Linear.Projection:inversePerspective from linear-1.19.1.3, B

Percentage Accurate: 76.5% → 100.0%

Time: 4.4s

Alternatives: 3

Speedup: 1.3×

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: 76.5% 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, 1.3× 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 78.0%

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

   1. div-sub N/A

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

   2. sub-neg N/A

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

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

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

   4. associate-/r* N/A

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

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

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

   6. associate-/r* N/A

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

   7. *-inverses N/A

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

   8. metadata-eval N/A

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

   9. distribute-neg-frac N/A

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

   10. associate-/l/ N/A

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

   11. distribute-frac-neg N/A

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

   12. *-inverses N/A

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

   13. metadata-eval N/A

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

   14. *-commutative N/A

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

   15. associate-/r* N/A

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

   16. /-lowering-/.f64 N/A

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

   17. metadata-eval 100.0%

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

3. Simplified 100.0%

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

Alternative 2: 74.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{-7}:\\ \;\;\;\;\frac{-0.5}{x}\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{-19}:\\ \;\;\;\;\frac{0.5}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -7.2e-7) (/ -0.5 x) (if (<= y 1.15e-19) (/ 0.5 y) (/ -0.5 x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -7.2e-7) {
		tmp = -0.5 / x;
	} else if (y <= 1.15e-19) {
		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 <= (-7.2d-7)) then
        tmp = (-0.5d0) / x
    else if (y <= 1.15d-19) 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 <= -7.2e-7) {
		tmp = -0.5 / x;
	} else if (y <= 1.15e-19) {
		tmp = 0.5 / y;
	} else {
		tmp = -0.5 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -7.2e-7:
		tmp = -0.5 / x
	elif y <= 1.15e-19:
		tmp = 0.5 / y
	else:
		tmp = -0.5 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -7.2e-7)
		tmp = Float64(-0.5 / x);
	elseif (y <= 1.15e-19)
		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 <= -7.2e-7)
		tmp = -0.5 / x;
	elseif (y <= 1.15e-19)
		tmp = 0.5 / y;
	else
		tmp = -0.5 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -7.2e-7], N[(-0.5 / x), $MachinePrecision], If[LessEqual[y, 1.15e-19], N[(0.5 / y), $MachinePrecision], N[(-0.5 / x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -7.19999999999999989e-7 or 1.1499999999999999e-19 < y

   1. Initial program 80.0%

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

      1. div-sub N/A

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

      2. sub-neg N/A

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

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

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

      4. associate-/r* N/A

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

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

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

      6. associate-/r* N/A

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

      7. *-inverses N/A

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

      8. metadata-eval N/A

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

      9. distribute-neg-frac N/A

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

      10. associate-/l/ N/A

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

      11. distribute-frac-neg N/A

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

      12. *-inverses N/A

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

      13. metadata-eval N/A

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

      14. *-commutative N/A

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

      15. associate-/r* N/A

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

      16. /-lowering-/.f64 N/A

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

      17. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around inf

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

      1. /-lowering-/.f64 81.4%

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

   7. Simplified 81.4%

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

   if -7.19999999999999989e-7 < y < 1.1499999999999999e-19

   1. Initial program 76.1%

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

      1. div-sub N/A

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

      2. sub-neg N/A

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

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

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

      4. associate-/r* N/A

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

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

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

      6. associate-/r* N/A

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

      7. *-inverses N/A

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

      8. metadata-eval N/A

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

      9. distribute-neg-frac N/A

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

      10. associate-/l/ N/A

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

      11. distribute-frac-neg N/A

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

      12. *-inverses N/A

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

      13. metadata-eval N/A

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

      14. *-commutative N/A

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

      15. associate-/r* N/A

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

      16. /-lowering-/.f64 N/A

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

      17. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 74.9%

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

   7. Simplified 74.9%

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

Alternative 3: 51.6% accurate, 3.0× 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 78.0%

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

   1. div-sub N/A

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

   2. sub-neg N/A

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

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

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

   4. associate-/r* N/A

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

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

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

   6. associate-/r* N/A

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

   7. *-inverses N/A

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

   8. metadata-eval N/A

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

   9. distribute-neg-frac N/A

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

   10. associate-/l/ N/A

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

   11. distribute-frac-neg N/A

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

   12. *-inverses N/A

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

   13. metadata-eval N/A

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

   14. *-commutative N/A

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

   15. associate-/r* N/A

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

   16. /-lowering-/.f64 N/A

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

   17. metadata-eval 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in y around inf

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

   1. /-lowering-/.f64 53.2%

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

7. Simplified 53.2%

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

Developer Target 1: 100.0% accurate, 1.3× 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 2024161 
(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)))