Linear.Projection:inversePerspective from linear-1.19.1.3, B

Percentage Accurate: 77.9% → 100.0%

Time: 4.3s

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: 77.9% 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 73.3%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lift--.f64 N/A

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

   5. div-sub N/A

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

   6. *-inverses N/A

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

   7. sub-div N/A

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

   8. associate-/l/ N/A

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

   9. lift-*.f64 N/A

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

   10. inv-pow N/A

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

   11. lower--.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. associate-/r* N/A

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

   14. lower-/.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. associate-/r* N/A

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

   17. *-inverses N/A

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

   18. metadata-eval N/A

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

   19. inv-pow N/A

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

   20. lift-*.f64 N/A

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

   21. *-commutative N/A

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

   22. associate-/r* N/A

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

   23. *-inverses N/A

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

   24. associate-/r* N/A

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

   25. lift-*.f64 N/A

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

4. Applied rewrites 100.0%

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

Alternative 2: 87.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{\left(x \cdot 2\right) \cdot y}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\frac{-0.5}{x}\\ \mathbf{elif}\;t\_0 \leq -1 \cdot 10^{-141} \lor \neg \left(t\_0 \leq 10^{-132} \lor \neg \left(t\_0 \leq 5 \cdot 10^{+302}\right)\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- x y) (* (* x 2.0) y))))
   (if (<= t_0 (- INFINITY))
     (/ -0.5 x)
     (if (or (<= t_0 -1e-141) (not (or (<= t_0 1e-132) (not (<= t_0 5e+302)))))
       t_0
       (/ 0.5 y)))))

C

double code(double x, double y) {
	double t_0 = (x - y) / ((x * 2.0) * y);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = -0.5 / x;
	} else if ((t_0 <= -1e-141) || !((t_0 <= 1e-132) || !(t_0 <= 5e+302))) {
		tmp = t_0;
	} else {
		tmp = 0.5 / y;
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = (x - y) / ((x * 2.0) * y);
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = -0.5 / x;
	} else if ((t_0 <= -1e-141) || !((t_0 <= 1e-132) || !(t_0 <= 5e+302))) {
		tmp = t_0;
	} else {
		tmp = 0.5 / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x - y) / ((x * 2.0) * y)
	tmp = 0
	if t_0 <= -math.inf:
		tmp = -0.5 / x
	elif (t_0 <= -1e-141) or not ((t_0 <= 1e-132) or not (t_0 <= 5e+302)):
		tmp = t_0
	else:
		tmp = 0.5 / y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x - y) / Float64(Float64(x * 2.0) * y))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(-0.5 / x);
	elseif ((t_0 <= -1e-141) || !((t_0 <= 1e-132) || !(t_0 <= 5e+302)))
		tmp = t_0;
	else
		tmp = Float64(0.5 / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x - y) / ((x * 2.0) * y);
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = -0.5 / x;
	elseif ((t_0 <= -1e-141) || ~(((t_0 <= 1e-132) || ~((t_0 <= 5e+302)))))
		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[(N[(x * 2.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(-0.5 / x), $MachinePrecision], If[Or[LessEqual[t$95$0, -1e-141], N[Not[Or[LessEqual[t$95$0, 1e-132], N[Not[LessEqual[t$95$0, 5e+302]], $MachinePrecision]]], $MachinePrecision]], t$95$0, N[(0.5 / y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (*.f64 (*.f64 x #s(literal 2 binary64)) y)) < -inf.0

   1. Initial program 8.5%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 57.8

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

   5. Applied rewrites 57.8%

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

   if -inf.0 < (/.f64 (-.f64 x y) (*.f64 (*.f64 x #s(literal 2 binary64)) y)) < -1e-141 or 9.9999999999999999e-133 < (/.f64 (-.f64 x y) (*.f64 (*.f64 x #s(literal 2 binary64)) y)) < 5e302

   1. Initial program 98.9%

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

   if -1e-141 < (/.f64 (-.f64 x y) (*.f64 (*.f64 x #s(literal 2 binary64)) y)) < 9.9999999999999999e-133 or 5e302 < (/.f64 (-.f64 x y) (*.f64 (*.f64 x #s(literal 2 binary64)) y))

   1. Initial program 9.1%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 58.9

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

   5. Applied rewrites 58.9%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{\left(x \cdot 2\right) \cdot y} \leq -\infty:\\ \;\;\;\;\frac{-0.5}{x}\\ \mathbf{elif}\;\frac{x - y}{\left(x \cdot 2\right) \cdot y} \leq -1 \cdot 10^{-141} \lor \neg \left(\frac{x - y}{\left(x \cdot 2\right) \cdot y} \leq 10^{-132} \lor \neg \left(\frac{x - y}{\left(x \cdot 2\right) \cdot y} \leq 5 \cdot 10^{+302}\right)\right):\\ \;\;\;\;\frac{x - y}{\left(x \cdot 2\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{-64} \lor \neg \left(x \leq 1.45 \cdot 10^{-25}\right):\\ \;\;\;\;\frac{0.5}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.4e-64) (not (<= x 1.45e-25))) (/ 0.5 y) (/ -0.5 x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.4e-64) || !(x <= 1.45e-25)) {
		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 <= (-1.4d-64)) .or. (.not. (x <= 1.45d-25))) 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 <= -1.4e-64) || !(x <= 1.45e-25)) {
		tmp = 0.5 / y;
	} else {
		tmp = -0.5 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.4e-64) or not (x <= 1.45e-25):
		tmp = 0.5 / y
	else:
		tmp = -0.5 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.4e-64) || !(x <= 1.45e-25))
		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 <= -1.4e-64) || ~((x <= 1.45e-25)))
		tmp = 0.5 / y;
	else
		tmp = -0.5 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.4e-64], N[Not[LessEqual[x, 1.45e-25]], $MachinePrecision]], N[(0.5 / y), $MachinePrecision], N[(-0.5 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.40000000000000002e-64 or 1.45e-25 < x

   1. Initial program 73.8%

      \[\frac{x - y}{\left(x \cdot 2\right) \cdot y} \]
   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 74.6

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

   5. Applied rewrites 74.6%

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

   if -1.40000000000000002e-64 < x < 1.45e-25

   1. Initial program 72.6%

      \[\frac{x - y}{\left(x \cdot 2\right) \cdot y} \]
   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 77.2

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

   5. Applied rewrites 77.2%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{-64} \lor \neg \left(x \leq 1.45 \cdot 10^{-25}\right):\\ \;\;\;\;\frac{0.5}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 51.6% 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 73.3%

   \[\frac{x - y}{\left(x \cdot 2\right) \cdot y} \]
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 49.4

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

5. Applied rewrites 49.4%

   \[\leadsto \color{blue}{\frac{-0.5}{x}} \]
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 2024318 
(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)))