Data.Colour.SRGB:transferFunction from colour-2.3.3

Percentage Accurate: 100.0% → 100.0%

Time: 6.6s

Alternatives: 10

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. *-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. associate--l+ N/A

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

   7. lower-fma.f64 N/A

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

   8. 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-*.f64 N/A

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

   2. associate-+r- N/A

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

   3. +-commutative N/A

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

   4. associate--l+ N/A

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

   5. lift--.f64 N/A

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

   6. +-commutative N/A

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

   7. lift--.f64 N/A

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

   8. sub-neg N/A

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

   9. lift-*.f64 N/A

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

   10. neg-mul-1 N/A

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

   11. distribute-rgt-out N/A

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

   12. lower-fma.f64 N/A

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

   13. 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. Add Preprocessing

Alternative 2: 86.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(x + 1\right) - x\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+290}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+307}:\\ \;\;\;\;y - x\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (* y (+ x 1.0)) x)))
   (if (<= t_0 -2e+290) (* x y) (if (<= t_0 4e+307) (- y x) (* x y)))))

C

double code(double x, double y) {
	double t_0 = (y * (x + 1.0)) - x;
	double tmp;
	if (t_0 <= -2e+290) {
		tmp = x * y;
	} else if (t_0 <= 4e+307) {
		tmp = y - 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) :: t_0
    real(8) :: tmp
    t_0 = (y * (x + 1.0d0)) - x
    if (t_0 <= (-2d+290)) then
        tmp = x * y
    else if (t_0 <= 4d+307) then
        tmp = y - x
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (y * (x + 1.0)) - x;
	double tmp;
	if (t_0 <= -2e+290) {
		tmp = x * y;
	} else if (t_0 <= 4e+307) {
		tmp = y - x;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y * (x + 1.0)) - x
	tmp = 0
	if t_0 <= -2e+290:
		tmp = x * y
	elif t_0 <= 4e+307:
		tmp = y - x
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y * Float64(x + 1.0)) - x)
	tmp = 0.0
	if (t_0 <= -2e+290)
		tmp = Float64(x * y);
	elseif (t_0 <= 4e+307)
		tmp = Float64(y - x);
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y * (x + 1.0)) - x;
	tmp = 0.0;
	if (t_0 <= -2e+290)
		tmp = x * y;
	elseif (t_0 <= 4e+307)
		tmp = y - x;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * N[(x + 1.0), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+290], N[(x * y), $MachinePrecision], If[LessEqual[t$95$0, 4e+307], N[(y - x), $MachinePrecision], N[(x * y), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (+.f64 x #s(literal 1 binary64)) y) x) < -2.00000000000000012e290 or 3.99999999999999994e307 < (-.f64 (*.f64 (+.f64 x #s(literal 1 binary64)) y) x)

   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 93.2

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

   5. Applied rewrites 93.2%

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

   6. Taylor expanded in x around inf

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

      1. lower-*.f64 89.6

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

   8. Applied rewrites 89.6%

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

   if -2.00000000000000012e290 < (-.f64 (*.f64 (+.f64 x #s(literal 1 binary64)) y) x) < 3.99999999999999994e307

   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 90.0%

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

      1. *-lft-identity 90.0

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

   7. Applied rewrites 90.0%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x + 1\right) - x \leq -2 \cdot 10^{+290}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \cdot \left(x + 1\right) - x \leq 4 \cdot 10^{+307}:\\ \;\;\;\;y - x\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\mathsf{fma}\left(y, x, -x\right)\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-11}:\\ \;\;\;\;y - x\\ \mathbf{else}:\\ \;\;\;\;x \cdot y - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.0) (fma y x (- x)) (if (<= x 1.6e-11) (- y x) (- (* x y) x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = fma(y, x, -x);
	} else if (x <= 1.6e-11) {
		tmp = y - x;
	} else {
		tmp = (x * y) - x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.0)
		tmp = fma(y, x, Float64(-x));
	elseif (x <= 1.6e-11)
		tmp = Float64(y - x);
	else
		tmp = Float64(Float64(x * y) - x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.0], N[(y * x + (-x)), $MachinePrecision], If[LessEqual[x, 1.6e-11], N[(y - x), $MachinePrecision], N[(N[(x * y), $MachinePrecision] - x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1

   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. *-commutative N/A

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

      2. lower-*.f64 98.8

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

   5. Applied rewrites 98.8%

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

      1. lift-*.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lower-fma.f64 98.8

