Data.Colour.CIE:cieLABView from colour-2.3.3, A

Percentage Accurate: 99.9% → 99.7%
Time: 4.4s
Alternatives: 4
Speedup: 4.4×

Specification

?
\[\begin{array}{l} \\ \frac{841}{108} \cdot x + \frac{4}{29} \end{array} \]
(FPCore (x) :precision binary64 (+ (* (/ 841.0 108.0) x) (/ 4.0 29.0)))
double code(double x) {
	return ((841.0 / 108.0) * x) + (4.0 / 29.0);
}

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((841.0d0 / 108.0d0) * x) + (4.0d0 / 29.0d0)
end function

public static double code(double x) {
	return ((841.0 / 108.0) * x) + (4.0 / 29.0);
}

def code(x):
	return ((841.0 / 108.0) * x) + (4.0 / 29.0)

function code(x)
	return Float64(Float64(Float64(841.0 / 108.0) * x) + Float64(4.0 / 29.0))
end

function tmp = code(x)
	tmp = ((841.0 / 108.0) * x) + (4.0 / 29.0);
end

code[x_] := N[(N[(N[(841.0 / 108.0), $MachinePrecision] * x), $MachinePrecision] + N[(4.0 / 29.0), $MachinePrecision]), $MachinePrecision]

\begin{array}{l}

\\
\frac{841}{108} \cdot x + \frac{4}{29}
\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 4 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: 99.9% accurate, 1.0× speedup?

\[\begin{array}{l} \\ \frac{841}{108} \cdot x + \frac{4}{29} \end{array} \]
(FPCore (x) :precision binary64 (+ (* (/ 841.0 108.0) x) (/ 4.0 29.0)))
double code(double x) {
	return ((841.0 / 108.0) * x) + (4.0 / 29.0);
}

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((841.0d0 / 108.0d0) * x) + (4.0d0 / 29.0d0)
end function

public static double code(double x) {
	return ((841.0 / 108.0) * x) + (4.0 / 29.0);
}

def code(x):
	return ((841.0 / 108.0) * x) + (4.0 / 29.0)

function code(x)
	return Float64(Float64(Float64(841.0 / 108.0) * x) + Float64(4.0 / 29.0))
end

function tmp = code(x)
	tmp = ((841.0 / 108.0) * x) + (4.0 / 29.0);
end

code[x_] := N[(N[(N[(841.0 / 108.0), $MachinePrecision] * x), $MachinePrecision] + N[(4.0 / 29.0), $MachinePrecision]), $MachinePrecision]

\begin{array}{l}

\\
\frac{841}{108} \cdot x + \frac{4}{29}
\end{array}

Alternative 1: 99.7% accurate, 0.7× speedup?

\[\begin{array}{l} \\ \mathsf{fma}\left(x, \frac{x}{\mathsf{fma}\left(-7.787037037037037, x, 0.13793103448275862\right)} \cdot -60.63794581618656, \frac{-0.019024970273483946}{\mathsf{fma}\left(x, 7.787037037037037, -0.13793103448275862\right)}\right) \end{array} \]
(FPCore (x)
 :precision binary64
 (fma
  x
  (* (/ x (fma -7.787037037037037 x 0.13793103448275862)) -60.63794581618656)
  (/ -0.019024970273483946 (fma x 7.787037037037037 -0.13793103448275862))))
double code(double x) {
	return fma(x, ((x / fma(-7.787037037037037, x, 0.13793103448275862)) * -60.63794581618656), (-0.019024970273483946 / fma(x, 7.787037037037037, -0.13793103448275862)));
}

function code(x)
	return fma(x, Float64(Float64(x / fma(-7.787037037037037, x, 0.13793103448275862)) * -60.63794581618656), Float64(-0.019024970273483946 / fma(x, 7.787037037037037, -0.13793103448275862)))
end

code[x_] := N[(x * N[(N[(x / N[(-7.787037037037037 * x + 0.13793103448275862), $MachinePrecision]), $MachinePrecision] * -60.63794581618656), $MachinePrecision] + N[(-0.019024970273483946 / N[(x * 7.787037037037037 + -0.13793103448275862), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\begin{array}{l}

\\
\mathsf{fma}\left(x, \frac{x}{\mathsf{fma}\left(-7.787037037037037, x, 0.13793103448275862\right)} \cdot -60.63794581618656, \frac{-0.019024970273483946}{\mathsf{fma}\left(x, 7.787037037037037, -0.13793103448275862\right)}\right)
\end{array}
Derivation

