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

Percentage Accurate: 99.7% → 99.8%

Time: 8.1s

Alternatives: 4

Speedup: 2.1×

Specification

Math

\[\begin{array}{l} \\ \left(\left(x - \frac{16}{116}\right) \cdot 3\right) \cdot y \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* (* (- x (/ 16.0 116.0)) 3.0) y))

C

double code(double x, double y) {
	return ((x - (16.0 / 116.0)) * 3.0) * y;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((x - (16.0d0 / 116.0d0)) * 3.0d0) * y
end function

Java

public static double code(double x, double y) {
	return ((x - (16.0 / 116.0)) * 3.0) * y;
}

Python

def code(x, y):
	return ((x - (16.0 / 116.0)) * 3.0) * y

Julia

function code(x, y)
	return Float64(Float64(Float64(x - Float64(16.0 / 116.0)) * 3.0) * y)
end

MATLAB

function tmp = code(x, y)
	tmp = ((x - (16.0 / 116.0)) * 3.0) * y;
end

Wolfram

code[x_, y_] := N[(N[(N[(x - N[(16.0 / 116.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision] * y), $MachinePrecision]

Initial Program: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(x - \frac{16}{116}\right) \cdot 3\right) \cdot y \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* (* (- x (/ 16.0 116.0)) 3.0) y))

C

double code(double x, double y) {
	return ((x - (16.0 / 116.0)) * 3.0) * y;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((x - (16.0d0 / 116.0d0)) * 3.0d0) * y
end function

Java

public static double code(double x, double y) {
	return ((x - (16.0 / 116.0)) * 3.0) * y;
}

Python

def code(x, y):
	return ((x - (16.0 / 116.0)) * 3.0) * y

Julia

function code(x, y)
	return Float64(Float64(Float64(x - Float64(16.0 / 116.0)) * 3.0) * y)
end

MATLAB

function tmp = code(x, y)
	tmp = ((x - (16.0 / 116.0)) * 3.0) * y;
end

Wolfram

code[x_, y_] := N[(N[(N[(x - N[(16.0 / 116.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision] * y), $MachinePrecision]

Alternative 1: 99.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, 3, -0.41379310344827586\right) \cdot y \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* (fma x 3.0 -0.41379310344827586) y))

C

double code(double x, double y) {
	return fma(x, 3.0, -0.41379310344827586) * y;
}

Julia

function code(x, y)
	return Float64(fma(x, 3.0, -0.41379310344827586) * y)
end

Wolfram

code[x_, y_] := N[(N[(x * 3.0 + -0.41379310344827586), $MachinePrecision] * y), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left(\left(x - \frac{16}{116}\right) \cdot 3\right) \cdot y \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \left(\left(x - \color{blue}{\frac{16}{116}}\right) \cdot 3\right) \cdot y \]

   2. lift--.f64 N/A

      \[\leadsto \left(\color{blue}{\left(x - \frac{16}{116}\right)} \cdot 3\right) \cdot y \]

   3. *-commutative N/A

      \[\leadsto \color{blue}{\left(3 \cdot \left(x - \frac{16}{116}\right)\right)} \cdot y \]

   4. lift--.f64 N/A

      \[\leadsto \left(3 \cdot \color{blue}{\left(x - \frac{16}{116}\right)}\right) \cdot y \]

   5. sub-neg N/A

      \[\leadsto \left(3 \cdot \color{blue}{\left(x + \left(\mathsf{neg}\left(\frac{16}{116}\right)\right)\right)}\right) \cdot y \]

   6. distribute-lft-in N/A

      \[\leadsto \color{blue}{\left(3 \cdot x + 3 \cdot \left(\mathsf{neg}\left(\frac{16}{116}\right)\right)\right)} \cdot y \]

   7. *-commutative N/A

      \[\leadsto \left(\color{blue}{x \cdot 3} + 3 \cdot \left(\mathsf{neg}\left(\frac{16}{116}\right)\right)\right) \cdot y \]

   8. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 3, 3 \cdot \left(\mathsf{neg}\left(\frac{16}{116}\right)\right)\right)} \cdot y \]

   9. lift-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(x, 3, 3 \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{16}{116}}\right)\right)\right) \cdot y \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(x, 3, 3 \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{4}{29}}\right)\right)\right) \cdot y \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(x, 3, 3 \cdot \color{blue}{\frac{-4}{29}}\right) \cdot y \]

   12. metadata-eval 99.8

       \[\leadsto \mathsf{fma}\left(x, 3, \color{blue}{-0.41379310344827586}\right) \cdot y \]

4. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, 3, -0.41379310344827586\right)} \cdot y \]
5. Add Preprocessing

