Linear.Projection:inversePerspective from linear-1.19.1.3, B

Percentage Accurate: 76.3% → 100.0%

Time: 3.5s

Alternatives: 3

Speedup: 1.3×

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: 76.3% 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, 1.3× 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 75.5%

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

   1. div-sub 74.9%

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

   2. associate-/r* 80.3%

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

   3. associate-/r* 80.3%

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

   4. *-inverses 80.3%

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

   5. metadata-eval 80.3%

      \[\leadsto \frac{\color{blue}{0.5}}{y} - \frac{y}{\left(x \cdot 2\right) \cdot y} \]

   6. associate-/l/ 100.0%

      \[\leadsto \frac{0.5}{y} - \color{blue}{\frac{\frac{y}{y}}{x \cdot 2}} \]

   7. *-inverses 100.0%

      \[\leadsto \frac{0.5}{y} - \frac{\color{blue}{1}}{x \cdot 2} \]

   8. *-inverses 100.0%

      \[\leadsto \frac{0.5}{y} - \frac{\color{blue}{\frac{x}{x}}}{x \cdot 2} \]

   9. *-commutative 100.0%

      \[\leadsto \frac{0.5}{y} - \frac{\frac{x}{x}}{\color{blue}{2 \cdot x}} \]

   10. associate-/r* 100.0%

       \[\leadsto \frac{0.5}{y} - \color{blue}{\frac{\frac{\frac{x}{x}}{2}}{x}} \]

   11. *-inverses 100.0%

       \[\leadsto \frac{0.5}{y} - \frac{\frac{\color{blue}{1}}{2}}{x} \]

   12. metadata-eval 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 74.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{+27}:\\ \;\;\;\;\frac{0.5}{y}\\ \mathbf{elif}\;x \leq -0.0031 \lor \neg \left(x \leq -2.9 \cdot 10^{-67}\right) \land x \leq 5.2 \cdot 10^{+48}:\\ \;\;\;\;\frac{-0.5}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.3e+27)
   (/ 0.5 y)
   (if (or (<= x -0.0031) (and (not (<= x -2.9e-67)) (<= x 5.2e+48)))
     (/ -0.5 x)
     (/ 0.5 y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.3e+27) {
		tmp = 0.5 / y;
	} else if ((x <= -0.0031) || (!(x <= -2.9e-67) && (x <= 5.2e+48))) {
		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 (x <= (-1.3d+27)) then
        tmp = 0.5d0 / y
    else if ((x <= (-0.0031d0)) .or. (.not. (x <= (-2.9d-67))) .and. (x <= 5.2d+48)) 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 (x <= -1.3e+27) {
		tmp = 0.5 / y;
	} else if ((x <= -0.0031) || (!(x <= -2.9e-67) && (x <= 5.2e+48))) {
		tmp = -0.5 / x;
	} else {
		tmp = 0.5 / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.3e+27:
		tmp = 0.5 / y
	elif (x <= -0.0031) or (not (x <= -2.9e-67) and (x <= 5.2e+48)):
		tmp = -0.5 / x
	else:
		tmp = 0.5 / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.3e+27)
		tmp = Float64(0.5 / y);
	elseif ((x <= -0.0031) || (!(x <= -2.9e-67) && (x <= 5.2e+48)))
		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 (x <= -1.3e+27)
		tmp = 0.5 / y;
	elseif ((x <= -0.0031) || (~((x <= -2.9e-67)) && (x <= 5.2e+48)))
		tmp = -0.5 / x;
	else
		tmp = 0.5 / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.3e+27], N[(0.5 / y), $MachinePrecision], If[Or[LessEqual[x, -0.0031], And[N[Not[LessEqual[x, -2.9e-67]], $MachinePrecision], LessEqual[x, 5.2e+48]]], N[(-0.5 / x), $MachinePrecision], N[(0.5 / y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.30000000000000004e27 or -0.00309999999999999989 < x < -2.90000000000000005e-67 or 5.1999999999999999e48 < x

   1. Initial program 73.7%

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

      1. div-sub 73.6%

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

      2. associate-/r* 84.4%

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

      3. associate-/r* 84.4%

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

      4. *-inverses 84.4%

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

      5. metadata-eval 84.4%

         \[\leadsto \frac{\color{blue}{0.5}}{y} - \frac{y}{\left(x \cdot 2\right) \cdot y} \]

      6. associate-/l/ 99.9%

         \[\leadsto \frac{0.5}{y} - \color{blue}{\frac{\frac{y}{y}}{x \cdot 2}} \]

