Data.Colour.CIE:lightness from colour-2.3.3

Percentage Accurate: 100.0% → 100.0%
Time: 2.4s
Alternatives: 3
Speedup: 1.3×

Specification

?
\[\begin{array}{l} \\ x \cdot 116 - 16 \end{array} \]
(FPCore (x) :precision binary64 (- (* x 116.0) 16.0))
double code(double x) {
	return (x * 116.0) - 16.0;
}

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x * 116.0d0) - 16.0d0
end function

public static double code(double x) {
	return (x * 116.0) - 16.0;
}

def code(x):
	return (x * 116.0) - 16.0

function code(x)
	return Float64(Float64(x * 116.0) - 16.0)
end

function tmp = code(x)
	tmp = (x * 116.0) - 16.0;
end

code[x_] := N[(N[(x * 116.0), $MachinePrecision] - 16.0), $MachinePrecision]

\begin{array}{l}

\\
x \cdot 116 - 16
\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: 100.0% accurate, 1.0× speedup?

\[\begin{array}{l} \\ x \cdot 116 - 16 \end{array} \]
(FPCore (x) :precision binary64 (- (* x 116.0) 16.0))
double code(double x) {
	return (x * 116.0) - 16.0;
}

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x * 116.0d0) - 16.0d0
end function

public static double code(double x) {
	return (x * 116.0) - 16.0;
}

def code(x):
	return (x * 116.0) - 16.0

function code(x)
	return Float64(Float64(x * 116.0) - 16.0)
end

function tmp = code(x)
	tmp = (x * 116.0) - 16.0;
end

code[x_] := N[(N[(x * 116.0), $MachinePrecision] - 16.0), $MachinePrecision]

\begin{array}{l}

\\
x \cdot 116 - 16
\end{array}

Alternative 1: 100.0% accurate, 1.3× speedup?

\[\begin{array}{l} \\ \mathsf{fma}\left(x, 116, -16\right) \end{array} \]
(FPCore (x) :precision binary64 (fma x 116.0 -16.0))
double code(double x) {
	return fma(x, 116.0, -16.0);
}

function code(x)
	return fma(x, 116.0, -16.0)
end

code[x_] := N[(x * 116.0 + -16.0), $MachinePrecision]

\begin{array}{l}

\\
\mathsf{fma}\left(x, 116, -16\right)
\end{array}
Derivation

  1. Initial program 100.0%

    \[x \cdot 116 - 16 \]
  2. Add Preprocessing
  3. Step-by-step derivation

    1. lift--.f64N/A

      \[\leadsto \color{blue}{x \cdot 116 - 16} \]

    2. sub-negN/A

      \[\leadsto \color{blue}{x \cdot 116 + \left(\mathsf{neg}\left(16\right)\right)} \]

    3. lift-*.f64N/A

      \[\leadsto \color{blue}{x \cdot 116} + \left(\mathsf{neg}\left(16\right)\right) \]

    4. lower-fma.f64N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 116, \mathsf{neg}\left(16\right)\right)} \]

    5. metadata-eval100.0

      \[\leadsto \mathsf{fma}\left(x, 116, \color{blue}{-16}\right) \]

  4. Applied rewrites100.0%

    \[\leadsto \color{blue}{\mathsf{fma}\left(x, 116, -16\right)} \]
  5. Add Preprocessing

Alternative 2: 97.9% accurate, 0.3× speedup?

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot 116 \leq -2000000:\\ \;\;\;\;x \cdot 116\\ \mathbf{elif}\;x \cdot 116 \leq 2:\\ \;\;\;\;-16\\ \mathbf{else}:\\ \;\;\;\;x \cdot 116\\ \end{array} \end{array} \]
(FPCore (x)
 :precision binary64
 (if (<= (* x 116.0) -2000000.0)
   (* x 116.0)
   (if (<= (* x 116.0) 2.0) -16.0 (* x 116.0))))
double code(double x) {
	double tmp;
	if ((x * 116.0) <= -2000000.0) {
		tmp = x * 116.0;
	} else if ((x * 116.0) <= 2.0) {
		tmp = -16.0;
	} else {
		tmp = x * 116.0;
	}
	return tmp;
}

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x * 116.0d0) <= (-2000000.0d0)) then
        tmp = x * 116.0d0
    else if ((x * 116.0d0) <= 2.0d0) then
        tmp = -16.0d0
    else
        tmp = x * 116.0d0
    end if
    code = tmp
end function

public static double code(double x) {
	double tmp;
	if ((x * 116.0) <= -2000000.0) {
		tmp = x * 116.0;
	} else if ((x * 116.0) <= 2.0) {
		tmp = -16.0;
	} else {
		tmp = x * 116.0;
	}
	return tmp;
}

def code(x):
	tmp = 0
	if (x * 116.0) <= -2000000.0:
		tmp = x * 116.0
	elif (x * 116.0) <= 2.0:
		tmp = -16.0
	else:
		tmp = x * 116.0
	return tmp

function code(x)
	tmp = 0.0
	if (Float64(x * 116.0) <= -2000000.0)
		tmp = Float64(x * 116.0);
	elseif (Float64(x * 116.0) <= 2.0)
		tmp = -16.0;
	else
		tmp = Float64(x * 116.0);
	end
	return tmp
end

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x * 116.0) <= -2000000.0)
		tmp = x * 116.0;
	elseif ((x * 116.0) <= 2.0)
		tmp = -16.0;
	else
		tmp = x * 116.0;
	end
	tmp_2 = tmp;
end

code[x_] := If[LessEqual[N[(x * 116.0), $MachinePrecision], -2000000.0], N[(x * 116.0), $MachinePrecision], If[LessEqual[N[(x * 116.0), $MachinePrecision], 2.0], -16.0, N[(x * 116.0), $MachinePrecision]]]

\begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;x \cdot 116 \leq -2000000:\\
\;\;\;\;x \cdot 116\\

\mathbf{elif}\;x \cdot 116 \leq 2:\\
\;\;\;\;-16\\

\mathbf{else}:\\
\;\;\;\;x \cdot 116\\


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

  2. if (*.f64 x #s(literal 116 binary64)) < -2e6 or 2 < (*.f64 x #s(literal 116 binary64))

    1. Initial program 100.0%

      \[x \cdot 116 - 16 \]
    2. Add Preprocessing

    3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{116 \cdot x} \]
    4. Step-by-step derivation

      1. lower-*.f6498.5

        \[\leadsto \color{blue}{116 \cdot x} \]

    5. Applied rewrites98.5%

      \[\leadsto \color{blue}{116 \cdot x} \]

    if -2e6 < (*.f64 x #s(literal 116 binary64)) < 2

    1. Initial program 100.0%

      \[x \cdot 116 - 16 \]
    2. Add Preprocessing

    3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-16} \]
    4. Step-by-step derivation

      1. Applied rewrites97.4%

        \[\leadsto \color{blue}{-16} \]
    5. Recombined 2 regimes into one program.

    6. Final simplification97.9%

      \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot 116 \leq -2000000:\\ \;\;\;\;x \cdot 116\\ \mathbf{elif}\;x \cdot 116 \leq 2:\\ \;\;\;\;-16\\ \mathbf{else}:\\ \;\;\;\;x \cdot 116\\ \end{array} \]
    7. Add Preprocessing

    Reproduce

    ?
    herbie shell --seed 2024227 
(FPCore (x)
  :name "Data.Colour.CIE:lightness from colour-2.3.3"
  :precision binary64
  (- (* x 116.0) 16.0))