Linear.Projection:inversePerspective from linear-1.19.1.3, C

Percentage Accurate: 77.4% → 100.0%

Time: 3.4s

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: 77.4% 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 81.4%

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

3. Taylor expanded in x around inf

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

   1. +-commutative N/A

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

   2. remove-double-neg N/A

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

   3. associate-*r/ N/A

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

   4. metadata-eval N/A

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

   5. distribute-frac-neg2 N/A

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

   6. mul-1-neg N/A

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

   7. unsub-neg N/A

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

   8. lower--.f64 N/A

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

   9. associate-*r/ N/A

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

   10. metadata-eval N/A

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

   11. lower-/.f64 N/A

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

   12. mul-1-neg N/A

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

   13. distribute-frac-neg2 N/A

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

   14. distribute-neg-frac N/A

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

   15. metadata-eval N/A

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

   16. metadata-eval N/A

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

   17. lower-/.f64 N/A

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

   18. metadata-eval 100.0

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

5. Applied rewrites 100.0%

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

Alternative 2: 68.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+61}:\\ \;\;\;\;\frac{0.5}{y}\\ \mathbf{elif}\;x \leq -7 \cdot 10^{-177}:\\ \;\;\;\;\frac{y + x}{\left(2 \cdot x\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -5e+61)
   (/ 0.5 y)
   (if (<= x -7e-177) (/ (+ y x) (* (* 2.0 x) y)) (/ 0.5 x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -5e+61) {
		tmp = 0.5 / y;
	} else if (x <= -7e-177) {
		tmp = (y + x) / ((2.0 * 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 (x <= (-5d+61)) then
        tmp = 0.5d0 / y
    else if (x <= (-7d-177)) then
        tmp = (y + x) / ((2.0d0 * 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 (x <= -5e+61) {
		tmp = 0.5 / y;
	} else if (x <= -7e-177) {
		tmp = (y + x) / ((2.0 * x) * y);
	} else {
		tmp = 0.5 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -5e+61:
		tmp = 0.5 / y
	elif x <= -7e-177:
		tmp = (y + x) / ((2.0 * x) * y)
	else:
		tmp = 0.5 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -5e+61)
		tmp = Float64(0.5 / y);
	elseif (x <= -7e-177)
		tmp = Float64(Float64(y + x) / Float64(Float64(2.0 * x) * y));
	else
		tmp = Float64(0.5 / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -5e+61)
		tmp = 0.5 / y;
	elseif (x <= -7e-177)
		tmp = (y + x) / ((2.0 * x) * y);
	else
		tmp = 0.5 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -5e+61], N[(0.5 / y), $MachinePrecision], If[LessEqual[x, -7e-177], N[(N[(y + x), $MachinePrecision] / N[(N[(2.0 * x), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(0.5 / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.00000000000000018e61

   1. Initial program 77.0%

      \[\frac{x + y}{\left(x \cdot 2\right) \cdot y} \]
   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 86.4

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

   5. Applied rewrites 86.4%

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

   if -5.00000000000000018e61 < x < -7.0000000000000003e-177

   1. Initial program 94.9%

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

   if -7.0000000000000003e-177 < x

   1. Initial program 79.3%

      \[\frac{x + y}{\left(x \cdot 2\right) \cdot y} \]
   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 55.1

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

   5. Applied rewrites 55.1%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+61}:\\ \;\;\;\;\frac{0.5}{y}\\ \mathbf{elif}\;x \leq -7 \cdot 10^{-177}:\\ \;\;\;\;\frac{y + x}{\left(2 \cdot x\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 63.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{-30}:\\ \;\;\;\;\frac{0.5}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x -7.2e-30) (/ 0.5 y) (/ 0.5 x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -7.2e-30) {
		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 <= (-7.2d-30)) 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 <= -7.2e-30) {
		tmp = 0.5 / y;
	} else {
		tmp = 0.5 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -7.2e-30:
		tmp = 0.5 / y
	else:
		tmp = 0.5 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -7.2e-30)
		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 <= -7.2e-30)
		tmp = 0.5 / y;
	else
		tmp = 0.5 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -7.2e-30], N[(0.5 / y), $MachinePrecision], N[(0.5 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -7.2000000000000006e-30

   1. Initial program 82.0%

      \[\frac{x + y}{\left(x \cdot 2\right) \cdot y} \]
   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 82.6

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

   5. Applied rewrites 82.6%

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

   if -7.2000000000000006e-30 < x

   1. Initial program 81.2%

      \[\frac{x + y}{\left(x \cdot 2\right) \cdot y} \]
   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 57.7

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

   5. Applied rewrites 57.7%

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

Alternative 4: 51.3% 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 81.4%

   \[\frac{x + y}{\left(x \cdot 2\right) \cdot y} \]
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 48.0

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

5. Applied rewrites 48.0%

   \[\leadsto \color{blue}{\frac{0.5}{x}} \]
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 2024295 
(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)))