Data.Colour.SRGB:transferFunction from colour-2.3.3

Percentage Accurate: 100.0% → 100.0%

Time: 4.2s

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

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

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 fma(y, x, Float64(y - x))
end

Wolfram

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

Alternative 2: 98.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2600000000:\\ \;\;\;\;\mathsf{fma}\left(x, y, y\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 \cdot y - x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, y, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2600000000.0)
   (fma x y y)
   (if (<= y 1.0) (- (* 1.0 y) x) (fma x y y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2600000000.0) {
		tmp = fma(x, y, y);
	} else if (y <= 1.0) {
		tmp = (1.0 * y) - x;
	} else {
		tmp = fma(x, y, y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2600000000.0)
		tmp = fma(x, y, y);
	elseif (y <= 1.0)
		tmp = Float64(Float64(1.0 * y) - x);
	else
		tmp = fma(x, y, y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2600000000.0], N[(x * y + y), $MachinePrecision], If[LessEqual[y, 1.0], N[(N[(1.0 * y), $MachinePrecision] - x), $MachinePrecision], N[(x * y + y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.6e9 or 1 < 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

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. lower-fma.f64 99.0

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

   5. Applied rewrites 99.0%

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

   if -2.6e9 < y < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 98.2%

      \[\leadsto \color{blue}{1} \cdot y - x \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 85.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot y - x\\ \mathbf{if}\;x \leq -3.05 \cdot 10^{-28}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 6 \cdot 10^{-36}:\\ \;\;\;\;\mathsf{fma}\left(x, y, y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (* x y) x)))
   (if (<= x -3.05e-28) t_0 (if (<= x 6e-36) (fma x y y) t_0))))

C

double code(double x, double y) {
	double t_0 = (x * y) - x;
	double tmp;
	if (x <= -3.05e-28) {
		tmp = t_0;
	} else if (x <= 6e-36) {
		tmp = fma(x, y, y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(x * y) - x)
	tmp = 0.0
	if (x <= -3.05e-28)
		tmp = t_0;
	elseif (x <= 6e-36)
		tmp = fma(x, y, y);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * y), $MachinePrecision] - x), $MachinePrecision]}, If[LessEqual[x, -3.05e-28], t$95$0, If[LessEqual[x, 6e-36], N[(x * y + y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.05e-28 or 6.0000000000000003e-36 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 95.8

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

   5. Applied rewrites 95.8%

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

   if -3.05e-28 < x < 6.0000000000000003e-36

   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

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. lower-fma.f64 80.6

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

   5. Applied rewrites 80.6%

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

Alternative 4: 85.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.25 \cdot 10^{-103}:\\ \;\;\;\;\mathsf{fma}\left(x, y, y\right)\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{-23}:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, y, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.25e-103) (fma x y y) (if (<= y 1.8e-23) (- x) (fma x y y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.25e-103) {
		tmp = fma(x, y, y);
	} else if (y <= 1.8e-23) {
		tmp = -x;
	} else {
		tmp = fma(x, y, y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.25e-103)
		tmp = fma(x, y, y);
	elseif (y <= 1.8e-23)
		tmp = Float64(-x);
	else
		tmp = fma(x, y, y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.25e-103], N[(x * y + y), $MachinePrecision], If[LessEqual[y, 1.8e-23], (-x), N[(x * y + y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.24999999999999992e-103 or 1.7999999999999999e-23 < 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

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. lower-fma.f64 92.4

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

   5. Applied rewrites 92.4%

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

   if -1.24999999999999992e-103 < y < 1.7999999999999999e-23

   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 78.9

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

   5. Applied rewrites 78.9%

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

Alternative 5: 61.7% accurate, 0.7× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x * y;
	} else if (y <= 1.0) {
		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 <= 1.0d0) 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 <= 1.0) {
		tmp = -x;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = x * y
	elif y <= 1.0:
		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 <= 1.0)
		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 <= 1.0)
		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, 1.0], (-x), N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < 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

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. lower-fma.f64 99.0

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

   5. Applied rewrites 99.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 56.1%

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

   if -1 < y < 1

   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 71.3

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

   5. Applied rewrites 71.3%

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

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 38.8% 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 36.9

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

5. Applied rewrites 36.9%

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

Reproduce

herbie shell --seed 2024242 
(FPCore (x y)
  :name "Data.Colour.SRGB:transferFunction from colour-2.3.3"
  :precision binary64
  (- (* (+ x 1.0) y) x))