Linear.Projection:inversePerspective from linear-1.19.1.3, C

Percentage Accurate: 77.8% → 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: 77.8% 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 73.6%

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

3. Taylor expanded in y 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: 64.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.2 \cdot 10^{-199}:\\ \;\;\;\;\frac{0.5}{y}\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{-96}:\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{elif}\;y \leq 7.8 \cdot 10^{+164}:\\ \;\;\;\;\frac{x + y}{\left(2 \cdot x\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 3.2e-199)
   (/ 0.5 y)
   (if (<= y 8.5e-96)
     (/ 0.5 x)
     (if (<= y 7.8e+164) (/ (+ x y) (* (* 2.0 x) y)) (/ 0.5 x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3.2e-199) {
		tmp = 0.5 / y;
	} else if (y <= 8.5e-96) {
		tmp = 0.5 / x;
	} else if (y <= 7.8e+164) {
		tmp = (x + y) / ((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 (y <= 3.2d-199) then
        tmp = 0.5d0 / y
    else if (y <= 8.5d-96) then
        tmp = 0.5d0 / x
    else if (y <= 7.8d+164) then
        tmp = (x + y) / ((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 (y <= 3.2e-199) {
		tmp = 0.5 / y;
	} else if (y <= 8.5e-96) {
		tmp = 0.5 / x;
	} else if (y <= 7.8e+164) {
		tmp = (x + y) / ((2.0 * x) * y);
	} else {
		tmp = 0.5 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 3.2e-199:
		tmp = 0.5 / y
	elif y <= 8.5e-96:
		tmp = 0.5 / x
	elif y <= 7.8e+164:
		tmp = (x + y) / ((2.0 * x) * y)
	else:
		tmp = 0.5 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3.2e-199)
		tmp = Float64(0.5 / y);
	elseif (y <= 8.5e-96)
		tmp = Float64(0.5 / x);
	elseif (y <= 7.8e+164)
		tmp = Float64(Float64(x + y) / 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 (y <= 3.2e-199)
		tmp = 0.5 / y;
	elseif (y <= 8.5e-96)
		tmp = 0.5 / x;
	elseif (y <= 7.8e+164)
		tmp = (x + y) / ((2.0 * x) * y);
	else
		tmp = 0.5 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3.2e-199], N[(0.5 / y), $MachinePrecision], If[LessEqual[y, 8.5e-96], N[(0.5 / x), $MachinePrecision], If[LessEqual[y, 7.8e+164], N[(N[(x + y), $MachinePrecision] / N[(N[(2.0 * x), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(0.5 / x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 3.1999999999999999e-199

   1. Initial program 72.6%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 49.3

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

   5. Applied rewrites 49.3%

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

   if 3.1999999999999999e-199 < y < 8.49999999999999983e-96 or 7.79999999999999971e164 < y

   1. Initial program 56.0%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 68.8

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

   5. Applied rewrites 68.8%

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

   if 8.49999999999999983e-96 < y < 7.79999999999999971e164

   1. Initial program 94.3%

      \[\frac{x + y}{\left(x \cdot 2\right) \cdot y} \]
   2. Add Preprocessing
3. Recombined 3 regimes into one program.

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.2 \cdot 10^{-199}:\\ \;\;\;\;\frac{0.5}{y}\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{-96}:\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{elif}\;y \leq 7.8 \cdot 10^{+164}:\\ \;\;\;\;\frac{x + y}{\left(2 \cdot x\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 57.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.2 \cdot 10^{-199}:\\ \;\;\;\;\frac{0.5}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 3.2e-199) (/ 0.5 y) (/ 0.5 x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3.2e-199) {
		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 <= 3.2d-199) 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 <= 3.2e-199) {
		tmp = 0.5 / y;
	} else {
		tmp = 0.5 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 3.2e-199:
		tmp = 0.5 / y
	else:
		tmp = 0.5 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3.2e-199)
		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 <= 3.2e-199)
		tmp = 0.5 / y;
	else
		tmp = 0.5 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3.2e-199], N[(0.5 / y), $MachinePrecision], N[(0.5 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.1999999999999999e-199

   1. Initial program 72.6%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 49.3

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

   5. Applied rewrites 49.3%

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

   if 3.1999999999999999e-199 < y

   1. Initial program 74.9%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 67.9

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

   5. Applied rewrites 67.9%

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

Alternative 4: 51.2% 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 73.6%

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

3. Taylor expanded in y around inf

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

   1. lower-/.f64 58.1

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

5. Applied rewrites 58.1%

   \[\leadsto \color{blue}{\frac{0.5}{x}} \]
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 2024277 
(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)))