Linear.Projection:inversePerspective from linear-1.19.1.3, C

Percentage Accurate: 77.2% → 100.0%

Time: 5.0s

Alternatives: 3

Speedup: 1.3×

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, 1.3× 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]

Derivation

1. Initial program 76.9%

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

   1. +-commutative 76.9%

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

   2. remove-double-neg 76.9%

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

   3. unsub-neg 76.9%

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

   4. div-sub 76.5%

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

   5. associate-/l/ 83.9%

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

   6. *-inverses 83.9%

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

   7. metadata-eval 83.9%

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

   8. *-commutative 83.9%

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

   9. associate-/r* 83.9%

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

   10. metadata-eval 83.9%

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

   11. metadata-eval 83.9%

       \[\leadsto \frac{\color{blue}{0.5}}{x} - \frac{-x}{\left(x \cdot 2\right) \cdot y} \]

   12. associate-/r* 100.0%

       \[\leadsto \frac{0.5}{x} - \color{blue}{\frac{\frac{-x}{x \cdot 2}}{y}} \]

   13. distribute-frac-neg 100.0%

       \[\leadsto \frac{0.5}{x} - \frac{\color{blue}{-\frac{x}{x \cdot 2}}}{y} \]

   14. associate-/r* 100.0%

       \[\leadsto \frac{0.5}{x} - \frac{-\color{blue}{\frac{\frac{x}{x}}{2}}}{y} \]

   15. *-inverses 100.0%

       \[\leadsto \frac{0.5}{x} - \frac{-\frac{\color{blue}{1}}{2}}{y} \]

   16. metadata-eval 100.0%

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

   17. metadata-eval 100.0%

       \[\leadsto \frac{0.5}{x} - \frac{-\color{blue}{\frac{-1}{-2}}}{y} \]

   18. metadata-eval 100.0%

       \[\leadsto \frac{0.5}{x} - \frac{-\frac{-1}{\color{blue}{-2}}}{y} \]

   19. metadata-eval 100.0%

       \[\leadsto \frac{0.5}{x} - \frac{-\frac{-1}{\color{blue}{-2}}}{y} \]

   20. metadata-eval 100.0%

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

   21. metadata-eval 100.0%

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

3. Simplified 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 61.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.5 \cdot 10^{-90}:\\ \;\;\;\;\frac{0.5}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 5.5e-90) (/ 0.5 y) (/ 0.5 x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 5.5e-90) {
		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 <= 5.5d-90) 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 <= 5.5e-90) {
		tmp = 0.5 / y;
	} else {
		tmp = 0.5 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 5.5e-90:
		tmp = 0.5 / y
	else:
		tmp = 0.5 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 5.5e-90)
		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 <= 5.5e-90)
		tmp = 0.5 / y;
	else
		tmp = 0.5 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 5.5e-90], N[(0.5 / y), $MachinePrecision], N[(0.5 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.5000000000000003e-90

   1. Initial program 74.9%

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

      1. +-commutative 74.9%

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

      2. remove-double-neg 74.9%

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

      3. unsub-neg 74.9%

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

      4. div-sub 74.3%

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

      5. associate-/l/ 80.5%

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

      6. *-inverses 80.5%

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

      7. metadata-eval 80.5%

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

      8. *-commutative 80.5%

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

      9. associate-/r* 80.5%

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

      10. metadata-eval 80.5%

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

      11. metadata-eval 80.5%

          \[\leadsto \frac{\color{blue}{0.5}}{x} - \frac{-x}{\left(x \cdot 2\right) \cdot y} \]

      12. associate-/r* 100.0%

          \[\leadsto \frac{0.5}{x} - \color{blue}{\frac{\frac{-x}{x \cdot 2}}{y}} \]

      13. distribute-frac-neg 100.0%

          \[\leadsto \frac{0.5}{x} - \frac{\color{blue}{-\frac{x}{x \cdot 2}}}{y} \]

      14. associate-/r* 100.0%

          \[\leadsto \frac{0.5}{x} - \frac{-\color{blue}{\frac{\frac{x}{x}}{2}}}{y} \]

      15. *-inverses 100.0%

          \[\leadsto \frac{0.5}{x} - \frac{-\frac{\color{blue}{1}}{2}}{y} \]

      16. metadata-eval 100.0%

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

      17. metadata-eval 100.0%

          \[\leadsto \frac{0.5}{x} - \frac{-\color{blue}{\frac{-1}{-2}}}{y} \]

      18. metadata-eval 100.0%

          \[\leadsto \frac{0.5}{x} - \frac{-\frac{-1}{\color{blue}{-2}}}{y} \]

