Linear.Projection:inversePerspective from linear-1.19.1.3, B

Percentage Accurate: 76.5% → 100.0%

Time: 4.4s

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: 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, 1.0× 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 75.1%

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

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

   3. associate-/l/ N/A

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

   4. lift--.f64 N/A

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

   5. div-sub N/A

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

   6. *-inverses N/A

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

   7. sub-div N/A

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

   8. associate-/l/ N/A

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

   9. lift-*.f64 N/A

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

   10. inv-pow N/A

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

   11. lower--.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. associate-/r* N/A

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

   14. lower-/.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. associate-/r* N/A

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

   17. *-inverses N/A

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

   18. metadata-eval N/A

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

   19. inv-pow N/A

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

   20. lift-*.f64 N/A

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

   21. *-commutative N/A

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

   22. associate-/r* N/A

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

   23. *-inverses N/A

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

   24. associate-/r* N/A

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

   25. lift-*.f64 N/A

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

4. Applied rewrites 100.0%

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

Alternative 2: 87.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{+147}:\\ \;\;\;\;\frac{0.5}{y}\\ \mathbf{elif}\;x \leq -1.1 \cdot 10^{-149}:\\ \;\;\;\;\frac{x - y}{\left(x \cdot 2\right) \cdot y}\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{-136}:\\ \;\;\;\;\frac{-0.5}{x}\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{+137}:\\ \;\;\;\;\frac{0.5}{y \cdot x} \cdot \left(x - y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -3.1e+147)
   (/ 0.5 y)
   (if (<= x -1.1e-149)
     (/ (- x y) (* (* x 2.0) y))
     (if (<= x 4.6e-136)
       (/ -0.5 x)
       (if (<= x 1.6e+137) (* (/ 0.5 (* y x)) (- x y)) (/ 0.5 y))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -3.1e+147) {
		tmp = 0.5 / y;
	} else if (x <= -1.1e-149) {
		tmp = (x - y) / ((x * 2.0) * y);
	} else if (x <= 4.6e-136) {
		tmp = -0.5 / x;
	} else if (x <= 1.6e+137) {
		tmp = (0.5 / (y * x)) * (x - y);
	} 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) :: tmp
    if (x <= (-3.1d+147)) then
        tmp = 0.5d0 / y
    else if (x <= (-1.1d-149)) then
        tmp = (x - y) / ((x * 2.0d0) * y)
    else if (x <= 4.6d-136) then
        tmp = (-0.5d0) / x
    else if (x <= 1.6d+137) then
        tmp = (0.5d0 / (y * x)) * (x - y)
    else
        tmp = 0.5d0 / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -3.1e+147) {
		tmp = 0.5 / y;
	} else if (x <= -1.1e-149) {
		tmp = (x - y) / ((x * 2.0) * y);
	} else if (x <= 4.6e-136) {
		tmp = -0.5 / x;
	} else if (x <= 1.6e+137) {
		tmp = (0.5 / (y * x)) * (x - y);
	} else {
		tmp = 0.5 / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -3.1e+147:
		tmp = 0.5 / y
	elif x <= -1.1e-149:
		tmp = (x - y) / ((x * 2.0) * y)
	elif x <= 4.6e-136:
		tmp = -0.5 / x
	elif x <= 1.6e+137:
		tmp = (0.5 / (y * x)) * (x - y)
	else:
		tmp = 0.5 / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -3.1e+147)
		tmp = Float64(0.5 / y);
	elseif (x <= -1.1e-149)
		tmp = Float64(Float64(x - y) / Float64(Float64(x * 2.0) * y));
	elseif (x <= 4.6e-136)
		tmp = Float64(-0.5 / x);
	elseif (x <= 1.6e+137)
		tmp = Float64(Float64(0.5 / Float64(y * x)) * Float64(x - y));
	else
		tmp = Float64(0.5 / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -3.1e+147)
		tmp = 0.5 / y;
	elseif (x <= -1.1e-149)
		tmp = (x - y) / ((x * 2.0) * y);
	elseif (x <= 4.6e-136)
		tmp = -0.5 / x;
	elseif (x <= 1.6e+137)
		tmp = (0.5 / (y * x)) * (x - y);
	else
		tmp = 0.5 / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -3.1e+147], N[(0.5 / y), $MachinePrecision], If[LessEqual[x, -1.1e-149], N[(N[(x - y), $MachinePrecision] / N[(N[(x * 2.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.6e-136], N[(-0.5 / x), $MachinePrecision], If[LessEqual[x, 1.6e+137], N[(N[(0.5 / N[(y * x), $MachinePrecision]), $MachinePrecision] * N[(x - y), $MachinePrecision]), $MachinePrecision], N[(0.5 / y), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -3.1e147 or 1.60000000000000009e137 < x

   1. Initial program 60.5%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 90.3

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

   5. Applied rewrites 90.3%

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

   if -3.1e147 < x < -1.0999999999999999e-149

   1. Initial program 87.8%

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

   if -1.0999999999999999e-149 < x < 4.59999999999999997e-136

   1. Initial program 65.9%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 92.9

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

   5. Applied rewrites 92.9%

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

   if 4.59999999999999997e-136 < x < 1.60000000000000009e137

   1. Initial program 90.2%

      \[\frac{x - y}{\left(x \cdot 2\right) \cdot y} \]
   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 91.7

