Linear.Projection:inversePerspective from linear-1.19.1.3, B

Percentage Accurate: 77.4% → 100.0%

Time: 4.8s

Alternatives: 5

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{y} - \frac{0.5}{x} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- (/ 0.5 y) (/ 0.5 x)))

C

double code(double x, double y) {
	return (0.5 / y) - (0.5 / x);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (0.5d0 / y) - (0.5d0 / x)
end function

Java

public static double code(double x, double y) {
	return (0.5 / y) - (0.5 / x);
}

Python

def code(x, y):
	return (0.5 / y) - (0.5 / x)

Julia

function code(x, y)
	return Float64(Float64(0.5 / y) - Float64(0.5 / x))
end

MATLAB

function tmp = code(x, y)
	tmp = (0.5 / y) - (0.5 / x);
end

Wolfram

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

Derivation

1. Initial program 74.9%

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

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. div-sub N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/l/ N/A

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

   6. *-inverses N/A

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

   7. lower--.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-/r* N/A

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

   10. lower-/.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. associate-/r* N/A

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

   13. *-inverses N/A

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

   14. metadata-eval N/A

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. associate-/r* N/A

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

   18. *-inverses N/A

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

   19. associate-/r* N/A

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

   20. lift-*.f64 N/A

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

   21. lower-/.f64 N/A

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

   22. lift-*.f64 N/A

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

   23. associate-/r* N/A

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

   24. *-inverses N/A

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

   25. metadata-eval 100.0

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

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\frac{0.5}{y} - \frac{0.5}{x}} \]
5. Add Preprocessing

Alternative 2: 86.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{\left(2 \cdot x\right) \cdot y}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+290}:\\ \;\;\;\;\frac{-0.5}{x}\\ \mathbf{elif}\;t\_0 \leq -5 \cdot 10^{-104}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{0.5}{y}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+299}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- x y) (* (* 2.0 x) y))))
   (if (<= t_0 -5e+290)
     (/ -0.5 x)
     (if (<= t_0 -5e-104)
       t_0
       (if (<= t_0 0.0) (/ 0.5 y) (if (<= t_0 2e+299) t_0 (/ 0.5 y)))))))

C

double code(double x, double y) {
	double t_0 = (x - y) / ((2.0 * x) * y);
	double tmp;
	if (t_0 <= -5e+290) {
		tmp = -0.5 / x;
	} else if (t_0 <= -5e-104) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = 0.5 / y;
	} else if (t_0 <= 2e+299) {
		tmp = t_0;
	} else {
		tmp = 0.5 / y;
	}
	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 = (x - y) / ((2.0d0 * x) * y)
    if (t_0 <= (-5d+290)) then
        tmp = (-0.5d0) / x
    else if (t_0 <= (-5d-104)) then
        tmp = t_0
    else if (t_0 <= 0.0d0) then
        tmp = 0.5d0 / y
    else if (t_0 <= 2d+299) then
        tmp = t_0
    else
        tmp = 0.5d0 / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x - y) / ((2.0 * x) * y);
	double tmp;
	if (t_0 <= -5e+290) {
		tmp = -0.5 / x;
	} else if (t_0 <= -5e-104) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = 0.5 / y;
	} else if (t_0 <= 2e+299) {
		tmp = t_0;
	} else {
		tmp = 0.5 / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x - y) / ((2.0 * x) * y)
	tmp = 0
	if t_0 <= -5e+290:
		tmp = -0.5 / x
	elif t_0 <= -5e-104:
		tmp = t_0
	elif t_0 <= 0.0:
		tmp = 0.5 / y
	elif t_0 <= 2e+299:
		tmp = t_0
	else:
		tmp = 0.5 / y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x - y) / Float64(Float64(2.0 * x) * y))
	tmp = 0.0
	if (t_0 <= -5e+290)
		tmp = Float64(-0.5 / x);
	elseif (t_0 <= -5e-104)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = Float64(0.5 / y);
	elseif (t_0 <= 2e+299)
		tmp = t_0;
	else
		tmp = Float64(0.5 / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x - y) / ((2.0 * x) * y);
	tmp = 0.0;
	if (t_0 <= -5e+290)
		tmp = -0.5 / x;
	elseif (t_0 <= -5e-104)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = 0.5 / y;
	elseif (t_0 <= 2e+299)
		tmp = t_0;
	else
		tmp = 0.5 / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x - y), $MachinePrecision] / N[(N[(2.0 * x), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+290], N[(-0.5 / x), $MachinePrecision], If[LessEqual[t$95$0, -5e-104], t$95$0, If[LessEqual[t$95$0, 0.0], N[(0.5 / y), $MachinePrecision], If[LessEqual[t$95$0, 2e+299], t$95$0, N[(0.5 / y), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (*.f64 (*.f64 x #s(literal 2 binary64)) y)) < -4.9999999999999998e290