  1. Initial program 99.8%

    \[\frac{841}{108} \cdot x + \frac{4}{29} \]
  2. Add Preprocessing

  4. Applied rewrites99.9%

    \[\leadsto \color{blue}{\mathsf{fma}\left(x, -60.63794581618656 \cdot \frac{x}{\mathsf{fma}\left(-7.787037037037037, x, 0.13793103448275862\right)}, \frac{-0.019024970273483946}{\mathsf{fma}\left(x, 7.787037037037037, -0.13793103448275862\right)}\right)} \]

  5. Final simplification99.9%

    \[\leadsto \mathsf{fma}\left(x, \frac{x}{\mathsf{fma}\left(-7.787037037037037, x, 0.13793103448275862\right)} \cdot -60.63794581618656, \frac{-0.019024970273483946}{\mathsf{fma}\left(x, 7.787037037037037, -0.13793103448275862\right)}\right) \]
  6. Add Preprocessing

Alternative 2: 97.4% accurate, 0.6× speedup?

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{841}{108} \cdot x\\ \mathbf{if}\;t\_0 \leq -50000:\\ \;\;\;\;7.787037037037037 \cdot x\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-14}:\\ \;\;\;\;0.13793103448275862\\ \mathbf{else}:\\ \;\;\;\;7.787037037037037 \cdot x\\ \end{array} \end{array} \]
(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (/ 841.0 108.0) x)))
   (if (<= t_0 -50000.0)
     (* 7.787037037037037 x)
     (if (<= t_0 5e-14) 0.13793103448275862 (* 7.787037037037037 x)))))
double code(double x) {
	double t_0 = (841.0 / 108.0) * x;
	double tmp;
	if (t_0 <= -50000.0) {
		tmp = 7.787037037037037 * x;
	} else if (t_0 <= 5e-14) {
		tmp = 0.13793103448275862;
	} else {
		tmp = 7.787037037037037 * x;
	}
	return tmp;
}

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (841.0d0 / 108.0d0) * x
    if (t_0 <= (-50000.0d0)) then
        tmp = 7.787037037037037d0 * x
    else if (t_0 <= 5d-14) then
        tmp = 0.13793103448275862d0
    else
        tmp = 7.787037037037037d0 * x
    end if
    code = tmp
end function

public static double code(double x) {
	double t_0 = (841.0 / 108.0) * x;
	double tmp;
	if (t_0 <= -50000.0) {
		tmp = 7.787037037037037 * x;
	} else if (t_0 <= 5e-14) {
		tmp = 0.13793103448275862;
	} else {
		tmp = 7.787037037037037 * x;
	}
	return tmp;
}

def code(x):
	t_0 = (841.0 / 108.0) * x
	tmp = 0
	if t_0 <= -50000.0:
		tmp = 7.787037037037037 * x
	elif t_0 <= 5e-14:
		tmp = 0.13793103448275862
	else:
		tmp = 7.787037037037037 * x
	return tmp

function code(x)
	t_0 = Float64(Float64(841.0 / 108.0) * x)
	tmp = 0.0
	if (t_0 <= -50000.0)
		tmp = Float64(7.787037037037037 * x);
	elseif (t_0 <= 5e-14)
		tmp = 0.13793103448275862;
	else
		tmp = Float64(7.787037037037037 * x);
	end
	return tmp
end

function tmp_2 = code(x)
	t_0 = (841.0 / 108.0) * x;
	tmp = 0.0;
	if (t_0 <= -50000.0)
		tmp = 7.787037037037037 * x;
	elseif (t_0 <= 5e-14)
		tmp = 0.13793103448275862;
	else
		tmp = 7.787037037037037 * x;
	end
	tmp_2 = tmp;
end

code[x_] := Block[{t$95$0 = N[(N[(841.0 / 108.0), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[t$95$0, -50000.0], N[(7.787037037037037 * x), $MachinePrecision], If[LessEqual[t$95$0, 5e-14], 0.13793103448275862, N[(7.787037037037037 * x), $MachinePrecision]]]]

\begin{array}{l}

\\
\begin{array}{l}
t_0 := \frac{841}{108} \cdot x\\
\mathbf{if}\;t\_0 \leq -50000:\\
\;\;\;\;7.787037037037037 \cdot x\\

\mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-14}:\\
\;\;\;\;0.13793103448275862\\

\mathbf{else}:\\
\;\;\;\;7.787037037037037 \cdot x\\


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

  2. if (*.f64 (/.f64 #s(literal 841 binary64) #s(literal 108 binary64)) x) < -5e4 or 5.0000000000000002e-14 < (*.f64 (/.f64 #s(literal 841 binary64) #s(literal 108 binary64)) x)

    1. Initial program 99.7%

      \[\frac{841}{108} \cdot x + \frac{4}{29} \]
    2. Add Preprocessing

    3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{841}{108} \cdot x} \]
    4. Step-by-step derivation

      1. *-commutativeN/A

        \[\leadsto \color{blue}{x \cdot \frac{841}{108}} \]

