Linear.Projection:inversePerspective from linear-1.19.1.3, C

Percentage Accurate: 77.2% → 100.0%

Time: 7.0s

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: 77.2% 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 73.5%

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

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-+.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. associate-/r* N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. lower-/.f64 N/A

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

   16. metadata-eval 89.4

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

4. Applied rewrites 89.4%

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

5. Taylor expanded in y around 0

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

7. Applied rewrites 100.0%

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

Alternative 2: 66.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{+77}:\\ \;\;\;\;\frac{0.5}{y}\\ \mathbf{elif}\;x \leq -4.4 \cdot 10^{-105}:\\ \;\;\;\;\frac{x + y}{\left(2 \cdot x\right) \cdot y}\\ \mathbf{elif}\;x \leq -8.5 \cdot 10^{-147}:\\ \;\;\;\;\frac{0.5}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -4.4e+77)
   (/ 0.5 y)
   (if (<= x -4.4e-105)
     (/ (+ x y) (* (* 2.0 x) y))
     (if (<= x -8.5e-147) (/ 0.5 y) (/ 0.5 x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -4.4e+77) {
		tmp = 0.5 / y;
	} else if (x <= -4.4e-105) {
		tmp = (x + y) / ((2.0 * x) * y);
	} else if (x <= -8.5e-147) {
		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 <= (-4.4d+77)) then
        tmp = 0.5d0 / y
    else if (x <= (-4.4d-105)) then
        tmp = (x + y) / ((2.0d0 * x) * y)
    else if (x <= (-8.5d-147)) 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 <= -4.4e+77) {
		tmp = 0.5 / y;
	} else if (x <= -4.4e-105) {
		tmp = (x + y) / ((2.0 * x) * y);
	} else if (x <= -8.5e-147) {
		tmp = 0.5 / y;
	} else {
		tmp = 0.5 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -4.4e+77:
		tmp = 0.5 / y
	elif x <= -4.4e-105:
		tmp = (x + y) / ((2.0 * x) * y)
	elif x <= -8.5e-147:
		tmp = 0.5 / y
	else:
		tmp = 0.5 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -4.4e+77)
		tmp = Float64(0.5 / y);
	elseif (x <= -4.4e-105)
		tmp = Float64(Float64(x + y) / Float64(Float64(2.0 * x) * y));
	elseif (x <= -8.5e-147)
		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 <= -4.4e+77)
		tmp = 0.5 / y;
	elseif (x <= -4.4e-105)
		tmp = (x + y) / ((2.0 * x) * y);
	elseif (x <= -8.5e-147)
		tmp = 0.5 / y;
	else
		tmp = 0.5 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -4.4e+77], N[(0.5 / y), $MachinePrecision], If[LessEqual[x, -4.4e-105], N[(N[(x + y), $MachinePrecision] / N[(N[(2.0 * x), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -8.5e-147], N[(0.5 / y), $MachinePrecision], N[(0.5 / x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.4000000000000001e77 or -4.40000000000000008e-105 < x < -8.5000000000000002e-147

   1. Initial program 71.9%

      \[\frac{x + y}{\left(x \cdot 2\right) \cdot y} \]
   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 73.1

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

   5. Applied rewrites 73.1%

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

   if -4.4000000000000001e77 < x < -4.40000000000000008e-105

   1. Initial program 93.8%

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

   if -8.5000000000000002e-147 < 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 56.9

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

   5. Applied rewrites 56.9%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{+77}:\\ \;\;\;\;\frac{0.5}{y}\\ \mathbf{elif}\;x \leq -4.4 \cdot 10^{-105}:\\ \;\;\;\;\frac{x + y}{\left(2 \cdot x\right) \cdot y}\\ \mathbf{elif}\;x \leq -8.5 \cdot 10^{-147}:\\ \;\;\;\;\frac{0.5}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \]
5. 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 2024230 
(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)))