Linear.Projection:inversePerspective from linear-1.19.1.3, B

Percentage Accurate: 77.1% → 100.0%

Time: 5.9s

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: 77.1% 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 81.9%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. div-sub N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/l/ N/A

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

   6. *-inverses N/A

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

   7. lower--.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-/r* N/A

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

   10. lower-/.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. associate-/r* N/A

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

   13. *-inverses N/A

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

   14. metadata-eval N/A

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. associate-/r* N/A

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

   18. *-inverses N/A

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

   19. associate-/r* N/A

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

   20. lift-*.f64 N/A

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

   21. lower-/.f64 N/A

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

   22. lift-*.f64 N/A

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

   23. associate-/r* N/A

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

   24. *-inverses N/A

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

   25. metadata-eval 100.0

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

4. Applied rewrites 100.0%

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

Alternative 2: 86.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{y \cdot \left(x \cdot 2\right)}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\frac{0.5}{y}\\ \mathbf{elif}\;t\_0 \leq -2 \cdot 10^{-108}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{0.5}{y}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+301}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- x y) (* y (* x 2.0)))))
   (if (<= t_0 (- INFINITY))
     (/ 0.5 y)
     (if (<= t_0 -2e-108)
       t_0
       (if (<= t_0 0.0) (/ 0.5 y) (if (<= t_0 2e+301) t_0 (/ 0.5 y)))))))

C

double code(double x, double y) {
	double t_0 = (x - y) / (y * (x * 2.0));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = 0.5 / y;
	} else if (t_0 <= -2e-108) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = 0.5 / y;
	} else if (t_0 <= 2e+301) {
		tmp = t_0;
	} else {
		tmp = 0.5 / y;
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = (x - y) / (y * (x * 2.0));
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = 0.5 / y;
	} else if (t_0 <= -2e-108) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = 0.5 / y;
	} else if (t_0 <= 2e+301) {
		tmp = t_0;
	} else {
		tmp = 0.5 / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x - y) / (y * (x * 2.0))
	tmp = 0
	if t_0 <= -math.inf:
		tmp = 0.5 / y
	elif t_0 <= -2e-108:
		tmp = t_0
	elif t_0 <= 0.0:
		tmp = 0.5 / y
	elif t_0 <= 2e+301:
		tmp = t_0
	else:
		tmp = 0.5 / y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x - y) / Float64(y * Float64(x * 2.0)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(0.5 / y);
	elseif (t_0 <= -2e-108)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = Float64(0.5 / y);
	elseif (t_0 <= 2e+301)
		tmp = t_0;
	else
		tmp = Float64(0.5 / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x - y) / (y * (x * 2.0));
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = 0.5 / y;
	elseif (t_0 <= -2e-108)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = 0.5 / y;
	elseif (t_0 <= 2e+301)
		tmp = t_0;
	else
		tmp = 0.5 / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x - y), $MachinePrecision] / N[(y * N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(0.5 / y), $MachinePrecision], If[LessEqual[t$95$0, -2e-108], t$95$0, If[LessEqual[t$95$0, 0.0], N[(0.5 / y), $MachinePrecision], If[LessEqual[t$95$0, 2e+301], t$95$0, N[(0.5 / y), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (*.f64 (*.f64 x #s(literal 2 binary64)) y)) < -inf.0 or -2.00000000000000008e-108 < (/.f64 (-.f64 x y) (*.f64 (*.f64 x #s(literal 2 binary64)) y)) < -0.0 or 2.00000000000000011e301 < (/.f64 (-.f64 x y) (*.f64 (*.f64 x #s(literal 2 binary64)) y))

   1. Initial program 8.0%

      \[\frac{x - y}{\left(x \cdot 2\right) \cdot y} \]
   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 64.8

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

   5. Applied rewrites 64.8%

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

   if -inf.0 < (/.f64 (-.f64 x y) (*.f64 (*.f64 x #s(literal 2 binary64)) y)) < -2.00000000000000008e-108 or -0.0 < (/.f64 (-.f64 x y) (*.f64 (*.f64 x #s(literal 2 binary64)) y)) < 2.00000000000000011e301

   1. Initial program 99.3%

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

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{y \cdot \left(x \cdot 2\right)} \leq -\infty:\\ \;\;\;\;\frac{0.5}{y}\\ \mathbf{elif}\;\frac{x - y}{y \cdot \left(x \cdot 2\right)} \leq -2 \cdot 10^{-108}:\\ \;\;\;\;\frac{x - y}{y \cdot \left(x \cdot 2\right)}\\ \mathbf{elif}\;\frac{x - y}{y \cdot \left(x \cdot 2\right)} \leq 0:\\ \;\;\;\;\frac{0.5}{y}\\ \mathbf{elif}\;\frac{x - y}{y \cdot \left(x \cdot 2\right)} \leq 2 \cdot 10^{+301}:\\ \;\;\;\;\frac{x - y}{y \cdot \left(x \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.4 \cdot 10^{+70}:\\ \;\;\;\;\frac{-0.5}{x}\\ \mathbf{elif}\;y \leq 7.4 \cdot 10^{-8}:\\ \;\;\;\;\frac{0.5}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -4.4e+70) (/ -0.5 x) (if (<= y 7.4e-8) (/ 0.5 y) (/ -0.5 x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -4.4e+70) {
		tmp = -0.5 / x;
	} else if (y <= 7.4e-8) {
		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 <= (-4.4d+70)) then
        tmp = (-0.5d0) / x
    else if (y <= 7.4d-8) 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 <= -4.4e+70) {
		tmp = -0.5 / x;
	} else if (y <= 7.4e-8) {
		tmp = 0.5 / y;
	} else {
		tmp = -0.5 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -4.4e+70:
		tmp = -0.5 / x
	elif y <= 7.4e-8:
		tmp = 0.5 / y
	else:
		tmp = -0.5 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -4.4e+70)
		tmp = Float64(-0.5 / x);
	elseif (y <= 7.4e-8)
		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 <= -4.4e+70)
		tmp = -0.5 / x;
	elseif (y <= 7.4e-8)
		tmp = 0.5 / y;
	else
		tmp = -0.5 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -4.4e+70], N[(-0.5 / x), $MachinePrecision], If[LessEqual[y, 7.4e-8], N[(0.5 / y), $MachinePrecision], N[(-0.5 / x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.40000000000000001e70 or 7.4000000000000001e-8 < y

   1. Initial program 82.2%

      \[\frac{x - y}{\left(x \cdot 2\right) \cdot y} \]
   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 82.5

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

   5. Applied rewrites 82.5%

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

   if -4.40000000000000001e70 < y < 7.4000000000000001e-8

   1. Initial program 81.6%

      \[\frac{x - y}{\left(x \cdot 2\right) \cdot y} \]
   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 73.9

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

   5. Applied rewrites 73.9%

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

Alternative 4: 50.9% 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 81.9%

   \[\frac{x - y}{\left(x \cdot 2\right) \cdot y} \]
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 52.1

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

5. Applied rewrites 52.1%

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

Developer Target 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{y} - \frac{0.5}{x} \end{array} \]

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]

Reproduce

herbie shell --seed 2024219 
(FPCore (x y)
  :name "Linear.Projection:inversePerspective from linear-1.19.1.3, B"
  :precision binary64

  :alt
  (! :herbie-platform default (- (/ 1/2 y) (/ 1/2 x)))

  (/ (- x y) (* (* x 2.0) y)))