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

Percentage Accurate: 99.9% → 99.7%

Time: 3.2s

Alternatives: 4

Speedup: 4.4×

Specification

Math

\[\begin{array}{l} \\ \frac{841}{108} \cdot x + \frac{4}{29} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ (* (/ 841.0 108.0) x) (/ 4.0 29.0)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{841}{108} \cdot x + \frac{4}{29} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ (* (/ 841.0 108.0) x) (/ 4.0 29.0)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (fma
  x
  (* (/ x (* -7.787037037037037 x)) -60.63794581618656)
  (/ -0.019024970273483946 -0.13793103448275862)))

C

double code(double x) {
	return fma(x, ((x / (-7.787037037037037 * x)) * -60.63794581618656), (-0.019024970273483946 / -0.13793103448275862));
}

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 99.9%

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

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(x, \frac{-707281}{11664} \cdot \frac{x}{\mathsf{fma}\left(\frac{-841}{108}, x, \frac{4}{29}\right)}, \frac{\frac{-16}{841}}{\color{blue}{\frac{-4}{29}}}\right) \]
6. Step-by-step derivation

7. Applied rewrites 97.4%

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

8. Taylor expanded in x around inf

   \[\leadsto \mathsf{fma}\left(x, \frac{-707281}{11664} \cdot \frac{x}{\color{blue}{\frac{-841}{108} \cdot x}}, \frac{\frac{-16}{841}}{\frac{-4}{29}}\right) \]
9. Step-by-step derivation

   1. lower-*.f64 99.9

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

10. Applied rewrites 99.9%

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

11. Final simplification 99.9%

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

Alternative 2: 97.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{841}{108} \cdot x\\ \mathbf{if}\;t\_0 \leq -10:\\ \;\;\;\;7.787037037037037 \cdot x\\ \mathbf{elif}\;t\_0 \leq 0.04:\\ \;\;\;\;0.13793103448275862\\ \mathbf{else}:\\ \;\;\;\;7.787037037037037 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (/ 841.0 108.0) x)))
   (if (<= t_0 -10.0)
     (* 7.787037037037037 x)
     (if (<= t_0 0.04) 0.13793103448275862 (* 7.787037037037037 x)))))

C

double code(double x) {
	double t_0 = (841.0 / 108.0) * x;
	double tmp;
	if (t_0 <= -10.0) {
		tmp = 7.787037037037037 * x;
	} else if (t_0 <= 0.04) {
		tmp = 0.13793103448275862;
	} else {
		tmp = 7.787037037037037 * x;
	}
	return tmp;
}

Fortran

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 <= (-10.0d0)) then
        tmp = 7.787037037037037d0 * x
    else if (t_0 <= 0.04d0) then
        tmp = 0.13793103448275862d0
    else
        tmp = 7.787037037037037d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = (841.0 / 108.0) * x;
	double tmp;
	if (t_0 <= -10.0) {
		tmp = 7.787037037037037 * x;
	} else if (t_0 <= 0.04) {
		tmp = 0.13793103448275862;
	} else {
		tmp = 7.787037037037037 * x;
	}
	return tmp;
}

Python

def code(x):
	t_0 = (841.0 / 108.0) * x
	tmp = 0
	if t_0 <= -10.0:
		tmp = 7.787037037037037 * x
	elif t_0 <= 0.04:
		tmp = 0.13793103448275862
	else:
		tmp = 7.787037037037037 * x
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(841.0 / 108.0) * x)
	tmp = 0.0
	if (t_0 <= -10.0)
		tmp = Float64(7.787037037037037 * x);
	elseif (t_0 <= 0.04)
		tmp = 0.13793103448275862;
	else
		tmp = Float64(7.787037037037037 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (841.0 / 108.0) * x;
	tmp = 0.0;
	if (t_0 <= -10.0)
		tmp = 7.787037037037037 * x;
	elseif (t_0 <= 0.04)
		tmp = 0.13793103448275862;
	else
		tmp = 7.787037037037037 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(841.0 / 108.0), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[t$95$0, -10.0], N[(7.787037037037037 * x), $MachinePrecision], If[LessEqual[t$95$0, 0.04], 0.13793103448275862, N[(7.787037037037037 * x), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 #s(literal 841 binary64) #s(literal 108 binary64)) x) < -10 or 0.0400000000000000008 < (*.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. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. flip3-+ N/A

         \[\leadsto \color{blue}{\frac{{\left(\frac{841}{108} \cdot x\right)}^{3} + {\left(\frac{4}{29}\right)}^{3}}{\left(\frac{841}{108} \cdot x\right) \cdot \left(\frac{841}{108} \cdot x\right) + \left(\frac{4}{29} \cdot \frac{4}{29} - \left(\frac{841}{108} \cdot x\right) \cdot \frac{4}{29}\right)}} \]

