Linear.Projection:inversePerspective from linear-1.19.1.3, B

Percentage Accurate: 76.0% → 100.0%
Time: 8.1s
Alternatives: 3
Speedup: 1.3×

Specification

?
\[\begin{array}{l} \\ \frac{x - y}{\left(x \cdot 2\right) \cdot y} \end{array} \]
(FPCore (x y) :precision binary64 (/ (- x y) (* (* x 2.0) y)))
double code(double x, double y) {
	return (x - y) / ((x * 2.0) * y);
}

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

public static double code(double x, double y) {
	return (x - y) / ((x * 2.0) * y);
}

def code(x, y):
	return (x - y) / ((x * 2.0) * y)

function code(x, y)
	return Float64(Float64(x - y) / Float64(Float64(x * 2.0) * y))
end

function tmp = code(x, y)
	tmp = (x - y) / ((x * 2.0) * y);
end

code[x_, y_] := N[(N[(x - y), $MachinePrecision] / N[(N[(x * 2.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]

\begin{array}{l}

\\
\frac{x - y}{\left(x \cdot 2\right) \cdot y}
\end{array}

Sampling outcomes in binary64 precision:

Local Percentage Accuracy vs ?

The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples.

Accuracy vs Speed?

Herbie found 3 alternatives:

AlternativeAccuracySpeedup
The accuracy (vertical axis) and speed (horizontal axis) of each alternatives. Up and to the right is better. The red square shows the initial program, and each blue circle shows an alternative.The line shows the best available speed-accuracy tradeoffs.

Initial Program: 76.0% accurate, 1.0× speedup?

\[\begin{array}{l} \\ \frac{x - y}{\left(x \cdot 2\right) \cdot y} \end{array} \]
(FPCore (x y) :precision binary64 (/ (- x y) (* (* x 2.0) y)))
double code(double x, double y) {
	return (x - y) / ((x * 2.0) * y);
}

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

public static double code(double x, double y) {
	return (x - y) / ((x * 2.0) * y);
}

def code(x, y):
	return (x - y) / ((x * 2.0) * y)

function code(x, y)
	return Float64(Float64(x - y) / Float64(Float64(x * 2.0) * y))
end

function tmp = code(x, y)
	tmp = (x - y) / ((x * 2.0) * y);
end

code[x_, y_] := N[(N[(x - y), $MachinePrecision] / N[(N[(x * 2.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]

\begin{array}{l}

\\
\frac{x - y}{\left(x \cdot 2\right) \cdot y}
\end{array}

Alternative 1: 100.0% accurate, 1.3× speedup?

\[\begin{array}{l} \\ \frac{0.5}{y} + \frac{-0.5}{x} \end{array} \]
(FPCore (x y) :precision binary64 (+ (/ 0.5 y) (/ -0.5 x)))
double code(double x, double y) {
	return (0.5 / y) + (-0.5 / x);
}

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

public static double code(double x, double y) {
	return (0.5 / y) + (-0.5 / x);
}

def code(x, y):
	return (0.5 / y) + (-0.5 / x)

function code(x, y)
	return Float64(Float64(0.5 / y) + Float64(-0.5 / x))
end

function tmp = code(x, y)
	tmp = (0.5 / y) + (-0.5 / x);
end

code[x_, y_] := N[(N[(0.5 / y), $MachinePrecision] + N[(-0.5 / x), $MachinePrecision]), $MachinePrecision]

\begin{array}{l}

\\
\frac{0.5}{y} + \frac{-0.5}{x}
\end{array}
Derivation

  1. Initial program 78.6%

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

  3. Simplified0

    \[\leadsto expr\]
  4. Add Preprocessing
  5. Add Preprocessing

Alternative 2: 74.0% accurate, 0.4× speedup?

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2850000000000:\\ \;\;\;\;\frac{0.5}{y}\\ \mathbf{elif}\;x \leq -5.9 \cdot 10^{-64}:\\ \;\;\;\;\frac{-0.5}{x}\\ \mathbf{elif}\;x \leq -6.6 \cdot 10^{-103}:\\ \;\;\;\;\frac{0.5}{y}\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{+16}:\\ \;\;\;\;\frac{-0.5}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{y}\\ \end{array} \end{array} \]
(FPCore (x y)
 :precision binary64
 (if (<= x -2850000000000.0)
   (/ 0.5 y)
   (if (<= x -5.9e-64)
     (/ -0.5 x)
     (if (<= x -6.6e-103)
       (/ 0.5 y)
       (if (<= x 1.3e+16) (/ -0.5 x) (/ 0.5 y))))))
double code(double x, double y) {
	double tmp;
	if (x <= -2850000000000.0) {
		tmp = 0.5 / y;
	} else if (x <= -5.9e-64) {
		tmp = -0.5 / x;
	} else if (x <= -6.6e-103) {
		tmp = 0.5 / y;
	} else if (x <= 1.3e+16) {
		tmp = -0.5 / x;
	} else {
		tmp = 0.5 / y;
	}
	return tmp;
}