   1. Initial program 14.3%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 56.5

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

   5. Applied rewrites 56.5%

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

   if -4.9999999999999998e290 < (/.f64 (-.f64 x y) (*.f64 (*.f64 x #s(literal 2 binary64)) y)) < -4.99999999999999979e-104 or 0.0 < (/.f64 (-.f64 x y) (*.f64 (*.f64 x #s(literal 2 binary64)) y)) < 2.0000000000000001e299

   1. Initial program 98.7%

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

   if -4.99999999999999979e-104 < (/.f64 (-.f64 x y) (*.f64 (*.f64 x #s(literal 2 binary64)) y)) < 0.0 or 2.0000000000000001e299 < (/.f64 (-.f64 x y) (*.f64 (*.f64 x #s(literal 2 binary64)) y))

   1. Initial program 6.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 61.5

         \[\leadsto \color{blue}{\frac{0.5}{y}} \]

   5. Applied rewrites 61.5%

      \[\leadsto \color{blue}{\frac{0.5}{y}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{\left(2 \cdot x\right) \cdot y} \leq -5 \cdot 10^{+290}:\\ \;\;\;\;\frac{-0.5}{x}\\ \mathbf{elif}\;\frac{x - y}{\left(2 \cdot x\right) \cdot y} \leq -5 \cdot 10^{-104}:\\ \;\;\;\;\frac{x - y}{\left(2 \cdot x\right) \cdot y}\\ \mathbf{elif}\;\frac{x - y}{\left(2 \cdot x\right) \cdot y} \leq 0:\\ \;\;\;\;\frac{0.5}{y}\\ \mathbf{elif}\;\frac{x - y}{\left(2 \cdot x\right) \cdot y} \leq 2 \cdot 10^{+299}:\\ \;\;\;\;\frac{x - y}{\left(2 \cdot x\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.5}{x \cdot y} \cdot \left(x - y\right)\\ t_1 := \frac{x - y}{\left(2 \cdot x\right) \cdot y}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+289}:\\ \;\;\;\;\frac{-0.5}{x}\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-104}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{0.5}{y}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+299}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (/ 0.5 (* x y)) (- x y))) (t_1 (/ (- x y) (* (* 2.0 x) y))))
   (if (<= t_1 -2e+289)
     (/ -0.5 x)
     (if (<= t_1 -5e-104)
       t_0
       (if (<= t_1 0.0) (/ 0.5 y) (if (<= t_1 2e+299) t_0 (/ 0.5 y)))))))

