binary64)) y) (-.f64 x y))

   1. Initial program 19.7%

      \[\frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
   2. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

      4. associate-/l* N/A

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

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

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

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

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

      7. --lowering--.f64 99.7%

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

   3. Simplified 99.7%

      \[\leadsto \color{blue}{y \cdot \left(x \cdot \frac{2}{x - y}\right)} \]
   4. Add Preprocessing

   if -9.99999999999999972e58 < (/.f64 (*.f64 (*.f64 x #s(literal 2 binary64)) y) (-.f64 x y)) < -1.99999999999999982e-307 or 0.0 < (/.f64 (*.f64 (*.f64 x #s(literal 2 binary64)) y) (-.f64 x y)) < 4.9999999999999997e59

   1. Initial program 99.2%

      \[\frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(x \cdot 2\right)}{x - y} \leq -1 \cdot 10^{+59}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{2}{x - y}\right)\\ \mathbf{elif}\;\frac{y \cdot \left(x \cdot 2\right)}{x - y} \leq -2 \cdot 10^{-307}:\\ \;\;\;\;\frac{y \cdot \left(x \cdot 2\right)}{x - y}\\ \mathbf{elif}\;\frac{y \cdot \left(x \cdot 2\right)}{x - y} \leq 0:\\ \;\;\;\;y \cdot \left(x \cdot \frac{2}{x - y}\right)\\ \mathbf{elif}\;\frac{y \cdot \left(x \cdot 2\right)}{x - y} \leq 5 \cdot 10^{+59}:\\ \;\;\;\;\frac{y \cdot \left(x \cdot 2\right)}{x - y}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{2}{x - y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.9 \cdot 10^{+187}:\\ \;\;\;\;x \cdot -2\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+201}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{2}{x - y}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot -2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -4.9e+187)
   (* x -2.0)
   (if (<= y 2e+201) (* y (* x (/ 2.0 (- x y)))) (* x -2.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -4.9e+187) {
		tmp = x * -2.0;
	} else if (y <= 2e+201) {
		tmp = y * (x * (2.0 / (x - y)));
	} else {
		tmp = x * -2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-4.9d+187)) then
        tmp = x * (-2.0d0)
    else if (y <= 2d+201) then
        tmp = y * (x * (2.0d0 / (x - y)))
    else
        tmp = x * (-2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -4.9e+187) {
		tmp = x * -2.0;
	} else if (y <= 2e+201) {
		tmp = y * (x * (2.0 / (x - y)));
	} else {
		tmp = x * -2.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -4.9e+187:
		tmp = x * -2.0
	elif y <= 2e+201:
		tmp = y * (x * (2.0 / (x - y)))
	else:
		tmp = x * -2.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -4.9e+187)
		tmp = Float64(x * -2.0);
	elseif (y <= 2e+201)
		tmp = Float64(y * Float64(x * Float64(2.0 / Float64(x - y))));
	else
		tmp = Float64(x * -2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -4.9e+187)
		tmp = x * -2.0;
	elseif (y <= 2e+201)
		tmp = y * (x * (2.0 / (x - y)));
	else
		tmp = x * -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -4.9e+187], N[(x * -2.0), $MachinePrecision], If[LessEqual[y, 2e+201], N[(y * N[(x * N[(2.0 / N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * -2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.9000000000000003e187 or 2.00000000000000008e201 < y

   1. Initial program 76.7%

      \[\frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
   2. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

      4. associate-/l* N/A

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

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

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

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

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

      7. --lowering--.f64 48.8%

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

   3. Simplified 48.8%

      \[\leadsto \color{blue}{y \cdot \left(x \cdot \frac{2}{x - y}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf

      \[\leadsto \color{blue}{-2 \cdot x} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto x \cdot \color{blue}{-2} \]

      2. *-lowering-*.f64 95.3%

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

   7. Simplified 95.3%

      \[\leadsto \color{blue}{x \cdot -2} \]

   if -4.9000000000000003e187 < y < 2.00000000000000008e201

   1. Initial program 80.2%

      \[\frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
   2. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

      4. associate-/l* N/A

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

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

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

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

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

      7. --lowering--.f64 96.1%

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

   3. Simplified 96.1%

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

Alternative 4: 75.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.8 \cdot 10^{+20}:\\ \;\;\;\;x \cdot -2\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{+18}:\\ \;\;\;\;y \cdot 2\\ \mathbf{else}:\\ \;\;\;\;x \cdot -2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -7.8e+20) (* x -2.0) (if (<= y 7.2e+18) (* y 2.0) (* x -2.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -7.8e+20) {
		tmp = x * -2.0;
	} else if (y <= 7.2e+18) {
		tmp = y * 2.0;
	} else {
		tmp = x * -2.0;
	}
	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+20)) then
        tmp = x * (-2.0d0)
    else if (y <= 7.2d+18) then
        tmp = y * 2.0d0
    else
        tmp = x * (-2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -7.8e+20) {
		tmp = x * -2.0;
	} else if (y <= 7.2e+18) {
		tmp = y * 2.0;
	} else {
		tmp = x * -2.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -7.8e+20:
		tmp = x * -2.0
	elif y <= 7.2e+18:
		tmp = y * 2.0
	else:
		tmp = x * -2.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -7.8e+20)
		tmp = Float64(x * -2.0);
	elseif (y <= 7.2e+18)
		tmp = Float64(y * 2.0);
	else
		tmp = Float64(x * -2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -7.8e+20)
		tmp = x * -2.0;
	elseif (y <= 7.2e+18)
		tmp = y * 2.0;
	else
		tmp = x * -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -7.8e+20], N[(x * -2.0), $MachinePrecision], If[LessEqual[y, 7.2e+18], N[(y * 2.0), $MachinePrecision], N[(x * -2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -7.8e20 or 7.2e18 < y

