sqrt sqr

Percentage Accurate: 50.2% → 100.0%
Time: 4.3s
Alternatives: 2
Speedup: 19.2×

Specification

?
\[\frac{x}{x} - \frac{1}{x} \cdot \sqrt{x \cdot x} \]
(FPCore (x)
  :precision binary64
  (- (/ x x) (* (/ 1.0 x) (sqrt (* x x)))))
double code(double x) {
	return (x / x) - ((1.0 / x) * sqrt((x * x)));
}

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    code = (x / x) - ((1.0d0 / x) * sqrt((x * x)))
end function

public static double code(double x) {
	return (x / x) - ((1.0 / x) * Math.sqrt((x * x)));
}

def code(x):
	return (x / x) - ((1.0 / x) * math.sqrt((x * x)))

function code(x)
	return Float64(Float64(x / x) - Float64(Float64(1.0 / x) * sqrt(Float64(x * x))))
end

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

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

\frac{x}{x} - \frac{1}{x} \cdot \sqrt{x \cdot x}

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 2 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: 50.2% accurate, 1.0× speedup?

\[\frac{x}{x} - \frac{1}{x} \cdot \sqrt{x \cdot x} \]
(FPCore (x)
  :precision binary64
  (- (/ x x) (* (/ 1.0 x) (sqrt (* x x)))))
double code(double x) {
	return (x / x) - ((1.0 / x) * sqrt((x * x)));
}

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    code = (x / x) - ((1.0d0 / x) * sqrt((x * x)))
end function

public static double code(double x) {
	return (x / x) - ((1.0 / x) * Math.sqrt((x * x)));
}

def code(x):
	return (x / x) - ((1.0 / x) * math.sqrt((x * x)))

function code(x)
	return Float64(Float64(x / x) - Float64(Float64(1.0 / x) * sqrt(Float64(x * x))))
end

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

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

\frac{x}{x} - \frac{1}{x} \cdot \sqrt{x \cdot x}

Alternative 1: 100.0% accurate, 1.9× speedup?

\[\begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-308}:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(1, x\right) - 1\\ \end{array} \]
(FPCore (x)
  :precision binary64
  (if (<= x -4e-308) 2.0 (- (copysign 1.0 x) 1.0)))
double code(double x) {
	double tmp;
	if (x <= -4e-308) {
		tmp = 2.0;
	} else {
		tmp = copysign(1.0, x) - 1.0;
	}
	return tmp;
}

public static double code(double x) {
	double tmp;
	if (x <= -4e-308) {
		tmp = 2.0;
	} else {
		tmp = Math.copySign(1.0, x) - 1.0;
	}
	return tmp;
}

def code(x):
	tmp = 0
	if x <= -4e-308:
		tmp = 2.0
	else:
		tmp = math.copysign(1.0, x) - 1.0
	return tmp

function code(x)
	tmp = 0.0
	if (x <= -4e-308)
		tmp = 2.0;
	else
		tmp = Float64(copysign(1.0, x) - 1.0);
	end
	return tmp
end

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -4e-308)
		tmp = 2.0;
	else
		tmp = (sign(x) * abs(1.0)) - 1.0;
	end
	tmp_2 = tmp;
end

code[x_] := If[LessEqual[x, -4e-308], 2.0, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] - 1.0), $MachinePrecision]]

\begin{array}{l}
\mathbf{if}\;x \leq -4 \cdot 10^{-308}:\\
\;\;\;\;2\\

\mathbf{else}:\\
\;\;\;\;\mathsf{copysign}\left(1, x\right) - 1\\


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

  2. if x < -4.0000000000000001e-308

    1. Initial program 50.2%

      \[\frac{x}{x} - \frac{1}{x} \cdot \sqrt{x \cdot x} \]

    2. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{2} \]
    3. Step-by-step derivation

      1. Applied rewrites50.5%

        \[\leadsto \color{blue}{2} \]

      if -4.0000000000000001e-308 < x

      1. Initial program 50.2%

        \[\frac{x}{x} - \frac{1}{x} \cdot \sqrt{x \cdot x} \]

      3. Applied rewrites51.9%

        \[\leadsto \color{blue}{\mathsf{copysign}\left(1, x\right) - 1} \]
    4. Recombined 2 regimes into one program.
    5. Add Preprocessing

    Alternative 2: 50.5% accurate, 19.2× speedup?

    \[2 \]
    (FPCore (x)
  :precision binary64
  2.0)
    double code(double x) {
	return 2.0;
}

    real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    code = 2.0d0
end function

    public static double code(double x) {
	return 2.0;
}

    def code(x):
	return 2.0

    function code(x)
	return 2.0
end

    function tmp = code(x)
	tmp = 2.0;
end

    code[x_] := 2.0

    2
    Derivation

    1. Initial program 50.2%

      \[\frac{x}{x} - \frac{1}{x} \cdot \sqrt{x \cdot x} \]

    2. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{2} \]
    3. Step-by-step derivation

      1. Applied rewrites50.5%

        \[\leadsto \color{blue}{2} \]
      2. Add Preprocessing

      Reproduce

      ?
      herbie shell --seed 2025326 
(FPCore (x)
  :name "sqrt sqr"
  :precision binary64
  (- (/ x x) (* (/ 1.0 x) (sqrt (* x x)))))