C

double code(double x, double y) {
	double t_0 = (0.5 / (x * y)) * (x - y);
	double t_1 = (x - y) / ((2.0 * x) * y);
	double tmp;
	if (t_1 <= -2e+289) {
		tmp = -0.5 / x;
	} else if (t_1 <= -5e-104) {
		tmp = t_0;
	} else if (t_1 <= 0.0) {
		tmp = 0.5 / y;
	} else if (t_1 <= 2e+299) {
		tmp = t_0;
	} else {
		tmp = 0.5 / y;
	}
	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 = (0.5d0 / (x * y)) * (x - y)
    t_1 = (x - y) / ((2.0d0 * x) * y)
    if (t_1 <= (-2d+289)) then
        tmp = (-0.5d0) / x
    else if (t_1 <= (-5d-104)) then
        tmp = t_0
    else if (t_1 <= 0.0d0) then
        tmp = 0.5d0 / y
    else if (t_1 <= 2d+299) then
        tmp = t_0
    else
        tmp = 0.5d0 / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (0.5 / (x * y)) * (x - y);
	double t_1 = (x - y) / ((2.0 * x) * y);
	double tmp;
	if (t_1 <= -2e+289) {
		tmp = -0.5 / x;
	} else if (t_1 <= -5e-104) {
		tmp = t_0;
	} else if (t_1 <= 0.0) {
		tmp = 0.5 / y;
	} else if (t_1 <= 2e+299) {
		tmp = t_0;
	} else {
		tmp = 0.5 / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (0.5 / (x * y)) * (x - y)
	t_1 = (x - y) / ((2.0 * x) * y)
	tmp = 0
	if t_1 <= -2e+289:
		tmp = -0.5 / x
	elif t_1 <= -5e-104:
		tmp = t_0
	elif t_1 <= 0.0:
		tmp = 0.5 / y
	elif t_1 <= 2e+299:
		tmp = t_0
	else:
		tmp = 0.5 / y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(0.5 / Float64(x * y)) * Float64(x - y))
	t_1 = Float64(Float64(x - y) / Float64(Float64(2.0 * x) * y))
	tmp = 0.0
	if (t_1 <= -2e+289)
		tmp = Float64(-0.5 / x);
	elseif (t_1 <= -5e-104)
		tmp = t_0;
	elseif (t_1 <= 0.0)
		tmp = Float64(0.5 / y);
	elseif (t_1 <= 2e+299)
		tmp = t_0;
	else
		tmp = Float64(0.5 / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (0.5 / (x * y)) * (x - y);
	t_1 = (x - y) / ((2.0 * x) * y);
	tmp = 0.0;
	if (t_1 <= -2e+289)
		tmp = -0.5 / x;
	elseif (t_1 <= -5e-104)
		tmp = t_0;
	elseif (t_1 <= 0.0)
		tmp = 0.5 / y;
	elseif (t_1 <= 2e+299)
		tmp = t_0;
	else
		tmp = 0.5 / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(0.5 / N[(x * y), $MachinePrecision]), $MachinePrecision] * N[(x - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x - y), $MachinePrecision] / N[(N[(2.0 * x), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+289], N[(-0.5 / x), $MachinePrecision], If[LessEqual[t$95$1, -5e-104], t$95$0, If[LessEqual[t$95$1, 0.0], N[(0.5 / y), $MachinePrecision], If[LessEqual[t$95$1, 2e+299], t$95$0, N[(0.5 / y), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (*.f64 (*.f64 x #s(literal 2 binary64)) y)) < -2.0000000000000001e289

   1. Initial program 19.6%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 59.2

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

   5. Applied rewrites 59.2%

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

   if -2.0000000000000001e289 < (/.f64 (-.f64 x y) (*.f64 (*.f64 x #s(literal 2 binary64)) y)) < -4.99999999999999979e-104 or 0.0 < (/.f64 (-.f64 x y) (*.f64 (*.f64 x #s(literal 2 binary64)) y)) < 2.0000000000000001e299

   1. Initial program 98.7%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. associate-/r* N/A

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

      10. *-inverses N/A

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

      11. associate-/r* N/A

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

      12. lift-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-/r* N/A

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

      16. *-inverses N/A

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

      17. metadata-eval N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 97.6

          \[\leadsto \frac{0.5}{\color{blue}{y \cdot x}} \cdot \left(x - y\right) \]

   4. Applied rewrites 97.6%

      \[\leadsto \color{blue}{\frac{0.5}{y \cdot x} \cdot \left(x - y\right)} \]

   if -4.99999999999999979e-104 < (/.f64 (-.f64 x y) (*.f64 (*.f64 x #s(literal 2 binary64)) y)) < 0.0 or 2.0000000000000001e299 < (/.f64 (-.f64 x y) (*.f64 (*.f64 x #s(literal 2 binary64)) y))

   1. Initial program 6.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 61.5

         \[\leadsto \color{blue}{\frac{0.5}{y}} \]

