Examples.Basics.BasicTests:f2 from sbv-4.4

Percentage Accurate: 93.5% → 100.0%
Time: 6.1s
Alternatives: 3
Speedup: 1.0×

Specification

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

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x * x) - (y * y)
end function

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

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

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

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

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

\begin{array}{l}

\\
x \cdot x - y \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: 93.5% accurate, 1.0× speedup?

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

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x * x) - (y * y)
end function

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

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

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

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

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

\begin{array}{l}

\\
x \cdot x - y \cdot y
\end{array}

Alternative 1: 100.0% accurate, 1.0× speedup?

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

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x + y) * (x - y)
end function

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

def code(x, y):
	return (x + y) * (x - y)

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

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

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

\begin{array}{l}

\\
\left(x + y\right) \cdot \left(x - y\right)
\end{array}
Derivation

  1. Initial program 94.1%

    \[x \cdot x - y \cdot y \]

  3. Simplified0

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

Alternative 2: 76.4% accurate, 0.3× speedup?

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -y \cdot y\\ \mathbf{if}\;x \cdot x \leq 6.5 \cdot 10^{-72}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \cdot x \leq 1.95 \cdot 10^{+61}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \cdot x \leq 8.5 \cdot 10^{+168}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (* y y))))
   (if (<= (* x x) 6.5e-72)
     t_0
     (if (<= (* x x) 1.95e+61)
       (* x x)
       (if (<= (* x x) 8.5e+168) t_0 (* x x))))))
double code(double x, double y) {
	double t_0 = -(y * y);
	double tmp;
	if ((x * x) <= 6.5e-72) {
		tmp = t_0;
	} else if ((x * x) <= 1.95e+61) {
		tmp = x * x;
	} else if ((x * x) <= 8.5e+168) {
		tmp = t_0;
	} else {
		tmp = x * x;
	}
	return tmp;
}

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 = -(y * y)
    if ((x * x) <= 6.5d-72) then
        tmp = t_0
    else if ((x * x) <= 1.95d+61) then
        tmp = x * x
    else if ((x * x) <= 8.5d+168) then
        tmp = t_0
    else
        tmp = x * x
    end if
    code = tmp
end function

public static double code(double x, double y) {
	double t_0 = -(y * y);
	double tmp;
	if ((x * x) <= 6.5e-72) {
		tmp = t_0;
	} else if ((x * x) <= 1.95e+61) {
		tmp = x * x;
	} else if ((x * x) <= 8.5e+168) {
		tmp = t_0;
	} else {
		tmp = x * x;
	}
	return tmp;
}

def code(x, y):
	t_0 = -(y * y)
	tmp = 0
	if (x * x) <= 6.5e-72:
		tmp = t_0
	elif (x * x) <= 1.95e+61:
		tmp = x * x
	elif (x * x) <= 8.5e+168:
		tmp = t_0
	else:
		tmp = x * x
	return tmp

function code(x, y)
	t_0 = Float64(-Float64(y * y))
	tmp = 0.0
	if (Float64(x * x) <= 6.5e-72)
		tmp = t_0;
	elseif (Float64(x * x) <= 1.95e+61)
		tmp = Float64(x * x);
	elseif (Float64(x * x) <= 8.5e+168)
		tmp = t_0;
	else
		tmp = Float64(x * x);
	end
	return tmp
end

function tmp_2 = code(x, y)
	t_0 = -(y * y);
	tmp = 0.0;
	if ((x * x) <= 6.5e-72)
		tmp = t_0;
	elseif ((x * x) <= 1.95e+61)
		tmp = x * x;
	elseif ((x * x) <= 8.5e+168)
		tmp = t_0;
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

code[x_, y_] := Block[{t$95$0 = (-N[(y * y), $MachinePrecision])}, If[LessEqual[N[(x * x), $MachinePrecision], 6.5e-72], t$95$0, If[LessEqual[N[(x * x), $MachinePrecision], 1.95e+61], N[(x * x), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 8.5e+168], t$95$0, N[(x * x), $MachinePrecision]]]]]

\begin{array}{l}

\\
\begin{array}{l}
t_0 := -y \cdot y\\
\mathbf{if}\;x \cdot x \leq 6.5 \cdot 10^{-72}:\\
\;\;\;\;t\_0\\

\mathbf{elif}\;x \cdot x \leq 1.95 \cdot 10^{+61}:\\
\;\;\;\;x \cdot x\\

\mathbf{elif}\;x \cdot x \leq 8.5 \cdot 10^{+168}:\\
\;\;\;\;t\_0\\

\mathbf{else}:\\
\;\;\;\;x \cdot x\\


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

  2. if (*.f64 x x) < 6.4999999999999997e-72 or 1.94999999999999994e61 < (*.f64 x x) < 8.50000000000000069e168

    1. Initial program 100.0%

      \[x \cdot x - y \cdot y \]

    3. Simplified0

      \[\leadsto expr\]
    4. Add Preprocessing

    5. Taylor expanded in x around 0 0

      \[\leadsto expr\]

    7. Simplified0

      \[\leadsto expr\]

    9. Applied egg-rr0

      \[\leadsto expr\]

    if 6.4999999999999997e-72 < (*.f64 x x) < 1.94999999999999994e61 or 8.50000000000000069e168 < (*.f64 x x)

    1. Initial program 87.7%

      \[x \cdot x - y \cdot y \]

    3. Simplified0

      \[\leadsto expr\]
    4. Add Preprocessing

    5. Taylor expanded in x around inf 0

      \[\leadsto expr\]

    7. Simplified0

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

Alternative 3: 54.0% accurate, 2.3× speedup?

\[\begin{array}{l} \\ x \cdot x \end{array} \]
(FPCore (x y) :precision binary64 (* x x))
double code(double x, double y) {
	return x * x;
}

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x * x
end function

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

def code(x, y):
	return x * x

function code(x, y)
	return Float64(x * x)
end

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

code[x_, y_] := N[(x * x), $MachinePrecision]

\begin{array}{l}

\\
x \cdot x
\end{array}
Derivation

  1. Initial program 94.1%

    \[x \cdot x - y \cdot y \]

  3. Simplified0

    \[\leadsto expr\]
  4. Add Preprocessing

  5. Taylor expanded in x around inf 0

    \[\leadsto expr\]

  7. Simplified0

    \[\leadsto expr\]
  8. Add Preprocessing

Reproduce

?
herbie shell --seed 2024110 
(FPCore (x y)
  :name "Examples.Basics.BasicTests:f2 from sbv-4.4"
  :precision binary64
  (- (* x x) (* y y)))