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-2850000000000.0d0)) then
        tmp = 0.5d0 / y
    else if (x <= (-5.9d-64)) then
        tmp = (-0.5d0) / x
    else if (x <= (-6.6d-103)) then
        tmp = 0.5d0 / y
    else if (x <= 1.3d+16) then
        tmp = (-0.5d0) / x
    else
        tmp = 0.5d0 / y
    end if
    code = tmp
end function

public static double code(double x, double y) {
	double tmp;
	if (x <= -2850000000000.0) {
		tmp = 0.5 / y;
	} else if (x <= -5.9e-64) {
		tmp = -0.5 / x;
	} else if (x <= -6.6e-103) {
		tmp = 0.5 / y;
	} else if (x <= 1.3e+16) {
		tmp = -0.5 / x;
	} else {
		tmp = 0.5 / y;
	}
	return tmp;
}

def code(x, y):
	tmp = 0
	if x <= -2850000000000.0:
		tmp = 0.5 / y
	elif x <= -5.9e-64:
		tmp = -0.5 / x
	elif x <= -6.6e-103:
		tmp = 0.5 / y
	elif x <= 1.3e+16:
		tmp = -0.5 / x
	else:
		tmp = 0.5 / y
	return tmp

function code(x, y)
	tmp = 0.0
	if (x <= -2850000000000.0)
		tmp = Float64(0.5 / y);
	elseif (x <= -5.9e-64)
		tmp = Float64(-0.5 / x);
	elseif (x <= -6.6e-103)
		tmp = Float64(0.5 / y);
	elseif (x <= 1.3e+16)
		tmp = Float64(-0.5 / x);
	else
		tmp = Float64(0.5 / y);
	end
	return tmp
end

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2850000000000.0)
		tmp = 0.5 / y;
	elseif (x <= -5.9e-64)
		tmp = -0.5 / x;
	elseif (x <= -6.6e-103)
		tmp = 0.5 / y;
	elseif (x <= 1.3e+16)
		tmp = -0.5 / x;
	else
		tmp = 0.5 / y;
	end
	tmp_2 = tmp;
end

code[x_, y_] := If[LessEqual[x, -2850000000000.0], N[(0.5 / y), $MachinePrecision], If[LessEqual[x, -5.9e-64], N[(-0.5 / x), $MachinePrecision], If[LessEqual[x, -6.6e-103], N[(0.5 / y), $MachinePrecision], If[LessEqual[x, 1.3e+16], N[(-0.5 / x), $MachinePrecision], N[(0.5 / y), $MachinePrecision]]]]]

\begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;x \leq -2850000000000:\\
\;\;\;\;\frac{0.5}{y}\\

\mathbf{elif}\;x \leq -5.9 \cdot 10^{-64}:\\
\;\;\;\;\frac{-0.5}{x}\\

\mathbf{elif}\;x \leq -6.6 \cdot 10^{-103}:\\
\;\;\;\;\frac{0.5}{y}\\

\mathbf{elif}\;x \leq 1.3 \cdot 10^{+16}:\\
\;\;\;\;\frac{-0.5}{x}\\

\mathbf{else}:\\
\;\;\;\;\frac{0.5}{y}\\


\end{array}
\end{array}
Derivation
  1. Split input into 2 regimes

  2. if x < -2.85e12 or -5.89999999999999995e-64 < x < -6.59999999999999979e-103 or 1.3e16 < x

    1. Initial program 79.6%

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

    3. Simplified0

      \[\leadsto expr\]
    4. Add Preprocessing

    5. Taylor expanded in y around 0 0

      \[\leadsto expr\]

    7. Simplified0

      \[\leadsto expr\]

    if -2.85e12 < x < -5.89999999999999995e-64 or -6.59999999999999979e-103 < x < 1.3e16

    1. Initial program 77.3%

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

    3. Simplified0

      \[\leadsto expr\]
    4. Add Preprocessing

    5. Taylor expanded in y around inf 0

      \[\leadsto expr\]

    7. Simplified0

      \[\leadsto expr\]
  3. Recombined 2 regimes into one program.
  4. Add Preprocessing

Alternative 3: 50.4% accurate, 3.0× speedup?

\[\begin{array}{l} \\ \frac{-0.5}{x} \end{array} \]
(FPCore (x y) :precision binary64 (/ -0.5 x))
double code(double x, double y) {
	return -0.5 / x;
}

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (-0.5d0) / x
end function

public static double code(double x, double y) {
	return -0.5 / x;
}

def code(x, y):
	return -0.5 / x

function code(x, y)
	return Float64(-0.5 / x)
end

function tmp = code(x, y)
	tmp = -0.5 / x;
end

code[x_, y_] := N[(-0.5 / x), $MachinePrecision]

\begin{array}{l}

\\
\frac{-0.5}{x}
\end{array}
Derivation

  1. Initial program 78.6%

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

  3. Simplified0

    \[\leadsto expr\]
  4. Add Preprocessing

  5. Taylor expanded in y around inf 0

    \[\leadsto expr\]

  7. Simplified0

    \[\leadsto expr\]
  8. Add Preprocessing

Developer target: 100.0% accurate, 1.3× speedup?

\[\begin{array}{l} \\ \frac{0.5}{y} - \frac{0.5}{x} \end{array} \]
(FPCore (x y) :precision binary64 (- (/ 0.5 y) (/ 0.5 x)))
double code(double x, double y) {
	return (0.5 / y) - (0.5 / x);
}

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

public static double code(double x, double y) {
	return (0.5 / y) - (0.5 / x);
}

def code(x, y):
	return (0.5 / y) - (0.5 / x)

function code(x, y)
	return Float64(Float64(0.5 / y) - Float64(0.5 / x))
end

function tmp = code(x, y)
	tmp = (0.5 / y) - (0.5 / x);
end

code[x_, y_] := N[(N[(0.5 / y), $MachinePrecision] - N[(0.5 / x), $MachinePrecision]), $MachinePrecision]

\begin{array}{l}

\\
\frac{0.5}{y} - \frac{0.5}{x}
\end{array}

Reproduce

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herbie shell --seed 2024110 
(FPCore (x y)
  :name "Linear.Projection:inversePerspective from linear-1.19.1.3, B"
  :precision binary64

  :alt
  (- (/ 0.5 y) (/ 0.5 x))

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