Alternative 2: 97.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \frac{16}{116}\\ t_1 := y \cdot \left(x \cdot 3\right)\\ \mathbf{if}\;t\_0 \leq -2:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq -0.1:\\ \;\;\;\;-0.41379310344827586 \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- x (/ 16.0 116.0))) (t_1 (* y (* x 3.0))))
   (if (<= t_0 -2.0) t_1 (if (<= t_0 -0.1) (* -0.41379310344827586 y) t_1))))

C

double code(double x, double y) {
	double t_0 = x - (16.0 / 116.0);
	double t_1 = y * (x * 3.0);
	double tmp;
	if (t_0 <= -2.0) {
		tmp = t_1;
	} else if (t_0 <= -0.1) {
		tmp = -0.41379310344827586 * y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x - (16.0d0 / 116.0d0)
    t_1 = y * (x * 3.0d0)
    if (t_0 <= (-2.0d0)) then
        tmp = t_1
    else if (t_0 <= (-0.1d0)) then
        tmp = (-0.41379310344827586d0) * y
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x - (16.0 / 116.0);
	double t_1 = y * (x * 3.0);
	double tmp;
	if (t_0 <= -2.0) {
		tmp = t_1;
	} else if (t_0 <= -0.1) {
		tmp = -0.41379310344827586 * y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x - (16.0 / 116.0)
	t_1 = y * (x * 3.0)
	tmp = 0
	if t_0 <= -2.0:
		tmp = t_1
	elif t_0 <= -0.1:
		tmp = -0.41379310344827586 * y
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x - Float64(16.0 / 116.0))
	t_1 = Float64(y * Float64(x * 3.0))
	tmp = 0.0
	if (t_0 <= -2.0)
		tmp = t_1;
	elseif (t_0 <= -0.1)
		tmp = Float64(-0.41379310344827586 * y);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x - (16.0 / 116.0);
	t_1 = y * (x * 3.0);
	tmp = 0.0;
	if (t_0 <= -2.0)
		tmp = t_1;
	elseif (t_0 <= -0.1)
		tmp = -0.41379310344827586 * y;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x - N[(16.0 / 116.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y * N[(x * 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2.0], t$95$1, If[LessEqual[t$95$0, -0.1], N[(-0.41379310344827586 * y), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (/.f64 #s(literal 16 binary64) #s(literal 116 binary64))) < -2 or -0.10000000000000001 < (-.f64 x (/.f64 #s(literal 16 binary64) #s(literal 116 binary64)))

   1. Initial program 99.8%

      \[\left(\left(x - \frac{16}{116}\right) \cdot 3\right) \cdot y \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\left(3 \cdot x\right)} \cdot y \]
   4. Step-by-step derivation

      1. lower-*.f64 96.6

         \[\leadsto \color{blue}{\left(3 \cdot x\right)} \cdot y \]

   5. Applied rewrites 96.6%

      \[\leadsto \color{blue}{\left(3 \cdot x\right)} \cdot y \]

   if -2 < (-.f64 x (/.f64 #s(literal 16 binary64) #s(literal 116 binary64))) < -0.10000000000000001

   1. Initial program 99.9%

      \[\left(\left(x - \frac{16}{116}\right) \cdot 3\right) \cdot y \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-12}{29}} \cdot y \]
   4. Step-by-step derivation

   5. Applied rewrites 98.4%

      \[\leadsto \color{blue}{-0.41379310344827586} \cdot y \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \frac{16}{116} \leq -2:\\ \;\;\;\;y \cdot \left(x \cdot 3\right)\\ \mathbf{elif}\;x - \frac{16}{116} \leq -0.1:\\ \;\;\;\;-0.41379310344827586 \cdot y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot 3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \frac{16}{116}\\ t_1 := 3 \cdot \left(x \cdot y\right)\\ \mathbf{if}\;t\_0 \leq -2:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq -0.1:\\ \;\;\;\;-0.41379310344827586 \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- x (/ 16.0 116.0))) (t_1 (* 3.0 (* x y))))
   (if (<= t_0 -2.0) t_1 (if (<= t_0 -0.1) (* -0.41379310344827586 y) t_1))))

