x / (x^2 + 1)

Percentage Accurate: 77.0% → 100.0%
Time: 1.6s
Alternatives: 3
Speedup: 1.0×

Specification

?
\[\begin{array}{l} \\ \frac{x}{x \cdot x + 1} \end{array} \]
(FPCore (x) :precision binary64 (/ x (+ (* x x) 1.0)))
double code(double x) {
	return x / ((x * x) + 1.0);
}

real(8) function code(x)
    real(8), intent (in) :: x
    code = x / ((x * x) + 1.0d0)
end function

public static double code(double x) {
	return x / ((x * x) + 1.0);
}

def code(x):
	return x / ((x * x) + 1.0)

function code(x)
	return Float64(x / Float64(Float64(x * x) + 1.0))
end

function tmp = code(x)
	tmp = x / ((x * x) + 1.0);
end

code[x_] := N[(x / N[(N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

\begin{array}{l}

\\
\frac{x}{x \cdot x + 1}
\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: 77.0% accurate, 1.0× speedup?

\[\begin{array}{l} \\ \frac{x}{x \cdot x + 1} \end{array} \]
(FPCore (x) :precision binary64 (/ x (+ (* x x) 1.0)))
double code(double x) {
	return x / ((x * x) + 1.0);
}

real(8) function code(x)
    real(8), intent (in) :: x
    code = x / ((x * x) + 1.0d0)
end function

public static double code(double x) {
	return x / ((x * x) + 1.0);
}

def code(x):
	return x / ((x * x) + 1.0)

function code(x)
	return Float64(x / Float64(Float64(x * x) + 1.0))
end

function tmp = code(x)
	tmp = x / ((x * x) + 1.0);
end

code[x_] := N[(x / N[(N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

\begin{array}{l}

\\
\frac{x}{x \cdot x + 1}
\end{array}

Alternative 1: 100.0% accurate, 0.6× speedup?

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+38} \lor \neg \left(x \leq 50000000\right):\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + x \cdot x}\\ \end{array} \end{array} \]
(FPCore (x)
 :precision binary64
 (if (or (<= x -2e+38) (not (<= x 50000000.0)))
   (/ 1.0 x)
   (/ x (+ 1.0 (* x x)))))
double code(double x) {
	double tmp;
	if ((x <= -2e+38) || !(x <= 50000000.0)) {
		tmp = 1.0 / x;
	} else {
		tmp = x / (1.0 + (x * x));
	}
	return tmp;
}

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-2d+38)) .or. (.not. (x <= 50000000.0d0))) then
        tmp = 1.0d0 / x
    else
        tmp = x / (1.0d0 + (x * x))
    end if
    code = tmp
end function

public static double code(double x) {
	double tmp;
	if ((x <= -2e+38) || !(x <= 50000000.0)) {
		tmp = 1.0 / x;
	} else {
		tmp = x / (1.0 + (x * x));
	}
	return tmp;
}

def code(x):
	tmp = 0
	if (x <= -2e+38) or not (x <= 50000000.0):
		tmp = 1.0 / x
	else:
		tmp = x / (1.0 + (x * x))
	return tmp

function code(x)
	tmp = 0.0
	if ((x <= -2e+38) || !(x <= 50000000.0))
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(x / Float64(1.0 + Float64(x * x)));
	end
	return tmp
end

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -2e+38) || ~((x <= 50000000.0)))
		tmp = 1.0 / x;
	else
		tmp = x / (1.0 + (x * x));
	end
	tmp_2 = tmp;
end

code[x_] := If[Or[LessEqual[x, -2e+38], N[Not[LessEqual[x, 50000000.0]], $MachinePrecision]], N[(1.0 / x), $MachinePrecision], N[(x / N[(1.0 + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

\begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;x \leq -2 \cdot 10^{+38} \lor \neg \left(x \leq 50000000\right):\\
\;\;\;\;\frac{1}{x}\\

\mathbf{else}:\\
\;\;\;\;\frac{x}{1 + x \cdot x}\\


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

  2. if x < -1.99999999999999995e38 or 5e7 < x

    1. Initial program 51.8%

      \[\frac{x}{x \cdot x + 1} \]

    2. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{\frac{1}{x}} \]

    if -1.99999999999999995e38 < x < 5e7

    1. Initial program 100.0%

      \[\frac{x}{x \cdot x + 1} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification100.0%

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+38} \lor \neg \left(x \leq 50000000\right):\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + x \cdot x}\\ \end{array} \]

Alternative 2: 98.9% accurate, 1.0× speedup?

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]
(FPCore (x)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 1.0))) (/ 1.0 x) x))
double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.0)) {
		tmp = 1.0 / x;
	} else {
		tmp = x;
	}
	return tmp;
}

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.0d0)) .or. (.not. (x <= 1.0d0))) then
        tmp = 1.0d0 / x
    else
        tmp = x
    end if
    code = tmp
end function

public static double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.0)) {
		tmp = 1.0 / x;
	} else {
		tmp = x;
	}
	return tmp;
}

def code(x):
	tmp = 0
	if (x <= -1.0) or not (x <= 1.0):
		tmp = 1.0 / x
	else:
		tmp = x
	return tmp

function code(x)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 1.0))
		tmp = Float64(1.0 / x);
	else
		tmp = x;
	end
	return tmp
end

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 1.0)))
		tmp = 1.0 / x;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

code[x_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], N[(1.0 / x), $MachinePrecision], x]

\begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1\right):\\
\;\;\;\;\frac{1}{x}\\

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


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

  2. if x < -1 or 1 < x

    1. Initial program 56.8%

      \[\frac{x}{x \cdot x + 1} \]

    2. Taylor expanded in x around inf 95.5%

      \[\leadsto \color{blue}{\frac{1}{x}} \]

    if -1 < x < 1

    1. Initial program 100.0%

      \[\frac{x}{x \cdot x + 1} \]

    2. Taylor expanded in x around 0 99.5%

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

  4. Final simplification97.6%

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3: 51.2% accurate, 7.0× speedup?

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

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

public static double code(double x) {
	return x;
}

def code(x):
	return x

function code(x)
	return x
end

function tmp = code(x)
	tmp = x;
end

code[x_] := x

\begin{array}{l}

\\
x
\end{array}
Derivation

  1. Initial program 78.9%

    \[\frac{x}{x \cdot x + 1} \]

  2. Taylor expanded in x around 0 53.0%

    \[\leadsto \color{blue}{x} \]

  3. Final simplification53.0%

    \[\leadsto x \]

Developer target: 99.9% accurate, 1.0× speedup?

\[\begin{array}{l} \\ \frac{1}{x + \frac{1}{x}} \end{array} \]
(FPCore (x) :precision binary64 (/ 1.0 (+ x (/ 1.0 x))))
double code(double x) {
	return 1.0 / (x + (1.0 / x));
}

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 / (x + (1.0d0 / x))
end function

public static double code(double x) {
	return 1.0 / (x + (1.0 / x));
}

def code(x):
	return 1.0 / (x + (1.0 / x))

function code(x)
	return Float64(1.0 / Float64(x + Float64(1.0 / x)))
end

function tmp = code(x)
	tmp = 1.0 / (x + (1.0 / x));
end

code[x_] := N[(1.0 / N[(x + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\begin{array}{l}

\\
\frac{1}{x + \frac{1}{x}}
\end{array}

Reproduce

?
herbie shell --seed 2023332 
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

  :herbie-target
  (/ 1.0 (+ x (/ 1.0 x)))

  (/ x (+ (* x x) 1.0)))