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

Percentage Accurate: 99.9% → 99.9%

Time: 4.8s

Alternatives: 3

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: 99.9% 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 99.9%

   \[\frac{841}{108} \cdot x + \frac{4}{29} \]
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 2: 97.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{841}{108} \cdot x\\ \mathbf{if}\;t\_0 \leq -0.2 \lor \neg \left(t\_0 \leq 2 \cdot 10^{-7}\right):\\ \;\;\;\;7.787037037037037 \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.13793103448275862\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (/ 841.0 108.0) x)))
   (if (or (<= t_0 -0.2) (not (<= t_0 2e-7)))
     (* 7.787037037037037 x)
     0.13793103448275862)))

C

double code(double x) {
	double t_0 = (841.0 / 108.0) * x;
	double tmp;
	if ((t_0 <= -0.2) || !(t_0 <= 2e-7)) {
		tmp = 7.787037037037037 * x;
	} else {
		tmp = 0.13793103448275862;
	}
	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 <= (-0.2d0)) .or. (.not. (t_0 <= 2d-7))) then
        tmp = 7.787037037037037d0 * x
    else
        tmp = 0.13793103448275862d0
    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 <= -0.2) || !(t_0 <= 2e-7)) {
		tmp = 7.787037037037037 * x;
	} else {
		tmp = 0.13793103448275862;
	}
	return tmp;
}

Python

def code(x):
	t_0 = (841.0 / 108.0) * x
	tmp = 0
	if (t_0 <= -0.2) or not (t_0 <= 2e-7):
		tmp = 7.787037037037037 * x
	else:
		tmp = 0.13793103448275862
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(841.0 / 108.0) * x)
	tmp = 0.0
	if ((t_0 <= -0.2) || !(t_0 <= 2e-7))
		tmp = Float64(7.787037037037037 * x);
	else
		tmp = 0.13793103448275862;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (841.0 / 108.0) * x;
	tmp = 0.0;
	if ((t_0 <= -0.2) || ~((t_0 <= 2e-7)))
		tmp = 7.787037037037037 * x;
	else
		tmp = 0.13793103448275862;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(841.0 / 108.0), $MachinePrecision] * x), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -0.2], N[Not[LessEqual[t$95$0, 2e-7]], $MachinePrecision]], N[(7.787037037037037 * x), $MachinePrecision], 0.13793103448275862]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 #s(literal 841 binary64) #s(literal 108 binary64)) x) < -0.20000000000000001 or 1.9999999999999999e-7 < (*.f64 (/.f64 #s(literal 841 binary64) #s(literal 108 binary64)) x)

   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. unpow1 N/A

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

      3. metadata-eval N/A

         \[\leadsto {\left(\frac{841}{108} \cdot x\right)}^{\color{blue}{\left(\frac{2}{2}\right)}} + \frac{4}{29} \]

      4. sqrt-pow1 N/A

         \[\leadsto \color{blue}{\sqrt{{\left(\frac{841}{108} \cdot x\right)}^{2}}} + \frac{4}{29} \]

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. sqrt-prod N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. associate-*r* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lift-/.f64 N/A

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

      18. metadata-eval N/A

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

      19. lower-sqrt.f64 23.3

          \[\leadsto \mathsf{fma}\left(\sqrt{\left(x \cdot x\right) \cdot 7.787037037037037}, \color{blue}{\sqrt{\frac{841}{108}}}, \frac{4}{29}\right) \]

      20. lift-/.f64 N/A

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

      21. metadata-eval 23.3

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

      22. lift-/.f64 N/A

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

      23. metadata-eval 23.3

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

   4. Applied rewrites 23.3%

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

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot {\left(\sqrt{\frac{841}{108}}\right)}^{2}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{{\left(\sqrt{\frac{841}{108}}\right)}^{2} \cdot x} \]

      2. unpow2 N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{841}{108}} \cdot \sqrt{\frac{841}{108}}\right)} \cdot x \]

      3. rem-square-sqrt N/A

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

      4. lower-*.f64 98.4

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

   7. Applied rewrites 98.4%

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

   if -0.20000000000000001 < (*.f64 (/.f64 #s(literal 841 binary64) #s(literal 108 binary64)) x) < 1.9999999999999999e-7

   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 98.0%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{841}{108} \cdot x \leq -0.2 \lor \neg \left(\frac{841}{108} \cdot x \leq 2 \cdot 10^{-7}\right):\\ \;\;\;\;7.787037037037037 \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.13793103448275862\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 49.6% 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 99.9%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 49.0%

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

Reproduce

herbie shell --seed 2024320 
(FPCore (x)
  :name "Data.Colour.CIE:cieLABView from colour-2.3.3, A"
  :precision binary64
  (+ (* (/ 841.0 108.0) x) (/ 4.0 29.0)))