      19. metadata-eval 100.0%

          \[\leadsto \frac{0.5}{x} - \frac{-\frac{-1}{\color{blue}{-2}}}{y} \]

      20. metadata-eval 100.0%

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

      21. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 57.5%

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

   if 5.5000000000000003e-90 < y

   1. Initial program 80.7%

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

      1. +-commutative 80.7%

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

      2. remove-double-neg 80.7%

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

      3. unsub-neg 80.7%

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

      4. div-sub 80.5%

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

      5. associate-/l/ 90.3%

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

      6. *-inverses 90.3%

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

      7. metadata-eval 90.3%

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

      8. *-commutative 90.3%

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

      9. associate-/r* 90.3%

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

      10. metadata-eval 90.3%

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

      11. metadata-eval 90.3%

          \[\leadsto \frac{\color{blue}{0.5}}{x} - \frac{-x}{\left(x \cdot 2\right) \cdot y} \]

      12. associate-/r* 100.0%

          \[\leadsto \frac{0.5}{x} - \color{blue}{\frac{\frac{-x}{x \cdot 2}}{y}} \]

      13. distribute-frac-neg 100.0%

          \[\leadsto \frac{0.5}{x} - \frac{\color{blue}{-\frac{x}{x \cdot 2}}}{y} \]

      14. associate-/r* 100.0%

          \[\leadsto \frac{0.5}{x} - \frac{-\color{blue}{\frac{\frac{x}{x}}{2}}}{y} \]

      15. *-inverses 100.0%

          \[\leadsto \frac{0.5}{x} - \frac{-\frac{\color{blue}{1}}{2}}{y} \]

      16. metadata-eval 100.0%

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

      17. metadata-eval 100.0%

          \[\leadsto \frac{0.5}{x} - \frac{-\color{blue}{\frac{-1}{-2}}}{y} \]

      18. metadata-eval 100.0%

          \[\leadsto \frac{0.5}{x} - \frac{-\frac{-1}{\color{blue}{-2}}}{y} \]

      19. metadata-eval 100.0%

          \[\leadsto \frac{0.5}{x} - \frac{-\frac{-1}{\color{blue}{-2}}}{y} \]

      20. metadata-eval 100.0%

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

      21. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 71.1%

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

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.5 \cdot 10^{-90}:\\ \;\;\;\;\frac{0.5}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 50.6% accurate, 3.0× 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 76.9%

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

   1. +-commutative 76.9%

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

   2. remove-double-neg 76.9%

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

   3. unsub-neg 76.9%

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

   4. div-sub 76.5%

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

   5. associate-/l/ 83.9%

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

   6. *-inverses 83.9%

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

   7. metadata-eval 83.9%

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

   8. *-commutative 83.9%

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

   9. associate-/r* 83.9%

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

   10. metadata-eval 83.9%

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

   11. metadata-eval 83.9%

       \[\leadsto \frac{\color{blue}{0.5}}{x} - \frac{-x}{\left(x \cdot 2\right) \cdot y} \]

   12. associate-/r* 100.0%

       \[\leadsto \frac{0.5}{x} - \color{blue}{\frac{\frac{-x}{x \cdot 2}}{y}} \]

   13. distribute-frac-neg 100.0%

       \[\leadsto \frac{0.5}{x} - \frac{\color{blue}{-\frac{x}{x \cdot 2}}}{y} \]

   14. associate-/r* 100.0%

       \[\leadsto \frac{0.5}{x} - \frac{-\color{blue}{\frac{\frac{x}{x}}{2}}}{y} \]

   15. *-inverses 100.0%

       \[\leadsto \frac{0.5}{x} - \frac{-\frac{\color{blue}{1}}{2}}{y} \]

   16. metadata-eval 100.0%

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

   17. metadata-eval 100.0%

       \[\leadsto \frac{0.5}{x} - \frac{-\color{blue}{\frac{-1}{-2}}}{y} \]

   18. metadata-eval 100.0%

       \[\leadsto \frac{0.5}{x} - \frac{-\frac{-1}{\color{blue}{-2}}}{y} \]

   19. metadata-eval 100.0%

       \[\leadsto \frac{0.5}{x} - \frac{-\frac{-1}{\color{blue}{-2}}}{y} \]

   20. metadata-eval 100.0%

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

   21. metadata-eval 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0 53.7%

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

6. Final simplification 53.7%

   \[\leadsto \frac{0.5}{x} \]
7. Add Preprocessing

Developer target: 100.0% accurate, 1.3× 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 2024046 
(FPCore (x y)
  :name "Linear.Projection:inversePerspective from linear-1.19.1.3, C"
  :precision binary64

  :alt
  (+ (/ 0.5 x) (/ 0.5 y))

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