Data.Colour.SRGB:transferFunction from colour-2.3.3

Percentage Accurate: 100.0% → 100.0%

Time: 6.6s

Alternatives: 6

Speedup: 1.2×

Specification

Math

\[\begin{array}{l} \\ \left(x + 1\right) \cdot y - x \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- (* (+ x 1.0) y) x))

C

double code(double x, double y) {
	return ((x + 1.0) * y) - x;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((x + 1.0d0) * y) - x
end function

Java

public static double code(double x, double y) {
	return ((x + 1.0) * y) - x;
}

Python

def code(x, y):
	return ((x + 1.0) * y) - x

Julia

function code(x, y)
	return Float64(Float64(Float64(x + 1.0) * y) - x)
end

MATLAB

function tmp = code(x, y)
	tmp = ((x + 1.0) * y) - x;
end

Wolfram

code[x_, y_] := N[(N[(N[(x + 1.0), $MachinePrecision] * y), $MachinePrecision] - x), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(x + 1\right) \cdot y - x \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- (* (+ x 1.0) y) x))

C

double code(double x, double y) {
	return ((x + 1.0) * y) - x;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((x + 1.0d0) * y) - x
end function

Java

public static double code(double x, double y) {
	return ((x + 1.0) * y) - x;
}

Python

def code(x, y):
	return ((x + 1.0) * y) - x

Julia

function code(x, y)
	return Float64(Float64(Float64(x + 1.0) * y) - x)
end

MATLAB

function tmp = code(x, y)
	tmp = ((x + 1.0) * y) - x;
end

Wolfram

code[x_, y_] := N[(N[(N[(x + 1.0), $MachinePrecision] * y), $MachinePrecision] - x), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, y + -1, y\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (fma x (+ y -1.0) y))

C

double code(double x, double y) {
	return fma(x, (y + -1.0), y);
}

Julia

function code(x, y)
	return fma(x, Float64(y + -1.0), y)
end

Wolfram

code[x_, y_] := N[(x * N[(y + -1.0), $MachinePrecision] + y), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(x + 1\right) \cdot y - x \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \color{blue}{\left(x + 1\right) \cdot y - x} \]

   2. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(x + 1\right) \cdot y} - x \]

   3. *-commutative N/A

      \[\leadsto \color{blue}{y \cdot \left(x + 1\right)} - x \]

   4. lift-+.f64 N/A

      \[\leadsto y \cdot \color{blue}{\left(x + 1\right)} - x \]

   5. +-commutative N/A

      \[\leadsto y \cdot \color{blue}{\left(1 + x\right)} - x \]

   6. distribute-rgt-in N/A

      \[\leadsto \color{blue}{\left(1 \cdot y + x \cdot y\right)} - x \]

   7. *-lft-identity N/A

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

   8. *-commutative N/A

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

   9. +-commutative N/A

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

   10. associate--l+ N/A

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

   11. lower-fma.f64 N/A

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

   12. lower--.f64 100.0

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

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, y - x\right)} \]
5. Step-by-step derivation

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift--.f64 N/A

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

   4. sub-neg N/A

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

   5. lift-neg.f64 N/A

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

   6. +-commutative N/A

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

   7. associate-+r+ N/A

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

   8. lift-neg.f64 N/A

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

   9. sub-neg N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. *-rgt-identity N/A

       \[\leadsto \left(x \cdot y - \color{blue}{x \cdot 1}\right) + y \]

   13. distribute-lft-out-- N/A

       \[\leadsto \color{blue}{x \cdot \left(y - 1\right)} + y \]

   14. lower-fma.f64 N/A

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

   15. lower--.f64 100.0

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

6. Applied rewrites 100.0%

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

7. Final simplification 100.0%

   \[\leadsto \mathsf{fma}\left(x, y + -1, y\right) \]
8. Add Preprocessing

Alternative 2: 98.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{+24}:\\ \;\;\;\;\mathsf{fma}\left(y, x, y\right)\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(x, -1, y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -6e+24) (fma y x y) (if (<= y 2.2e-5) (fma x -1.0 y) (fma y x y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -6e+24) {
		tmp = fma(y, x, y);
	} else if (y <= 2.2e-5) {
		tmp = fma(x, -1.0, y);
	} else {
		tmp = fma(y, x, y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -6e+24)
		tmp = fma(y, x, y);
	elseif (y <= 2.2e-5)
		tmp = fma(x, -1.0, y);
	else
		tmp = fma(y, x, y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -6e+24], N[(y * x + y), $MachinePrecision], If[LessEqual[y, 2.2e-5], N[(x * -1.0 + y), $MachinePrecision], N[(y * x + y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -5.9999999999999999e24 or 2.1999999999999999e-5 < y

   1. Initial program 100.0%

      \[\left(x + 1\right) \cdot y - x \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(1 + x\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto y \cdot \color{blue}{\left(x + 1\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{y \cdot x + y \cdot 1} \]

      3. *-rgt-identity N/A

         \[\leadsto y \cdot x + \color{blue}{y} \]

      4. lower-fma.f64 99.8

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

   5. Applied rewrites 99.8%

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

   if -5.9999999999999999e24 < y < 2.1999999999999999e-5

   1. Initial program 100.0%

      \[\left(x + 1\right) \cdot y - x \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(x + 1\right) \cdot y - x} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(x + 1\right) \cdot y} - x \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{y \cdot \left(x + 1\right)} - x \]