   5. Applied rewrites 61.5%

      \[\leadsto \color{blue}{\frac{0.5}{y}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{\left(2 \cdot x\right) \cdot y} \leq -2 \cdot 10^{+289}:\\ \;\;\;\;\frac{-0.5}{x}\\ \mathbf{elif}\;\frac{x - y}{\left(2 \cdot x\right) \cdot y} \leq -5 \cdot 10^{-104}:\\ \;\;\;\;\frac{0.5}{x \cdot y} \cdot \left(x - y\right)\\ \mathbf{elif}\;\frac{x - y}{\left(2 \cdot x\right) \cdot y} \leq 0:\\ \;\;\;\;\frac{0.5}{y}\\ \mathbf{elif}\;\frac{x - y}{\left(2 \cdot x\right) \cdot y} \leq 2 \cdot 10^{+299}:\\ \;\;\;\;\frac{0.5}{x \cdot y} \cdot \left(x - y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.8 \cdot 10^{+17}:\\ \;\;\;\;\frac{-0.5}{x}\\ \mathbf{elif}\;y \leq 4 \cdot 10^{-37}:\\ \;\;\;\;\frac{0.5}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -7.8e+17) (/ -0.5 x) (if (<= y 4e-37) (/ 0.5 y) (/ -0.5 x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -7.8e+17) {
		tmp = -0.5 / x;
	} else if (y <= 4e-37) {
		tmp = 0.5 / y;
	} else {
		tmp = -0.5 / x;
	}
	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+17)) then
        tmp = (-0.5d0) / x
    else if (y <= 4d-37) then
        tmp = 0.5d0 / y
    else
        tmp = (-0.5d0) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -7.8e+17) {
		tmp = -0.5 / x;
	} else if (y <= 4e-37) {
		tmp = 0.5 / y;
	} else {
		tmp = -0.5 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -7.8e+17:
		tmp = -0.5 / x
	elif y <= 4e-37:
		tmp = 0.5 / y
	else:
		tmp = -0.5 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -7.8e+17)
		tmp = Float64(-0.5 / x);
	elseif (y <= 4e-37)
		tmp = Float64(0.5 / y);
	else
		tmp = Float64(-0.5 / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -7.8e+17)
		tmp = -0.5 / x;
	elseif (y <= 4e-37)
		tmp = 0.5 / y;
	else
		tmp = -0.5 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -7.8e+17], N[(-0.5 / x), $MachinePrecision], If[LessEqual[y, 4e-37], N[(0.5 / y), $MachinePrecision], N[(-0.5 / x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -7.8e17 or 4.00000000000000027e-37 < y

   1. Initial program 71.8%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 74.7

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

   5. Applied rewrites 74.7%

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

   if -7.8e17 < y < 4.00000000000000027e-37

   1. Initial program 78.3%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 81.1

         \[\leadsto \color{blue}{\frac{0.5}{y}} \]

   5. Applied rewrites 81.1%

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

Alternative 5: 50.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \frac{-0.5}{x} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ -0.5 x))

C

double code(double x, double y) {
	return -0.5 / x;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (-0.5d0) / x
end function

Java

public static double code(double x, double y) {
	return -0.5 / x;
}

Python

def code(x, y):
	return -0.5 / x

Julia

function code(x, y)
	return Float64(-0.5 / x)
end

MATLAB

function tmp = code(x, y)
	tmp = -0.5 / x;
end

Wolfram

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

Derivation

1. Initial program 74.9%

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

3. Taylor expanded in y around inf

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

   1. lower-/.f64 47.9

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

5. Applied rewrites 47.9%

   \[\leadsto \color{blue}{\frac{-0.5}{x}} \]
6. Add Preprocessing

Developer Target 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{y} - \frac{0.5}{x} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- (/ 0.5 y) (/ 0.5 x)))

C

double code(double x, double y) {
	return (0.5 / y) - (0.5 / x);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (0.5d0 / y) - (0.5d0 / x)
end function

Java

public static double code(double x, double y) {
	return (0.5 / y) - (0.5 / x);
}

Python

def code(x, y):
	return (0.5 / y) - (0.5 / x)

Julia

function code(x, y)
	return Float64(Float64(0.5 / y) - Float64(0.5 / x))
end

MATLAB

function tmp = code(x, y)
	tmp = (0.5 / y) - (0.5 / x);
end

Wolfram

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

Reproduce

herbie shell --seed 2024255 
(FPCore (x y)
  :name "Linear.Projection:inversePerspective from linear-1.19.1.3, B"
  :precision binary64

  :alt
  (! :herbie-platform default (- (/ 1/2 y) (/ 1/2 x)))

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