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

   4. Applied rewrites 91.7%

      \[\leadsto \color{blue}{\frac{0.5}{y \cdot x} \cdot \left(x - y\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 3: 87.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.5}{y \cdot x} \cdot \left(x - y\right)\\ \mathbf{if}\;x \leq -2.1 \cdot 10^{+135}:\\ \;\;\;\;\frac{0.5}{y}\\ \mathbf{elif}\;x \leq -2.9 \cdot 10^{-148}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{-136}:\\ \;\;\;\;\frac{-0.5}{x}\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{+137}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (/ 0.5 (* y x)) (- x y))))
   (if (<= x -2.1e+135)
     (/ 0.5 y)
     (if (<= x -2.9e-148)
       t_0
       (if (<= x 4.6e-136) (/ -0.5 x) (if (<= x 1.6e+137) t_0 (/ 0.5 y)))))))

C

double code(double x, double y) {
	double t_0 = (0.5 / (y * x)) * (x - y);
	double tmp;
	if (x <= -2.1e+135) {
		tmp = 0.5 / y;
	} else if (x <= -2.9e-148) {
		tmp = t_0;
	} else if (x <= 4.6e-136) {
		tmp = -0.5 / x;
	} else if (x <= 1.6e+137) {
		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 = (0.5d0 / (y * x)) * (x - y)
    if (x <= (-2.1d+135)) then
        tmp = 0.5d0 / y
    else if (x <= (-2.9d-148)) then
        tmp = t_0
    else if (x <= 4.6d-136) then
        tmp = (-0.5d0) / x
    else if (x <= 1.6d+137) 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 / (y * x)) * (x - y);
	double tmp;
	if (x <= -2.1e+135) {
		tmp = 0.5 / y;
	} else if (x <= -2.9e-148) {
		tmp = t_0;
	} else if (x <= 4.6e-136) {
		tmp = -0.5 / x;
	} else if (x <= 1.6e+137) {
		tmp = t_0;
	} else {
		tmp = 0.5 / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (0.5 / (y * x)) * (x - y)
	tmp = 0
	if x <= -2.1e+135:
		tmp = 0.5 / y
	elif x <= -2.9e-148:
		tmp = t_0
	elif x <= 4.6e-136:
		tmp = -0.5 / x
	elif x <= 1.6e+137:
		tmp = t_0
	else:
		tmp = 0.5 / y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(0.5 / Float64(y * x)) * Float64(x - y))
	tmp = 0.0
	if (x <= -2.1e+135)
		tmp = Float64(0.5 / y);
	elseif (x <= -2.9e-148)
		tmp = t_0;
	elseif (x <= 4.6e-136)
		tmp = Float64(-0.5 / x);
	elseif (x <= 1.6e+137)
		tmp = t_0;
	else
		tmp = Float64(0.5 / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (0.5 / (y * x)) * (x - y);
	tmp = 0.0;
	if (x <= -2.1e+135)
		tmp = 0.5 / y;
	elseif (x <= -2.9e-148)
		tmp = t_0;
	elseif (x <= 4.6e-136)
		tmp = -0.5 / x;
	elseif (x <= 1.6e+137)
		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[(y * x), $MachinePrecision]), $MachinePrecision] * N[(x - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.1e+135], N[(0.5 / y), $MachinePrecision], If[LessEqual[x, -2.9e-148], t$95$0, If[LessEqual[x, 4.6e-136], N[(-0.5 / x), $MachinePrecision], If[LessEqual[x, 1.6e+137], t$95$0, N[(0.5 / y), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.1000000000000001e135 or 1.60000000000000009e137 < x

   1. Initial program 61.8%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 89.6

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

   5. Applied rewrites 89.6%

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

   if -2.1000000000000001e135 < x < -2.8999999999999998e-148 or 4.59999999999999997e-136 < x < 1.60000000000000009e137

   1. Initial program 89.4%

      \[\frac{x - y}{\left(x \cdot 2\right) \cdot y} \]
   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 90.2

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

   4. Applied rewrites 90.2%

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

   if -2.8999999999999998e-148 < x < 4.59999999999999997e-136

   1. Initial program 65.9%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 92.9

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

   5. Applied rewrites 92.9%

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

Alternative 4: 74.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.85 \cdot 10^{+69} \lor \neg \left(x \leq 2.6 \cdot 10^{-64}\right):\\ \;\;\;\;\frac{0.5}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.85e+69) (not (<= x 2.6e-64))) (/ 0.5 y) (/ -0.5 x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.85e+69) || !(x <= 2.6e-64)) {
		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 ((x <= (-1.85d+69)) .or. (.not. (x <= 2.6d-64))) 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 ((x <= -1.85e+69) || !(x <= 2.6e-64)) {
		tmp = 0.5 / y;
	} else {
		tmp = -0.5 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.85e+69) or not (x <= 2.6e-64):
		tmp = 0.5 / y
	else:
		tmp = -0.5 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.85e+69) || !(x <= 2.6e-64))
		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 ((x <= -1.85e+69) || ~((x <= 2.6e-64)))
		tmp = 0.5 / y;
	else
		tmp = -0.5 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.85e+69], N[Not[LessEqual[x, 2.6e-64]], $MachinePrecision]], N[(0.5 / y), $MachinePrecision], N[(-0.5 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.8499999999999999e69 or 2.6e-64 < x

   1. Initial program 73.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 77.3

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

   5. Applied rewrites 77.3%

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

   if -1.8499999999999999e69 < x < 2.6e-64

   1. Initial program 77.6%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 85.8

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

   5. Applied rewrites 85.8%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.85 \cdot 10^{+69} \lor \neg \left(x \leq 2.6 \cdot 10^{-64}\right):\\ \;\;\;\;\frac{0.5}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{x}\\ \end{array} \]
5. 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 75.1%

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

3. Taylor expanded in x around 0

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

   1. lower-/.f64 52.4

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

5. Applied rewrites 52.4%

   \[\leadsto \color{blue}{\frac{-0.5}{x}} \]
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 2024313 
(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)))