      4. lift-+.f64 N/A

         \[\leadsto y \cdot \color{blue}{\left(x + 1\right)} - x \]

      5. +-commutative N/A

         \[\leadsto y \cdot \color{blue}{\left(1 + x\right)} - x \]

      6. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(1 \cdot y + x \cdot y\right)} - x \]

      7. *-lft-identity N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. associate--l+ N/A

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

      11. lower-fma.f64 N/A

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

      12. lower--.f64 100.0

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

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, y - x\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. sub-neg N/A

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

      5. lift-neg.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-+r+ N/A

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

      8. lift-neg.f64 N/A

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

      9. sub-neg N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. *-rgt-identity N/A

          \[\leadsto \left(x \cdot y - \color{blue}{x \cdot 1}\right) + y \]

      13. distribute-lft-out-- N/A

          \[\leadsto \color{blue}{x \cdot \left(y - 1\right)} + y \]

      14. lower-fma.f64 N/A

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

      15. lower--.f64 100.0

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{-1}, y\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 98.2%

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{-1}, y\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 87.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(y, x, y\right)\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{-28}:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -4.5e-13) (fma y x y) (if (<= y 1.7e-28) (- x) (fma y x y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -4.5e-13) {
		tmp = fma(y, x, y);
	} else if (y <= 1.7e-28) {
		tmp = -x;
	} else {
		tmp = fma(y, x, y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -4.5e-13)
		tmp = fma(y, x, y);
	elseif (y <= 1.7e-28)
		tmp = Float64(-x);
	else
		tmp = fma(y, x, y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -4.5e-13], N[(y * x + y), $MachinePrecision], If[LessEqual[y, 1.7e-28], (-x), N[(y * x + y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.5e-13 or 1.7e-28 < y

   1. Initial program 100.0%

      \[\left(x + 1\right) \cdot y - x \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(1 + x\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto y \cdot \color{blue}{\left(x + 1\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{y \cdot x + y \cdot 1} \]

      3. *-rgt-identity N/A

         \[\leadsto y \cdot x + \color{blue}{y} \]

      4. lower-fma.f64 97.3

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

   5. Applied rewrites 97.3%

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

   if -4.5e-13 < y < 1.7e-28

   1. Initial program 100.0%

      \[\left(x + 1\right) \cdot y - x \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{-1 \cdot x} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(x\right)} \]

      2. lower-neg.f64 77.7

         \[\leadsto \color{blue}{-x} \]

   5. Applied rewrites 77.7%

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

Alternative 4: 61.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 1150000000:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0) (* x y) (if (<= y 1150000000.0) (- x) (* x y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x * y;
	} else if (y <= 1150000000.0) {
		tmp = -x;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.0d0)) then
        tmp = x * y
    else if (y <= 1150000000.0d0) then
        tmp = -x
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x * y;
	} else if (y <= 1150000000.0) {
		tmp = -x;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = x * y
	elif y <= 1150000000.0:
		tmp = -x
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = Float64(x * y);
	elseif (y <= 1150000000.0)
		tmp = Float64(-x);
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.0)
		tmp = x * y;
	elseif (y <= 1150000000.0)
		tmp = -x;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], N[(x * y), $MachinePrecision], If[LessEqual[y, 1150000000.0], (-x), N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1.15e9 < y

   1. Initial program 100.0%

      \[\left(x + 1\right) \cdot y - x \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(1 + x\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto y \cdot \color{blue}{\left(x + 1\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{y \cdot x + y \cdot 1} \]

      3. *-rgt-identity N/A

         \[\leadsto y \cdot x + \color{blue}{y} \]

      4. lower-fma.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 53.5%

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

   if -1 < y < 1.15e9

   1. Initial program 100.0%

      \[\left(x + 1\right) \cdot y - x \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{-1 \cdot x} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(x\right)} \]

      2. lower-neg.f64 72.4

         \[\leadsto \color{blue}{-x} \]

   5. Applied rewrites 72.4%

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

Alternative 5: 100.0% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 (- (fma y x y) x))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\left(x + 1\right) \cdot y - x \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(x + 1\right) \cdot y} - x \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{y \cdot \left(x + 1\right)} - x \]

   3. lift-+.f64 N/A

      \[\leadsto y \cdot \color{blue}{\left(x + 1\right)} - x \]

   4. distribute-lft-in N/A

      \[\leadsto \color{blue}{\left(y \cdot x + y \cdot 1\right)} - x \]

   5. *-rgt-identity N/A

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

   6. lower-fma.f64 100.0

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

4. Applied rewrites 100.0%

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

Alternative 6: 38.4% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ -x \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return -x

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := (-x)

Derivation

1. Initial program 100.0%

   \[\left(x + 1\right) \cdot y - x \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{-1 \cdot x} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(x\right)} \]

   2. lower-neg.f64 35.6

      \[\leadsto \color{blue}{-x} \]

5. Applied rewrites 35.6%

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

Reproduce

herbie shell --seed 2024220 
(FPCore (x y)
  :name "Data.Colour.SRGB:transferFunction from colour-2.3.3"
  :precision binary64
  (- (* (+ x 1.0) y) x))