Linear.Projection:inversePerspective from linear-1.19.1.3, B

Percentage Accurate: 77.9% → 100.0%

Time: 3.5s

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 78.0%

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

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

   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. lower--.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-/r* N/A

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

   10. *-inverses N/A

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

   11. metadata-eval N/A

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. associate-/r* N/A

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

   15. *-inverses N/A

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

   16. *-inverses N/A

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

   17. lift-*.f64 N/A

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

   18. *-commutative N/A

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

   19. associate-/r* N/A

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

   20. associate-/r* N/A

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

   21. lift-*.f64 N/A

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

   22. lower-/.f64 N/A

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

   23. lift-*.f64 N/A

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

   24. associate-/r* N/A

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

   25. *-inverses N/A

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

   26. 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: 88.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{\left(x \cdot 2\right) \cdot y}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+303}:\\ \;\;\;\;\frac{0.5}{y}\\ \mathbf{elif}\;t\_0 \leq -5 \cdot 10^{-119} \lor \neg \left(t\_0 \leq 0 \lor \neg \left(t\_0 \leq 5 \cdot 10^{+304}\right)\right):\\ \;\;\;\;\frac{x - y}{\left(x + x\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- x y) (* (* x 2.0) y))))
   (if (<= t_0 -2e+303)
     (/ 0.5 y)
     (if (or (<= t_0 -5e-119) (not (or (<= t_0 0.0) (not (<= t_0 5e+304)))))
       (/ (- x y) (* (+ x x) y))
       (/ -0.5 x)))))

C

double code(double x, double y) {
	double t_0 = (x - y) / ((x * 2.0) * y);
	double tmp;
	if (t_0 <= -2e+303) {
		tmp = 0.5 / y;
	} else if ((t_0 <= -5e-119) || !((t_0 <= 0.0) || !(t_0 <= 5e+304))) {
		tmp = (x - y) / ((x + 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) :: t_0
    real(8) :: tmp
    t_0 = (x - y) / ((x * 2.0d0) * y)
    if (t_0 <= (-2d+303)) then
        tmp = 0.5d0 / y
    else if ((t_0 <= (-5d-119)) .or. (.not. (t_0 <= 0.0d0) .or. (.not. (t_0 <= 5d+304)))) then
        tmp = (x - y) / ((x + x) * y)
    else
        tmp = (-0.5d0) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x - y) / ((x * 2.0) * y);
	double tmp;
	if (t_0 <= -2e+303) {
		tmp = 0.5 / y;
	} else if ((t_0 <= -5e-119) || !((t_0 <= 0.0) || !(t_0 <= 5e+304))) {
		tmp = (x - y) / ((x + x) * y);
	} else {
		tmp = -0.5 / x;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x - y) / ((x * 2.0) * y)
	tmp = 0
	if t_0 <= -2e+303:
		tmp = 0.5 / y
	elif (t_0 <= -5e-119) or not ((t_0 <= 0.0) or not (t_0 <= 5e+304)):
		tmp = (x - y) / ((x + x) * y)
	else:
		tmp = -0.5 / x
	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 <= -2e+303)
		tmp = Float64(0.5 / y);
	elseif ((t_0 <= -5e-119) || !((t_0 <= 0.0) || !(t_0 <= 5e+304)))
		tmp = Float64(Float64(x - y) / Float64(Float64(x + x) * y));
	else
		tmp = Float64(-0.5 / x);
	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 <= -2e+303)
		tmp = 0.5 / y;
	elseif ((t_0 <= -5e-119) || ~(((t_0 <= 0.0) || ~((t_0 <= 5e+304)))))
		tmp = (x - y) / ((x + x) * y);
	else
		tmp = -0.5 / x;
	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, -2e+303], N[(0.5 / y), $MachinePrecision], If[Or[LessEqual[t$95$0, -5e-119], N[Not[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, 5e+304]], $MachinePrecision]]], $MachinePrecision]], N[(N[(x - y), $MachinePrecision] / N[(N[(x + x), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(-0.5 / x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (*.f64 (*.f64 x #s(literal 2 binary64)) y)) < -2e303

   1. Initial program 10.3%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 48.7

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

   5. Applied rewrites 48.7%

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

   if -2e303 < (/.f64 (-.f64 x y) (*.f64 (*.f64 x #s(literal 2 binary64)) y)) < -4.99999999999999993e-119 or -0.0 < (/.f64 (-.f64 x y) (*.f64 (*.f64 x #s(literal 2 binary64)) y)) < 4.9999999999999997e304

   1. Initial program 98.8%

      \[\frac{x - y}{\left(x \cdot 2\right) \cdot y} \]
   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. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 98.8

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

   4. Applied rewrites 98.8%

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

   if -4.99999999999999993e-119 < (/.f64 (-.f64 x y) (*.f64 (*.f64 x #s(literal 2 binary64)) y)) < -0.0 or 4.9999999999999997e304 < (/.f64 (-.f64 x y) (*.f64 (*.f64 x #s(literal 2 binary64)) y))

   1. Initial program 5.7%

      \[\frac{x - y}{\left(x \cdot 2\right) \cdot y} \]
   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 58.7

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

   5. Applied rewrites 58.7%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{\left(x \cdot 2\right) \cdot y} \leq -2 \cdot 10^{+303}:\\ \;\;\;\;\frac{0.5}{y}\\ \mathbf{elif}\;\frac{x - y}{\left(x \cdot 2\right) \cdot y} \leq -5 \cdot 10^{-119} \lor \neg \left(\frac{x - y}{\left(x \cdot 2\right) \cdot y} \leq 0 \lor \neg \left(\frac{x - y}{\left(x \cdot 2\right) \cdot y} \leq 5 \cdot 10^{+304}\right)\right):\\ \;\;\;\;\frac{x - y}{\left(x + x\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{+37} \lor \neg \left(x \leq 4.4 \cdot 10^{-19}\right):\\ \;\;\;\;\frac{0.5}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -4e+37) (not (<= x 4.4e-19))) (/ 0.5 y) (/ -0.5 x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -4e+37) || !(x <= 4.4e-19)) {
		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 <= (-4d+37)) .or. (.not. (x <= 4.4d-19))) 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 <= -4e+37) || !(x <= 4.4e-19)) {
		tmp = 0.5 / y;
	} else {
		tmp = -0.5 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -4e+37) or not (x <= 4.4e-19):
		tmp = 0.5 / y
	else:
		tmp = -0.5 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -4e+37) || !(x <= 4.4e-19))
		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 <= -4e+37) || ~((x <= 4.4e-19)))
		tmp = 0.5 / y;
	else
		tmp = -0.5 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -4e+37], N[Not[LessEqual[x, 4.4e-19]], $MachinePrecision]], N[(0.5 / y), $MachinePrecision], N[(-0.5 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.99999999999999982e37 or 4.3999999999999997e-19 < x

   1. Initial program 76.1%

      \[\frac{x - y}{\left(x \cdot 2\right) \cdot y} \]
   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 80.1

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

   5. Applied rewrites 80.1%

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

   if -3.99999999999999982e37 < x < 4.3999999999999997e-19

   1. Initial program 79.8%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 83.2

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

   5. Applied rewrites 83.2%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{+37} \lor \neg \left(x \leq 4.4 \cdot 10^{-19}\right):\\ \;\;\;\;\frac{0.5}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 50.8% 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 78.0%

   \[\frac{x - y}{\left(x \cdot 2\right) \cdot y} \]
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 52.3

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

5. Applied rewrites 52.3%

   \[\leadsto \color{blue}{\frac{-0.5}{x}} \]
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 2024337 
(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)))