      2. lower-*.f6498.1

        \[\leadsto \color{blue}{x \cdot 7.787037037037037} \]

    5. Applied rewrites98.1%

      \[\leadsto \color{blue}{x \cdot 7.787037037037037} \]

    if -5e4 < (*.f64 (/.f64 #s(literal 841 binary64) #s(literal 108 binary64)) x) < 5.0000000000000002e-14

    1. Initial program 100.0%

      \[\frac{841}{108} \cdot x + \frac{4}{29} \]
    2. Add Preprocessing

    3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{4}{29}} \]
    4. Step-by-step derivation

      1. Applied rewrites99.1%

        \[\leadsto \color{blue}{0.13793103448275862} \]
    5. Recombined 2 regimes into one program.

    6. Final simplification98.5%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{841}{108} \cdot x \leq -50000:\\ \;\;\;\;7.787037037037037 \cdot x\\ \mathbf{elif}\;\frac{841}{108} \cdot x \leq 5 \cdot 10^{-14}:\\ \;\;\;\;0.13793103448275862\\ \mathbf{else}:\\ \;\;\;\;7.787037037037037 \cdot x\\ \end{array} \]
    7. Add Preprocessing

    Alternative 3: 99.9% accurate, 4.4× speedup?

    \[\begin{array}{l} \\ \mathsf{fma}\left(x, 7.787037037037037, 0.13793103448275862\right) \end{array} \]
    (FPCore (x) :precision binary64 (fma x 7.787037037037037 0.13793103448275862))
    double code(double x) {
	return fma(x, 7.787037037037037, 0.13793103448275862);
}

    function code(x)
	return fma(x, 7.787037037037037, 0.13793103448275862)
end

    code[x_] := N[(x * 7.787037037037037 + 0.13793103448275862), $MachinePrecision]

    \begin{array}{l}

\\
\mathsf{fma}\left(x, 7.787037037037037, 0.13793103448275862\right)
\end{array}
    Derivation

    1. Initial program 99.8%

      \[\frac{841}{108} \cdot x + \frac{4}{29} \]
    2. Add Preprocessing
    3. Step-by-step derivation

      1. lift-+.f64N/A

        \[\leadsto \color{blue}{\frac{841}{108} \cdot x + \frac{4}{29}} \]

      2. lift-*.f64N/A

        \[\leadsto \color{blue}{\frac{841}{108} \cdot x} + \frac{4}{29} \]

      3. *-commutativeN/A

        \[\leadsto \color{blue}{x \cdot \frac{841}{108}} + \frac{4}{29} \]

      4. lower-fma.f6499.9

        \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{841}{108}, \frac{4}{29}\right)} \]

      5. lift-/.f64N/A

        \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{841}{108}}, \frac{4}{29}\right) \]

      6. metadata-eval99.9

        \[\leadsto \mathsf{fma}\left(x, \color{blue}{7.787037037037037}, \frac{4}{29}\right) \]

      7. lift-/.f64N/A

        \[\leadsto \mathsf{fma}\left(x, \frac{841}{108}, \color{blue}{\frac{4}{29}}\right) \]

      8. metadata-eval99.9

        \[\leadsto \mathsf{fma}\left(x, 7.787037037037037, \color{blue}{0.13793103448275862}\right) \]

    4. Applied rewrites99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 7.787037037037037, 0.13793103448275862\right)} \]
    5. Add Preprocessing

    Alternative 4: 50.2% accurate, 31.0× speedup?

    \[\begin{array}{l} \\ 0.13793103448275862 \end{array} \]
    (FPCore (x) :precision binary64 0.13793103448275862)
    double code(double x) {
	return 0.13793103448275862;
}

    real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.13793103448275862d0
end function

    public static double code(double x) {
	return 0.13793103448275862;
}

    def code(x):
	return 0.13793103448275862

    function code(x)
	return 0.13793103448275862
end

    function tmp = code(x)
	tmp = 0.13793103448275862;
end

    code[x_] := 0.13793103448275862

    \begin{array}{l}

\\
0.13793103448275862
\end{array}
    Derivation

    1. Initial program 99.8%

      \[\frac{841}{108} \cdot x + \frac{4}{29} \]
    2. Add Preprocessing

    3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{4}{29}} \]
    4. Step-by-step derivation

      1. Applied rewrites44.2%

        \[\leadsto \color{blue}{0.13793103448275862} \]
      2. Add Preprocessing

      Reproduce

      ?
      herbie shell --seed 2024277 
(FPCore (x)
  :name "Data.Colour.CIE:cieLABView from colour-2.3.3, A"
  :precision binary64
  (+ (* (/ 841.0 108.0) x) (/ 4.0 29.0)))