Linear.Projection:inversePerspective from linear-1.19.1.3, B

Percentage Accurate: 76.6% → 100.0%

Time: 5.1s

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: 76.6% 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 72.0%

   \[\frac{x - y}{\left(x \cdot 2\right) \cdot y} \]
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: 86.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{\left(x + x\right) \cdot y}\\ \mathbf{if}\;y \leq -5.2 \cdot 10^{+124}:\\ \;\;\;\;\frac{-0.5}{x}\\ \mathbf{elif}\;y \leq -1.55 \cdot 10^{-215}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{-205}:\\ \;\;\;\;\frac{0.5}{y}\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{-198} \lor \neg \left(y \leq 7.5 \cdot 10^{+103}\right):\\ \;\;\;\;\frac{-0.5}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- x y) (* (+ x x) y))))
   (if (<= y -5.2e+124)
     (/ -0.5 x)
     (if (<= y -1.55e-215)
       t_0
       (if (<= y 2.4e-205)
         (/ 0.5 y)
         (if (or (<= y 3.6e-198) (not (<= y 7.5e+103))) (/ -0.5 x) t_0))))))

C

double code(double x, double y) {
	double t_0 = (x - y) / ((x + x) * y);
	double tmp;
	if (y <= -5.2e+124) {
		tmp = -0.5 / x;
	} else if (y <= -1.55e-215) {
		tmp = t_0;
	} else if (y <= 2.4e-205) {
		tmp = 0.5 / y;
	} else if ((y <= 3.6e-198) || !(y <= 7.5e+103)) {
		tmp = -0.5 / x;
	} else {
		tmp = t_0;
	}
	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 + x) * y)
    if (y <= (-5.2d+124)) then
        tmp = (-0.5d0) / x
    else if (y <= (-1.55d-215)) then
        tmp = t_0
    else if (y <= 2.4d-205) then
        tmp = 0.5d0 / y
    else if ((y <= 3.6d-198) .or. (.not. (y <= 7.5d+103))) then
        tmp = (-0.5d0) / x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x - y) / ((x + x) * y);
	double tmp;
	if (y <= -5.2e+124) {
		tmp = -0.5 / x;
	} else if (y <= -1.55e-215) {
		tmp = t_0;
	} else if (y <= 2.4e-205) {
		tmp = 0.5 / y;
	} else if ((y <= 3.6e-198) || !(y <= 7.5e+103)) {
		tmp = -0.5 / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x - y) / ((x + x) * y)
	tmp = 0
	if y <= -5.2e+124:
		tmp = -0.5 / x
	elif y <= -1.55e-215:
		tmp = t_0
	elif y <= 2.4e-205:
		tmp = 0.5 / y
	elif (y <= 3.6e-198) or not (y <= 7.5e+103):
		tmp = -0.5 / x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x - y) / Float64(Float64(x + x) * y))
	tmp = 0.0
	if (y <= -5.2e+124)
		tmp = Float64(-0.5 / x);
	elseif (y <= -1.55e-215)
		tmp = t_0;
	elseif (y <= 2.4e-205)
		tmp = Float64(0.5 / y);
	elseif ((y <= 3.6e-198) || !(y <= 7.5e+103))
		tmp = Float64(-0.5 / x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x - y) / ((x + x) * y);
	tmp = 0.0;
	if (y <= -5.2e+124)
		tmp = -0.5 / x;
	elseif (y <= -1.55e-215)
		tmp = t_0;
	elseif (y <= 2.4e-205)
		tmp = 0.5 / y;
	elseif ((y <= 3.6e-198) || ~((y <= 7.5e+103)))
		tmp = -0.5 / x;
	else
		tmp = t_0;
	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]}, If[LessEqual[y, -5.2e+124], N[(-0.5 / x), $MachinePrecision], If[LessEqual[y, -1.55e-215], t$95$0, If[LessEqual[y, 2.4e-205], N[(0.5 / y), $MachinePrecision], If[Or[LessEqual[y, 3.6e-198], N[Not[LessEqual[y, 7.5e+103]], $MachinePrecision]], N[(-0.5 / x), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.2000000000000001e124 or 2.4000000000000002e-205 < y < 3.59999999999999998e-198 or 7.49999999999999922e103 < y

   1. Initial program 61.5%

      \[\frac{x - y}{\left(x \cdot 2\right) \cdot y} \]
   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 84.8

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

   5. Applied rewrites 84.8%

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

   if -5.2000000000000001e124 < y < -1.54999999999999997e-215 or 3.59999999999999998e-198 < y < 7.49999999999999922e103

   1. Initial program 85.7%

      \[\frac{x - y}{\left(x \cdot 2\right) \cdot y} \]
   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 85.7

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

   4. Applied rewrites 85.7%

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

   if -1.54999999999999997e-215 < y < 2.4000000000000002e-205

   1. Initial program 59.6%

      \[\frac{x - y}{\left(x \cdot 2\right) \cdot y} \]
   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 97.7

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

   5. Applied rewrites 97.7%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{+124}:\\ \;\;\;\;\frac{-0.5}{x}\\ \mathbf{elif}\;y \leq -1.55 \cdot 10^{-215}:\\ \;\;\;\;\frac{x - y}{\left(x + x\right) \cdot y}\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{-205}:\\ \;\;\;\;\frac{0.5}{y}\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{-198} \lor \neg \left(y \leq 7.5 \cdot 10^{+103}\right):\\ \;\;\;\;\frac{-0.5}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{\left(x + x\right) \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 69.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -29 \lor \neg \left(y \leq 2.4 \cdot 10^{-205}\right):\\ \;\;\;\;\frac{-0.5}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -29.0) (not (<= y 2.4e-205))) (/ -0.5 x) (/ 0.5 y)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -29.0) || !(y <= 2.4e-205)) {
		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 <= (-29.0d0)) .or. (.not. (y <= 2.4d-205))) 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 <= -29.0) || !(y <= 2.4e-205)) {
		tmp = -0.5 / x;
	} else {
		tmp = 0.5 / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -29.0) or not (y <= 2.4e-205):
		tmp = -0.5 / x
	else:
		tmp = 0.5 / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -29.0) || !(y <= 2.4e-205))
		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 <= -29.0) || ~((y <= 2.4e-205)))
		tmp = -0.5 / x;
	else
		tmp = 0.5 / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -29.0], N[Not[LessEqual[y, 2.4e-205]], $MachinePrecision]], N[(-0.5 / x), $MachinePrecision], N[(0.5 / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -29 or 2.4000000000000002e-205 < y

   1. Initial program 73.1%

      \[\frac{x - y}{\left(x \cdot 2\right) \cdot y} \]
   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 74.9

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

   5. Applied rewrites 74.9%

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

   if -29 < y < 2.4000000000000002e-205

   1. Initial program 69.3%

      \[\frac{x - y}{\left(x \cdot 2\right) \cdot y} \]
   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 83.9

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

   5. Applied rewrites 83.9%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -29 \lor \neg \left(y \leq 2.4 \cdot 10^{-205}\right):\\ \;\;\;\;\frac{-0.5}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 50.6% 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 72.0%

   \[\frac{x - y}{\left(x \cdot 2\right) \cdot y} \]
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 58.0

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

5. Applied rewrites 58.0%

   \[\leadsto \color{blue}{\frac{-0.5}{x}} \]
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 2024338 
(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)))