Linear.Projection:inversePerspective from linear-1.19.1.3, B

Percentage Accurate: 77.3% → 100.0%

Time: 3.6s

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: 77.3% 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 81.6%

   \[\frac{x - y}{\left(x \cdot 2\right) \cdot y} \]
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: 85.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{\left(x + x\right) \cdot y}\\ t_1 := \frac{x - y}{\left(x \cdot 2\right) \cdot y}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+286}:\\ \;\;\;\;\frac{0.5}{y}\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-127}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{0.5}{y}\\ \mathbf{elif}\;t\_1 \leq 10^{+305}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- x y) (* (+ x x) y))) (t_1 (/ (- x y) (* (* x 2.0) y))))
   (if (<= t_1 -4e+286)
     (/ 0.5 y)
     (if (<= t_1 -2e-127)
       t_0
       (if (<= t_1 0.0) (/ 0.5 y) (if (<= t_1 1e+305) t_0 (/ -0.5 x)))))))

C

double code(double x, double y) {
	double t_0 = (x - y) / ((x + x) * y);
	double t_1 = (x - y) / ((x * 2.0) * y);
	double tmp;
	if (t_1 <= -4e+286) {
		tmp = 0.5 / y;
	} else if (t_1 <= -2e-127) {
		tmp = t_0;
	} else if (t_1 <= 0.0) {
		tmp = 0.5 / y;
	} else if (t_1 <= 1e+305) {
		tmp = t_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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (x - y) / ((x + x) * y)
    t_1 = (x - y) / ((x * 2.0d0) * y)
    if (t_1 <= (-4d+286)) then
        tmp = 0.5d0 / y
    else if (t_1 <= (-2d-127)) then
        tmp = t_0
    else if (t_1 <= 0.0d0) then
        tmp = 0.5d0 / y
    else if (t_1 <= 1d+305) then
        tmp = t_0
    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 + x) * y);
	double t_1 = (x - y) / ((x * 2.0) * y);
	double tmp;
	if (t_1 <= -4e+286) {
		tmp = 0.5 / y;
	} else if (t_1 <= -2e-127) {
		tmp = t_0;
	} else if (t_1 <= 0.0) {
		tmp = 0.5 / y;
	} else if (t_1 <= 1e+305) {
		tmp = t_0;
	} else {
		tmp = -0.5 / x;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x - y) / ((x + x) * y)
	t_1 = (x - y) / ((x * 2.0) * y)
	tmp = 0
	if t_1 <= -4e+286:
		tmp = 0.5 / y
	elif t_1 <= -2e-127:
		tmp = t_0
	elif t_1 <= 0.0:
		tmp = 0.5 / y
	elif t_1 <= 1e+305:
		tmp = t_0
	else:
		tmp = -0.5 / x
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x - y) / Float64(Float64(x + x) * y))
	t_1 = Float64(Float64(x - y) / Float64(Float64(x * 2.0) * y))
	tmp = 0.0
	if (t_1 <= -4e+286)
		tmp = Float64(0.5 / y);
	elseif (t_1 <= -2e-127)
		tmp = t_0;
	elseif (t_1 <= 0.0)
		tmp = Float64(0.5 / y);
	elseif (t_1 <= 1e+305)
		tmp = t_0;
	else
		tmp = Float64(-0.5 / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x - y) / ((x + x) * y);
	t_1 = (x - y) / ((x * 2.0) * y);
	tmp = 0.0;
	if (t_1 <= -4e+286)
		tmp = 0.5 / y;
	elseif (t_1 <= -2e-127)
		tmp = t_0;
	elseif (t_1 <= 0.0)
		tmp = 0.5 / y;
	elseif (t_1 <= 1e+305)
		tmp = t_0;
	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 + x), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x - y), $MachinePrecision] / N[(N[(x * 2.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+286], N[(0.5 / y), $MachinePrecision], If[LessEqual[t$95$1, -2e-127], t$95$0, If[LessEqual[t$95$1, 0.0], N[(0.5 / y), $MachinePrecision], If[LessEqual[t$95$1, 1e+305], t$95$0, 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)) < -4.00000000000000013e286 or -2.0000000000000001e-127 < (/.f64 (-.f64 x y) (*.f64 (*.f64 x #s(literal 2 binary64)) y)) < -0.0

   1. Initial program 8.9%

      \[\frac{x - y}{\left(x \cdot 2\right) \cdot y} \]
   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 65.7

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

   5. Applied rewrites 65.7%

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

   if -4.00000000000000013e286 < (/.f64 (-.f64 x y) (*.f64 (*.f64 x #s(literal 2 binary64)) y)) < -2.0000000000000001e-127 or -0.0 < (/.f64 (-.f64 x y) (*.f64 (*.f64 x #s(literal 2 binary64)) y)) < 9.9999999999999994e304

   1. Initial program 98.5%

      \[\frac{x - y}{\left(x \cdot 2\right) \cdot y} \]
   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 98.5

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

   4. Applied rewrites 98.5%

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

   if 9.9999999999999994e304 < (/.f64 (-.f64 x y) (*.f64 (*.f64 x #s(literal 2 binary64)) y))

   1. Initial program 7.6%

      \[\frac{x - y}{\left(x \cdot 2\right) \cdot y} \]
   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 56.6

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

   5. Applied rewrites 56.6%

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

Alternative 3: 75.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.6 \cdot 10^{-12} \lor \neg \left(y \leq 1750000000\right):\\ \;\;\;\;\frac{-0.5}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -4.6e-12) (not (<= y 1750000000.0))) (/ -0.5 x) (/ 0.5 y)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -4.6e-12) || !(y <= 1750000000.0)) {
		tmp = -0.5 / x;
	} else {
		tmp = 0.5 / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-4.6d-12)) .or. (.not. (y <= 1750000000.0d0))) then
        tmp = (-0.5d0) / x
    else
        tmp = 0.5d0 / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -4.6e-12) || !(y <= 1750000000.0)) {
		tmp = -0.5 / x;
	} else {
		tmp = 0.5 / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -4.6e-12) or not (y <= 1750000000.0):
		tmp = -0.5 / x
	else:
		tmp = 0.5 / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -4.6e-12) || !(y <= 1750000000.0))
		tmp = Float64(-0.5 / x);
	else
		tmp = Float64(0.5 / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -4.6e-12) || ~((y <= 1750000000.0)))
		tmp = -0.5 / x;
	else
		tmp = 0.5 / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -4.6e-12], N[Not[LessEqual[y, 1750000000.0]], $MachinePrecision]], N[(-0.5 / x), $MachinePrecision], N[(0.5 / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.59999999999999979e-12 or 1.75e9 < y

   1. Initial program 83.5%

      \[\frac{x - y}{\left(x \cdot 2\right) \cdot y} \]
   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 81.4

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

   5. Applied rewrites 81.4%

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

   if -4.59999999999999979e-12 < y < 1.75e9

   1. Initial program 80.0%

      \[\frac{x - y}{\left(x \cdot 2\right) \cdot y} \]
   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 82.6

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

   5. Applied rewrites 82.6%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.6 \cdot 10^{-12} \lor \neg \left(y \leq 1750000000\right):\\ \;\;\;\;\frac{-0.5}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 51.2% 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 81.6%

   \[\frac{x - y}{\left(x \cdot 2\right) \cdot y} \]
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 48.0

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

5. Applied rewrites 48.0%

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

Developer Target 1: 100.0% accurate, 1.0× speedup

Math

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

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 2024339 
(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)))