Linear.Projection:inversePerspective from linear-1.19.1.3, C

Percentage Accurate: 76.5% → 100.0%

Time: 2.6s

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: 76.5% 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 77.8%

   \[\frac{x + y}{\left(x \cdot 2\right) \cdot y} \]
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: 68.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.02 \cdot 10^{-196}:\\ \;\;\;\;\frac{0.5}{y}\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{+139}:\\ \;\;\;\;\frac{x + y}{\left(x + x\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.02e-196)
   (/ 0.5 y)
   (if (<= y 1.4e+139) (/ (+ x y) (* (+ x x) y)) (/ 0.5 x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.02e-196) {
		tmp = 0.5 / y;
	} else if (y <= 1.4e+139) {
		tmp = (x + y) / ((x + x) * 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 <= 1.02d-196) then
        tmp = 0.5d0 / y
    else if (y <= 1.4d+139) then
        tmp = (x + y) / ((x + x) * 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 <= 1.02e-196) {
		tmp = 0.5 / y;
	} else if (y <= 1.4e+139) {
		tmp = (x + y) / ((x + x) * y);
	} else {
		tmp = 0.5 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.02e-196:
		tmp = 0.5 / y
	elif y <= 1.4e+139:
		tmp = (x + y) / ((x + x) * y)
	else:
		tmp = 0.5 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.02e-196)
		tmp = Float64(0.5 / y);
	elseif (y <= 1.4e+139)
		tmp = Float64(Float64(x + y) / Float64(Float64(x + x) * y));
	else
		tmp = Float64(0.5 / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.02e-196)
		tmp = 0.5 / y;
	elseif (y <= 1.4e+139)
		tmp = (x + y) / ((x + x) * y);
	else
		tmp = 0.5 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.02e-196], N[(0.5 / y), $MachinePrecision], If[LessEqual[y, 1.4e+139], N[(N[(x + y), $MachinePrecision] / N[(N[(x + x), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(0.5 / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.0200000000000001e-196

   1. Initial program 76.4%

      \[\frac{x + y}{\left(x \cdot 2\right) \cdot y} \]
   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 56.8

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

   5. Applied rewrites 56.8%

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

   if 1.0200000000000001e-196 < y < 1.3999999999999999e139

   1. Initial program 84.3%

      \[\frac{x + y}{\left(x \cdot 2\right) \cdot y} \]
   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 84.3

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

   4. Applied rewrites 84.3%

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

   if 1.3999999999999999e139 < y

   1. Initial program 67.9%

      \[\frac{x + y}{\left(x \cdot 2\right) \cdot y} \]
   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 94.8

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

   5. Applied rewrites 94.8%

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

Alternative 3: 61.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{-118}:\\ \;\;\;\;\frac{0.5}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x -3.8e-118) (/ 0.5 y) (/ 0.5 x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -3.8e-118) {
		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 <= (-3.8d-118)) 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 <= -3.8e-118) {
		tmp = 0.5 / y;
	} else {
		tmp = 0.5 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -3.8e-118:
		tmp = 0.5 / y
	else:
		tmp = 0.5 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -3.8e-118)
		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 <= -3.8e-118)
		tmp = 0.5 / y;
	else
		tmp = 0.5 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -3.8e-118], N[(0.5 / y), $MachinePrecision], N[(0.5 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.8000000000000001e-118

   1. Initial program 84.2%

      \[\frac{x + y}{\left(x \cdot 2\right) \cdot y} \]
   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 68.4

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

   5. Applied rewrites 68.4%

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

   if -3.8000000000000001e-118 < x

   1. Initial program 74.9%

      \[\frac{x + y}{\left(x \cdot 2\right) \cdot y} \]
   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 54.3

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

   5. Applied rewrites 54.3%

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

Alternative 4: 51.5% 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 77.8%

   \[\frac{x + y}{\left(x \cdot 2\right) \cdot y} \]
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 48.1

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

5. Applied rewrites 48.1%

   \[\leadsto \color{blue}{\frac{0.5}{x}} \]
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 2024342 
(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)))