      3. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(\frac{841}{108} \cdot x\right) \cdot \left(\frac{841}{108} \cdot x\right) + \left(\frac{4}{29} \cdot \frac{4}{29} - \left(\frac{841}{108} \cdot x\right) \cdot \frac{4}{29}\right)}{{\left(\frac{841}{108} \cdot x\right)}^{3} + {\left(\frac{4}{29}\right)}^{3}}}} \]

      4. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\frac{\left(\frac{841}{108} \cdot x\right) \cdot \left(\frac{841}{108} \cdot x\right) + \left(\frac{4}{29} \cdot \frac{4}{29} - \left(\frac{841}{108} \cdot x\right) \cdot \frac{4}{29}\right)}{{\left(\frac{841}{108} \cdot x\right)}^{3} + {\left(\frac{4}{29}\right)}^{3}}\right)}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{-1}}{\mathsf{neg}\left(\frac{\left(\frac{841}{108} \cdot x\right) \cdot \left(\frac{841}{108} \cdot x\right) + \left(\frac{4}{29} \cdot \frac{4}{29} - \left(\frac{841}{108} \cdot x\right) \cdot \frac{4}{29}\right)}{{\left(\frac{841}{108} \cdot x\right)}^{3} + {\left(\frac{4}{29}\right)}^{3}}\right)} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-1}{\mathsf{neg}\left(\frac{\left(\frac{841}{108} \cdot x\right) \cdot \left(\frac{841}{108} \cdot x\right) + \left(\frac{4}{29} \cdot \frac{4}{29} - \left(\frac{841}{108} \cdot x\right) \cdot \frac{4}{29}\right)}{{\left(\frac{841}{108} \cdot x\right)}^{3} + {\left(\frac{4}{29}\right)}^{3}}\right)}} \]

      7. clear-num N/A

         \[\leadsto \frac{-1}{\mathsf{neg}\left(\color{blue}{\frac{1}{\frac{{\left(\frac{841}{108} \cdot x\right)}^{3} + {\left(\frac{4}{29}\right)}^{3}}{\left(\frac{841}{108} \cdot x\right) \cdot \left(\frac{841}{108} \cdot x\right) + \left(\frac{4}{29} \cdot \frac{4}{29} - \left(\frac{841}{108} \cdot x\right) \cdot \frac{4}{29}\right)}}}\right)} \]

   4. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\frac{-1}{\frac{-1}{\mathsf{fma}\left(x, 7.787037037037037, 0.13793103448275862\right)}}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{-1}{\color{blue}{\frac{\frac{-108}{841}}{x}}} \]
   6. Step-by-step derivation

      1. lower-/.f64 97.6

         \[\leadsto \frac{-1}{\color{blue}{\frac{-0.12841854934601665}{x}}} \]

   7. Applied rewrites 97.6%

      \[\leadsto \frac{-1}{\color{blue}{\frac{-0.12841854934601665}{x}}} \]

   8. Taylor expanded in x around inf

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

      1. lower-*.f64 98.0

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

   10. Applied rewrites 98.0%

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

   if -10 < (*.f64 (/.f64 #s(literal 841 binary64) #s(literal 108 binary64)) x) < 0.0400000000000000008

   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

   5. Applied rewrites 96.7%

      \[\leadsto \color{blue}{0.13793103448275862} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 99.9% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, 7.787037037037037, 0.13793103448275862\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (fma x 7.787037037037037 0.13793103448275862))

C

double code(double x) {
	return fma(x, 7.787037037037037, 0.13793103448275862);
}

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 99.9

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

   5. lift-/.f64 N/A

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

   6. metadata-eval 99.9

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

   7. lift-/.f64 N/A

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

   8. metadata-eval 99.9

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

4. Applied rewrites 99.9%

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

Alternative 4: 49.9% accurate, 31.0× speedup

Math

\[\begin{array}{l} \\ 0.13793103448275862 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.13793103448275862)

C

double code(double x) {
	return 0.13793103448275862;
}

Fortran

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

Java

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

Python

def code(x):
	return 0.13793103448275862

Julia

function code(x)
	return 0.13793103448275862
end

MATLAB

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

Wolfram

code[x_] := 0.13793103448275862

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

5. Applied rewrites 49.8%

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

Reproduce

herbie shell --seed 2024308 
(FPCore (x)
  :name "Data.Colour.CIE:cieLABView from colour-2.3.3, A"
  :precision binary64
  (+ (* (/ 841.0 108.0) x) (/ 4.0 29.0)))