C

double code(double x, double y) {
	double t_0 = x - (16.0 / 116.0);
	double t_1 = 3.0 * (x * y);
	double tmp;
	if (t_0 <= -2.0) {
		tmp = t_1;
	} else if (t_0 <= -0.1) {
		tmp = -0.41379310344827586 * y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x - (16.0d0 / 116.0d0)
    t_1 = 3.0d0 * (x * y)
    if (t_0 <= (-2.0d0)) then
        tmp = t_1
    else if (t_0 <= (-0.1d0)) then
        tmp = (-0.41379310344827586d0) * y
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x - (16.0 / 116.0);
	double t_1 = 3.0 * (x * y);
	double tmp;
	if (t_0 <= -2.0) {
		tmp = t_1;
	} else if (t_0 <= -0.1) {
		tmp = -0.41379310344827586 * y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x - (16.0 / 116.0)
	t_1 = 3.0 * (x * y)
	tmp = 0
	if t_0 <= -2.0:
		tmp = t_1
	elif t_0 <= -0.1:
		tmp = -0.41379310344827586 * y
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x - Float64(16.0 / 116.0))
	t_1 = Float64(3.0 * Float64(x * y))
	tmp = 0.0
	if (t_0 <= -2.0)
		tmp = t_1;
	elseif (t_0 <= -0.1)
		tmp = Float64(-0.41379310344827586 * y);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x - (16.0 / 116.0);
	t_1 = 3.0 * (x * y);
	tmp = 0.0;
	if (t_0 <= -2.0)
		tmp = t_1;
	elseif (t_0 <= -0.1)
		tmp = -0.41379310344827586 * y;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x - N[(16.0 / 116.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 * N[(x * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2.0], t$95$1, If[LessEqual[t$95$0, -0.1], N[(-0.41379310344827586 * y), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (/.f64 #s(literal 16 binary64) #s(literal 116 binary64))) < -2 or -0.10000000000000001 < (-.f64 x (/.f64 #s(literal 16 binary64) #s(literal 116 binary64)))

   1. Initial program 99.8%

      \[\left(\left(x - \frac{16}{116}\right) \cdot 3\right) \cdot y \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{3 \cdot \left(x \cdot y\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{3 \cdot \left(x \cdot y\right)} \]

      2. lower-*.f64 96.5

         \[\leadsto 3 \cdot \color{blue}{\left(x \cdot y\right)} \]

   5. Applied rewrites 96.5%

      \[\leadsto \color{blue}{3 \cdot \left(x \cdot y\right)} \]

   if -2 < (-.f64 x (/.f64 #s(literal 16 binary64) #s(literal 116 binary64))) < -0.10000000000000001

   1. Initial program 99.9%

      \[\left(\left(x - \frac{16}{116}\right) \cdot 3\right) \cdot y \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-12}{29}} \cdot y \]
   4. Step-by-step derivation

   5. Applied rewrites 98.4%

      \[\leadsto \color{blue}{-0.41379310344827586} \cdot y \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 50.7% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ -0.41379310344827586 \cdot y \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* -0.41379310344827586 y))

C

double code(double x, double y) {
	return -0.41379310344827586 * y;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (-0.41379310344827586d0) * y
end function

Java

public static double code(double x, double y) {
	return -0.41379310344827586 * y;
}

Python

def code(x, y):
	return -0.41379310344827586 * y

Julia

function code(x, y)
	return Float64(-0.41379310344827586 * y)
end

MATLAB

function tmp = code(x, y)
	tmp = -0.41379310344827586 * y;
end

Wolfram

code[x_, y_] := N[(-0.41379310344827586 * y), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left(\left(x - \frac{16}{116}\right) \cdot 3\right) \cdot y \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{-12}{29}} \cdot y \]
4. Step-by-step derivation

5. Applied rewrites 47.1%

   \[\leadsto \color{blue}{-0.41379310344827586} \cdot y \]
6. Add Preprocessing

Developer Target 1: 99.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ y \cdot \left(x \cdot 3 - 0.41379310344827586\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* y (- (* x 3.0) 0.41379310344827586)))

C

double code(double x, double y) {
	return y * ((x * 3.0) - 0.41379310344827586);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = y * ((x * 3.0d0) - 0.41379310344827586d0)
end function

Java

public static double code(double x, double y) {
	return y * ((x * 3.0) - 0.41379310344827586);
}

Python

def code(x, y):
	return y * ((x * 3.0) - 0.41379310344827586)

Julia

function code(x, y)
	return Float64(y * Float64(Float64(x * 3.0) - 0.41379310344827586))
end

MATLAB

function tmp = code(x, y)
	tmp = y * ((x * 3.0) - 0.41379310344827586);
end

Wolfram

code[x_, y_] := N[(y * N[(N[(x * 3.0), $MachinePrecision] - 0.41379310344827586), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024216 
(FPCore (x y)
  :name "Data.Colour.CIE:cieLAB from colour-2.3.3, A"
  :precision binary64

  :alt
  (! :herbie-platform default (* y (- (* x 3) 20689655172413793/50000000000000000)))

  (* (* (- x (/ 16.0 116.0)) 3.0) y))