      7. *-inverses 99.9%

         \[\leadsto \frac{0.5}{y} - \frac{\color{blue}{1}}{x \cdot 2} \]

      8. *-inverses 99.9%

         \[\leadsto \frac{0.5}{y} - \frac{\color{blue}{\frac{x}{x}}}{x \cdot 2} \]

      9. *-commutative 99.9%

         \[\leadsto \frac{0.5}{y} - \frac{\frac{x}{x}}{\color{blue}{2 \cdot x}} \]

      10. associate-/r* 99.9%

          \[\leadsto \frac{0.5}{y} - \color{blue}{\frac{\frac{\frac{x}{x}}{2}}{x}} \]

      11. *-inverses 99.9%

          \[\leadsto \frac{0.5}{y} - \frac{\frac{\color{blue}{1}}{2}}{x} \]

      12. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around 0 81.5%

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

   if -1.30000000000000004e27 < x < -0.00309999999999999989 or -2.90000000000000005e-67 < x < 5.1999999999999999e48

   1. Initial program 77.0%

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

      1. div-sub 76.0%

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

      2. associate-/r* 76.7%

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

      3. associate-/r* 76.7%

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

      4. *-inverses 76.7%

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

      5. metadata-eval 76.7%

         \[\leadsto \frac{\color{blue}{0.5}}{y} - \frac{y}{\left(x \cdot 2\right) \cdot y} \]

      6. associate-/l/ 100.0%

         \[\leadsto \frac{0.5}{y} - \color{blue}{\frac{\frac{y}{y}}{x \cdot 2}} \]

      7. *-inverses 100.0%

         \[\leadsto \frac{0.5}{y} - \frac{\color{blue}{1}}{x \cdot 2} \]

      8. *-inverses 100.0%

         \[\leadsto \frac{0.5}{y} - \frac{\color{blue}{\frac{x}{x}}}{x \cdot 2} \]

      9. *-commutative 100.0%

         \[\leadsto \frac{0.5}{y} - \frac{\frac{x}{x}}{\color{blue}{2 \cdot x}} \]

      10. associate-/r* 100.0%

          \[\leadsto \frac{0.5}{y} - \color{blue}{\frac{\frac{\frac{x}{x}}{2}}{x}} \]

      11. *-inverses 100.0%

          \[\leadsto \frac{0.5}{y} - \frac{\frac{\color{blue}{1}}{2}}{x} \]

      12. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around inf 80.4%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{+27}:\\ \;\;\;\;\frac{0.5}{y}\\ \mathbf{elif}\;x \leq -0.0031 \lor \neg \left(x \leq -2.9 \cdot 10^{-67}\right) \land x \leq 5.2 \cdot 10^{+48}:\\ \;\;\;\;\frac{-0.5}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{y}\\ \end{array} \]

Alternative 3: 51.2% accurate, 3.0× 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 75.5%

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

   1. div-sub 74.9%

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

   2. associate-/r* 80.3%

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

   3. associate-/r* 80.3%

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

   4. *-inverses 80.3%

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

   5. metadata-eval 80.3%

      \[\leadsto \frac{\color{blue}{0.5}}{y} - \frac{y}{\left(x \cdot 2\right) \cdot y} \]

   6. associate-/l/ 100.0%

      \[\leadsto \frac{0.5}{y} - \color{blue}{\frac{\frac{y}{y}}{x \cdot 2}} \]

   7. *-inverses 100.0%

      \[\leadsto \frac{0.5}{y} - \frac{\color{blue}{1}}{x \cdot 2} \]

   8. *-inverses 100.0%

      \[\leadsto \frac{0.5}{y} - \frac{\color{blue}{\frac{x}{x}}}{x \cdot 2} \]

   9. *-commutative 100.0%

      \[\leadsto \frac{0.5}{y} - \frac{\frac{x}{x}}{\color{blue}{2 \cdot x}} \]

   10. associate-/r* 100.0%

       \[\leadsto \frac{0.5}{y} - \color{blue}{\frac{\frac{\frac{x}{x}}{2}}{x}} \]

   11. *-inverses 100.0%

       \[\leadsto \frac{0.5}{y} - \frac{\frac{\color{blue}{1}}{2}}{x} \]

   12. metadata-eval 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in y around inf 52.7%

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

5. Final simplification 52.7%

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

Developer target: 100.0% accurate, 1.3× 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 2023214 
(FPCore (x y)
  :name "Linear.Projection:inversePerspective from linear-1.19.1.3, B"
  :precision binary64

  :herbie-target
  (- (/ 0.5 y) (/ 0.5 x))

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