   1. Initial program 79.4%

      \[\frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
   2. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

      4. associate-/l* N/A

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

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

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

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

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

      7. --lowering--.f64 71.3%

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

   3. Simplified 71.3%

      \[\leadsto \color{blue}{y \cdot \left(x \cdot \frac{2}{x - y}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf

      \[\leadsto \color{blue}{-2 \cdot x} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto x \cdot \color{blue}{-2} \]

      2. *-lowering-*.f64 84.2%

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

   7. Simplified 84.2%

      \[\leadsto \color{blue}{x \cdot -2} \]

   if -7.8e20 < y < 7.2e18

   1. Initial program 79.7%

      \[\frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
   2. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

      4. associate-/l* N/A

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

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

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

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

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

      7. --lowering--.f64 99.7%

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

   3. Simplified 99.7%

      \[\leadsto \color{blue}{y \cdot \left(x \cdot \frac{2}{x - y}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{2}\right) \]
   6. Step-by-step derivation

   7. Simplified 76.4%

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

Alternative 5: 50.9% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ x \cdot -2 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* x -2.0))

C

double code(double x, double y) {
	return x * -2.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x * (-2.0d0)
end function

Java

public static double code(double x, double y) {
	return x * -2.0;
}

Python

def code(x, y):
	return x * -2.0

Julia

function code(x, y)
	return Float64(x * -2.0)
end

MATLAB

function tmp = code(x, y)
	tmp = x * -2.0;
end

Wolfram

code[x_, y_] := N[(x * -2.0), $MachinePrecision]

Derivation

1. Initial program 79.6%

   \[\frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
2. Step-by-step derivation

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

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

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

   4. associate-/l* N/A

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

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

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

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

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

   7. --lowering--.f64 87.8%

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

3. Simplified 87.8%

   \[\leadsto \color{blue}{y \cdot \left(x \cdot \frac{2}{x - y}\right)} \]
4. Add Preprocessing

5. Taylor expanded in y around inf

   \[\leadsto \color{blue}{-2 \cdot x} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto x \cdot \color{blue}{-2} \]

   2. *-lowering-*.f64 49.6%

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

7. Simplified 49.6%

   \[\leadsto \color{blue}{x \cdot -2} \]
8. Add Preprocessing

Developer Target 1: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2 \cdot x}{x - y} \cdot y\\ \mathbf{if}\;x < -1.7210442634149447 \cdot 10^{+81}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x < 83645045635564430:\\ \;\;\;\;\frac{x \cdot 2}{\frac{x - y}{y}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (/ (* 2.0 x) (- x y)) y)))
   (if (< x -1.7210442634149447e+81)
     t_0
     (if (< x 83645045635564430.0) (/ (* x 2.0) (/ (- x y) y)) t_0))))

C

double code(double x, double y) {
	double t_0 = ((2.0 * x) / (x - y)) * y;
	double tmp;
	if (x < -1.7210442634149447e+81) {
		tmp = t_0;
	} else if (x < 83645045635564430.0) {
		tmp = (x * 2.0) / ((x - y) / 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 = ((2.0d0 * x) / (x - y)) * y
    if (x < (-1.7210442634149447d+81)) then
        tmp = t_0
    else if (x < 83645045635564430.0d0) then
        tmp = (x * 2.0d0) / ((x - y) / y)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = ((2.0 * x) / (x - y)) * y;
	double tmp;
	if (x < -1.7210442634149447e+81) {
		tmp = t_0;
	} else if (x < 83645045635564430.0) {
		tmp = (x * 2.0) / ((x - y) / y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = ((2.0 * x) / (x - y)) * y
	tmp = 0
	if x < -1.7210442634149447e+81:
		tmp = t_0
	elif x < 83645045635564430.0:
		tmp = (x * 2.0) / ((x - y) / y)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(2.0 * x) / Float64(x - y)) * y)
	tmp = 0.0
	if (x < -1.7210442634149447e+81)
		tmp = t_0;
	elseif (x < 83645045635564430.0)
		tmp = Float64(Float64(x * 2.0) / Float64(Float64(x - y) / y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = ((2.0 * x) / (x - y)) * y;
	tmp = 0.0;
	if (x < -1.7210442634149447e+81)
		tmp = t_0;
	elseif (x < 83645045635564430.0)
		tmp = (x * 2.0) / ((x - y) / y);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(2.0 * x), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]}, If[Less[x, -1.7210442634149447e+81], t$95$0, If[Less[x, 83645045635564430.0], N[(N[(x * 2.0), $MachinePrecision] / N[(N[(x - y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Reproduce

herbie shell --seed 2024155 
(FPCore (x y)
  :name "Linear.Projection:perspective from linear-1.19.1.3, B"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< x -1721044263414944700000000000000000000000000000000000000000000000000000000000000000) (* (/ (* 2 x) (- x y)) y) (if (< x 83645045635564430) (/ (* x 2) (/ (- x y) y)) (* (/ (* 2 x) (- x y)) y))))

  (/ (* (* x 2.0) y) (- x y)))