Linear.Projection:inversePerspective from linear-1.19.1.3, C

Percentage Accurate: 76.6% → 100.0%

Time: 4.4s

Alternatives: 4

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: 76.6% 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 72.0%

   \[\frac{x + y}{\left(x \cdot 2\right) \cdot y} \]
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-add N/A

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

   4. lower-+.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-/r* N/A

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

   10. *-inverses N/A

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

   11. metadata-eval N/A

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. associate-/r* N/A

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. associate-/r* N/A

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

   18. *-inverses N/A

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

   19. *-inverses N/A

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

   20. associate-/r* N/A

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

   21. lift-*.f64 N/A

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

   22. lower-/.f64 N/A

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

   23. lift-*.f64 N/A

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

   24. associate-/r* N/A

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

   25. *-inverses N/A

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

   26. 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: 68.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.4 \cdot 10^{-205}:\\ \;\;\;\;\frac{0.5}{y}\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{-189} \lor \neg \left(y \leq 8.5 \cdot 10^{+103}\right):\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y}{\left(x + x\right) \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.4e-205)
   (/ 0.5 y)
   (if (or (<= y 9.5e-189) (not (<= y 8.5e+103)))
     (/ 0.5 x)
     (/ (+ x y) (* (+ x x) y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.4e-205) {
		tmp = 0.5 / y;
	} else if ((y <= 9.5e-189) || !(y <= 8.5e+103)) {
		tmp = 0.5 / x;
	} else {
		tmp = (x + y) / ((x + x) * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2.4d-205) then
        tmp = 0.5d0 / y
    else if ((y <= 9.5d-189) .or. (.not. (y <= 8.5d+103))) then
        tmp = 0.5d0 / x
    else
        tmp = (x + y) / ((x + x) * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.4e-205) {
		tmp = 0.5 / y;
	} else if ((y <= 9.5e-189) || !(y <= 8.5e+103)) {
		tmp = 0.5 / x;
	} else {
		tmp = (x + y) / ((x + x) * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.4e-205:
		tmp = 0.5 / y
	elif (y <= 9.5e-189) or not (y <= 8.5e+103):
		tmp = 0.5 / x
	else:
		tmp = (x + y) / ((x + x) * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.4e-205)
		tmp = Float64(0.5 / y);
	elseif ((y <= 9.5e-189) || !(y <= 8.5e+103))
		tmp = Float64(0.5 / x);
	else
		tmp = Float64(Float64(x + y) / Float64(Float64(x + x) * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.4e-205)
		tmp = 0.5 / y;
	elseif ((y <= 9.5e-189) || ~((y <= 8.5e+103)))
		tmp = 0.5 / x;
	else
		tmp = (x + y) / ((x + x) * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.4e-205], N[(0.5 / y), $MachinePrecision], If[Or[LessEqual[y, 9.5e-189], N[Not[LessEqual[y, 8.5e+103]], $MachinePrecision]], N[(0.5 / x), $MachinePrecision], N[(N[(x + y), $MachinePrecision] / N[(N[(x + x), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 2.4000000000000002e-205

   1. Initial program 73.7%

      \[\frac{x + y}{\left(x \cdot 2\right) \cdot y} \]
   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 54.9

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

   5. Applied rewrites 54.9%

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

   if 2.4000000000000002e-205 < y < 9.499999999999999e-189 or 8.4999999999999992e103 < y

   1. Initial program 57.4%

      \[\frac{x + y}{\left(x \cdot 2\right) \cdot y} \]
   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 83.6

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

   5. Applied rewrites 83.6%

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

   if 9.499999999999999e-189 < y < 8.4999999999999992e103

   1. Initial program 84.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 84.7

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

   4. Applied rewrites 84.7%

      \[\leadsto \frac{x + y}{\color{blue}{\left(x + x\right)} \cdot y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.4 \cdot 10^{-205}:\\ \;\;\;\;\frac{0.5}{y}\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{-189} \lor \neg \left(y \leq 8.5 \cdot 10^{+103}\right):\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y}{\left(x + x\right) \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 61.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 8.5 \cdot 10^{-125}:\\ \;\;\;\;\frac{0.5}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 8.5e-125) (/ 0.5 y) (/ 0.5 x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 8.5e-125) {
		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 <= 8.5d-125) 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 <= 8.5e-125) {
		tmp = 0.5 / y;
	} else {
		tmp = 0.5 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 8.5e-125:
		tmp = 0.5 / y
	else:
		tmp = 0.5 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 8.5e-125)
		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 <= 8.5e-125)
		tmp = 0.5 / y;
	else
		tmp = 0.5 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 8.5e-125], N[(0.5 / y), $MachinePrecision], N[(0.5 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 8.5000000000000002e-125

   1. Initial program 70.9%

      \[\frac{x + y}{\left(x \cdot 2\right) \cdot y} \]
   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 53.2

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

   5. Applied rewrites 53.2%

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

   if 8.5000000000000002e-125 < y

   1. Initial program 74.1%

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

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

   5. Applied rewrites 75.4%

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

Alternative 4: 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 72.0%

   \[\frac{x + y}{\left(x \cdot 2\right) \cdot y} \]
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 57.8

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

5. Applied rewrites 57.8%

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

Developer Target 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024338 
(FPCore (x y)
  :name "Linear.Projection:inversePerspective from linear-1.19.1.3, C"
  :precision binary64

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

  (/ (+ x y) (* (* x 2.0) y)))