Linear.Projection:perspective from linear-1.19.1.3, B

Percentage Accurate: 77.7% → 99.0%

Time: 5.0s

Alternatives: 4

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: 77.7% 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: 99.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{x - y} \cdot \frac{x}{0.5}\\ \mathbf{if}\;y \leq -2.05 \cdot 10^{-73}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{-100}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{2}{x - y}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (/ y (- x y)) (/ x 0.5))))
   (if (<= y -2.05e-73)
     t_0
     (if (<= y 5.8e-100) (* y (* x (/ 2.0 (- x y)))) t_0))))

C

double code(double x, double y) {
	double t_0 = (y / (x - y)) * (x / 0.5);
	double tmp;
	if (y <= -2.05e-73) {
		tmp = t_0;
	} else if (y <= 5.8e-100) {
		tmp = y * (x * (2.0 / (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)) * (x / 0.5d0)
    if (y <= (-2.05d-73)) then
        tmp = t_0
    else if (y <= 5.8d-100) then
        tmp = y * (x * (2.0d0 / (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)) * (x / 0.5);
	double tmp;
	if (y <= -2.05e-73) {
		tmp = t_0;
	} else if (y <= 5.8e-100) {
		tmp = y * (x * (2.0 / (x - y)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y / (x - y)) * (x / 0.5)
	tmp = 0
	if y <= -2.05e-73:
		tmp = t_0
	elif y <= 5.8e-100:
		tmp = y * (x * (2.0 / (x - y)))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y / Float64(x - y)) * Float64(x / 0.5))
	tmp = 0.0
	if (y <= -2.05e-73)
		tmp = t_0;
	elseif (y <= 5.8e-100)
		tmp = Float64(y * Float64(x * Float64(2.0 / Float64(x - y))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y / (x - y)) * (x / 0.5);
	tmp = 0.0;
	if (y <= -2.05e-73)
		tmp = t_0;
	elseif (y <= 5.8e-100)
		tmp = y * (x * (2.0 / (x - y)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y / N[(x - y), $MachinePrecision]), $MachinePrecision] * N[(x / 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.05e-73], t$95$0, If[LessEqual[y, 5.8e-100], N[(y * N[(x * N[(2.0 / N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.05000000000000008e-73 or 5.79999999999999951e-100 < y

   1. Initial program 80.0%

      \[\frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
   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 79.5%

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

   3. Simplified 79.5%

      \[\leadsto \color{blue}{y \cdot \left(x \cdot \frac{2}{x - y}\right)} \]
   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. div-inv N/A

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

      5. times-frac N/A

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

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

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

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

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

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

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

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

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

      10. metadata-eval 99.9%

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

   6. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{y}{x - y} \cdot \frac{x}{0.5}} \]

   if -2.05000000000000008e-73 < y < 5.79999999999999951e-100

   1. Initial program 72.3%

      \[\frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
   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 99.7%

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

   3. Simplified 99.7%

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

Alternative 2: 93.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.12 \cdot 10^{+103}:\\ \;\;\;\;x \cdot -2\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{+199}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{2}{x - y}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot -2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.12e+103)
   (* x -2.0)
   (if (<= y 9.5e+199) (* y (* x (/ 2.0 (- x y)))) (* x -2.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.12e+103) {
		tmp = x * -2.0;
	} else if (y <= 9.5e+199) {
		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 <= (-1.12d+103)) then
        tmp = x * (-2.0d0)
    else if (y <= 9.5d+199) 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 <= -1.12e+103) {
		tmp = x * -2.0;
	} else if (y <= 9.5e+199) {
		tmp = y * (x * (2.0 / (x - y)));
	} else {
		tmp = x * -2.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.12e+103:
		tmp = x * -2.0
	elif y <= 9.5e+199:
		tmp = y * (x * (2.0 / (x - y)))
	else:
		tmp = x * -2.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.12e+103)
		tmp = Float64(x * -2.0);
	elseif (y <= 9.5e+199)
		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 <= -1.12e+103)
		tmp = x * -2.0;
	elseif (y <= 9.5e+199)
		tmp = y * (x * (2.0 / (x - y)));
	else
		tmp = x * -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.12e+103], N[(x * -2.0), $MachinePrecision], If[LessEqual[y, 9.5e+199], 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 < -1.12000000000000007e103 or 9.49999999999999954e199 < y

   1. Initial program 78.5%

      \[\frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
   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 59.9%

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

   3. Simplified 59.9%

      \[\leadsto \color{blue}{y \cdot \left(x \cdot \frac{2}{x - y}\right)} \]
   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 92.8%

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

   7. Simplified 92.8%

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

   if -1.12000000000000007e103 < y < 9.49999999999999954e199

   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 96.7%

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

   3. Simplified 96.7%

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

Alternative 3: 74.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.05 \cdot 10^{-73}:\\ \;\;\;\;x \cdot -2\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{+49}:\\ \;\;\;\;y \cdot 2\\ \mathbf{else}:\\ \;\;\;\;x \cdot -2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.05e-73) (* x -2.0) (if (<= y 1.8e+49) (* y 2.0) (* x -2.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.05e-73) {
		tmp = x * -2.0;
	} else if (y <= 1.8e+49) {
		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 <= (-2.05d-73)) then
        tmp = x * (-2.0d0)
    else if (y <= 1.8d+49) 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 <= -2.05e-73) {
		tmp = x * -2.0;
	} else if (y <= 1.8e+49) {
		tmp = y * 2.0;
	} else {
		tmp = x * -2.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.05e-73:
		tmp = x * -2.0
	elif y <= 1.8e+49:
		tmp = y * 2.0
	else:
		tmp = x * -2.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.05e-73)
		tmp = Float64(x * -2.0);
	elseif (y <= 1.8e+49)
		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 <= -2.05e-73)
		tmp = x * -2.0;
	elseif (y <= 1.8e+49)
		tmp = y * 2.0;
	else
		tmp = x * -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.05e-73], N[(x * -2.0), $MachinePrecision], If[LessEqual[y, 1.8e+49], N[(y * 2.0), $MachinePrecision], N[(x * -2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.05000000000000008e-73 or 1.79999999999999998e49 < y

   1. Initial program 80.1%

      \[\frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
   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 75.4%

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

   3. Simplified 75.4%

      \[\leadsto \color{blue}{y \cdot \left(x \cdot \frac{2}{x - y}\right)} \]
   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 79.3%

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

   7. Simplified 79.3%

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

   if -2.05000000000000008e-73 < y < 1.79999999999999998e49

   1. Initial program 74.0%

      \[\frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
   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 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 78.8%

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

Alternative 4: 50.8% 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 77.1%

   \[\frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
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 87.1%

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

3. Simplified 87.1%

   \[\leadsto \color{blue}{y \cdot \left(x \cdot \frac{2}{x - y}\right)} \]
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 52.5%

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

7. Simplified 52.5%

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

Developer Target 1: 99.6% 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 2024161 
(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)))