Linear.Projection:inversePerspective from linear-1.19.1.3, C

Percentage Accurate: 76.9% → 100.0%

Time: 5.7s

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: 76.9% 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 78.5%

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

3. Taylor expanded in x around 0

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

   1. /-lowering-/.f64 N/A

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

   2. +-commutative N/A

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

   3. accelerator-lowering-fma.f64 N/A

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

   4. /-lowering-/.f64 86.8

      \[\leadsto \frac{\mathsf{fma}\left(0.5, \color{blue}{\frac{x}{y}}, 0.5\right)}{x} \]

5. Simplified 86.8%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.5, \frac{x}{y}, 0.5\right)}{x}} \]
6. Step-by-step derivation

   1. clear-num N/A

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

   2. associate-/r/ N/A

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

   3. distribute-rgt-in N/A

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

   4. *-commutative N/A

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

   5. div-inv N/A

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

   6. associate-*l* N/A

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

   7. associate-/r/ N/A

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

   8. div-inv N/A

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

   9. metadata-eval N/A

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

   10. *-commutative N/A

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

   11. un-div-inv N/A

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

   12. times-frac N/A

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

   13. *-rgt-identity N/A

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

   14. associate-/l/ N/A

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

   15. *-inverses N/A

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

   16. associate-/r* N/A

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

   17. metadata-eval N/A

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

   18. div-inv N/A

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

   19. clear-num N/A

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

   20. div-inv N/A

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

   21. metadata-eval N/A

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

   22. +-lowering-+.f64 N/A

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

   23. /-lowering-/.f64 N/A

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

   24. metadata-eval N/A

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

7. Applied egg-rr 100.0%

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

Alternative 2: 68.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.4 \cdot 10^{+223}:\\ \;\;\;\;\frac{0.5}{y}\\ \mathbf{elif}\;x \leq -5.5 \cdot 10^{-164}:\\ \;\;\;\;\frac{y + x}{y \cdot \left(x \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -6.4e+223)
   (/ 0.5 y)
   (if (<= x -5.5e-164) (/ (+ y x) (* y (* x 2.0))) (/ 0.5 x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -6.4e+223) {
		tmp = 0.5 / y;
	} else if (x <= -5.5e-164) {
		tmp = (y + x) / (y * (x * 2.0));
	} 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 <= (-6.4d+223)) then
        tmp = 0.5d0 / y
    else if (x <= (-5.5d-164)) then
        tmp = (y + x) / (y * (x * 2.0d0))
    else
        tmp = 0.5d0 / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -6.4e+223) {
		tmp = 0.5 / y;
	} else if (x <= -5.5e-164) {
		tmp = (y + x) / (y * (x * 2.0));
	} else {
		tmp = 0.5 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -6.4e+223:
		tmp = 0.5 / y
	elif x <= -5.5e-164:
		tmp = (y + x) / (y * (x * 2.0))
	else:
		tmp = 0.5 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -6.4e+223)
		tmp = Float64(0.5 / y);
	elseif (x <= -5.5e-164)
		tmp = Float64(Float64(y + x) / Float64(y * Float64(x * 2.0)));
	else
		tmp = Float64(0.5 / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -6.4e+223)
		tmp = 0.5 / y;
	elseif (x <= -5.5e-164)
		tmp = (y + x) / (y * (x * 2.0));
	else
		tmp = 0.5 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -6.4e+223], N[(0.5 / y), $MachinePrecision], If[LessEqual[x, -5.5e-164], N[(N[(y + x), $MachinePrecision] / N[(y * N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.4000000000000003e223

   1. Initial program 78.3%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 94.6

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

   5. Simplified 94.6%

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

   if -6.4000000000000003e223 < x < -5.50000000000000027e-164

   1. Initial program 94.0%

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

   if -5.50000000000000027e-164 < x

   1. Initial program 72.8%

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

   3. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 54.7

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

   5. Simplified 54.7%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.4 \cdot 10^{+223}:\\ \;\;\;\;\frac{0.5}{y}\\ \mathbf{elif}\;x \leq -5.5 \cdot 10^{-164}:\\ \;\;\;\;\frac{y + x}{y \cdot \left(x \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 59.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.45 \cdot 10^{-165}:\\ \;\;\;\;\frac{0.5}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 1.45e-165) (/ 0.5 y) (/ 0.5 x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.45e-165) {
		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 <= 1.45d-165) 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 <= 1.45e-165) {
		tmp = 0.5 / y;
	} else {
		tmp = 0.5 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.45e-165:
		tmp = 0.5 / y
	else:
		tmp = 0.5 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.45e-165)
		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 <= 1.45e-165)
		tmp = 0.5 / y;
	else
		tmp = 0.5 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.45e-165], N[(0.5 / y), $MachinePrecision], N[(0.5 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.45e-165

   1. Initial program 74.1%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 65.4

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

   5. Simplified 65.4%

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

   if 1.45e-165 < y

   1. Initial program 84.4%

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

   3. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 67.3

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

   5. Simplified 67.3%

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

Alternative 4: 50.7% 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 78.5%

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

3. Taylor expanded in x around 0

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

   1. /-lowering-/.f64 49.0

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

5. Simplified 49.0%

   \[\leadsto \color{blue}{\frac{0.5}{x}} \]
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 2024198 
(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)))