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

   7. Applied rewrites 98.8%

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

   if -1 < x < 1.59999999999999997e-11

   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 99.8%

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

      1. *-lft-identity 99.8

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

   7. Applied rewrites 99.8%

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

   if 1.59999999999999997e-11 < 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. *-commutative N/A

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

      2. lower-*.f64 99.5

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

   5. Applied rewrites 99.5%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\mathsf{fma}\left(y, x, -x\right)\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-11}:\\ \;\;\;\;y - x\\ \mathbf{else}:\\ \;\;\;\;x \cdot y - x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;x \cdot \left(y + -1\right)\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-11}:\\ \;\;\;\;y - x\\ \mathbf{else}:\\ \;\;\;\;x \cdot y - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.0) (* x (+ y -1.0)) (if (<= x 1.6e-11) (- y x) (- (* x y) x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = x * (y + -1.0);
	} else if (x <= 1.6e-11) {
		tmp = y - x;
	} else {
		tmp = (x * y) - x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = x * (y + (-1.0d0))
    else if (x <= 1.6d-11) then
        tmp = y - x
    else
        tmp = (x * y) - x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = x * (y + -1.0);
	} else if (x <= 1.6e-11) {
		tmp = y - x;
	} else {
		tmp = (x * y) - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.0:
		tmp = x * (y + -1.0)
	elif x <= 1.6e-11:
		tmp = y - x
	else:
		tmp = (x * y) - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(x * Float64(y + -1.0));
	elseif (x <= 1.6e-11)
		tmp = Float64(y - x);
	else
		tmp = Float64(Float64(x * y) - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = x * (y + -1.0);
	elseif (x <= 1.6e-11)
		tmp = y - x;
	else
		tmp = (x * y) - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.0], N[(x * N[(y + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.6e-11], N[(y - x), $MachinePrecision], N[(N[(x * y), $MachinePrecision] - x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1

   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. *-commutative N/A

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

      2. lower-*.f64 98.8

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

   5. Applied rewrites 98.8%

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

      1. lift-*.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. neg-mul-1 N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 98.8

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

   7. Applied rewrites 98.8%

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

   if -1 < x < 1.59999999999999997e-11

   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 99.8%

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

      1. *-lft-identity 99.8

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

   7. Applied rewrites 99.8%

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

   if 1.59999999999999997e-11 < 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. *-commutative N/A

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

      2. lower-*.f64 99.5

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

   5. Applied rewrites 99.5%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;x \cdot \left(y + -1\right)\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-11}:\\ \;\;\;\;y - x\\ \mathbf{else}:\\ \;\;\;\;x \cdot y - x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(y + -1\right)\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-11}:\\ \;\;\;\;y - x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (+ y -1.0))))
   (if (<= x -1.0) t_0 (if (<= x 1.6e-11) (- y x) t_0))))

C

double code(double x, double y) {
	double t_0 = x * (y + -1.0);
	double tmp;
	if (x <= -1.0) {
		tmp = t_0;
	} else if (x <= 1.6e-11) {
		tmp = y - x;
	} else {
		tmp = t_0;
	}
	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) :: tmp
    t_0 = x * (y + (-1.0d0))
    if (x <= (-1.0d0)) then
        tmp = t_0
    else if (x <= 1.6d-11) then
        tmp = y - x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x * (y + -1.0);
	double tmp;
	if (x <= -1.0) {
		tmp = t_0;
	} else if (x <= 1.6e-11) {
		tmp = y - x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x * (y + -1.0)
	tmp = 0
	if x <= -1.0:
		tmp = t_0
	elif x <= 1.6e-11:
		tmp = y - x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x * Float64(y + -1.0))
	tmp = 0.0
	if (x <= -1.0)
		tmp = t_0;
	elseif (x <= 1.6e-11)
		tmp = Float64(y - x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x * (y + -1.0);
	tmp = 0.0;
	if (x <= -1.0)
		tmp = t_0;
	elseif (x <= 1.6e-11)
		tmp = y - x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x * N[(y + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.0], t$95$0, If[LessEqual[x, 1.6e-11], N[(y - x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1.59999999999999997e-11 < 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. *-commutative N/A

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

      2. lower-*.f64 99.1

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

   5. Applied rewrites 99.1%

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

      1. lift-*.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. neg-mul-1 N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 99.1

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

   7. Applied rewrites 99.1%

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

   if -1 < x < 1.59999999999999997e-11

   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 99.8%

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

      1. *-lft-identity 99.8

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

   7. Applied rewrites 99.8%

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

Alternative 6: 99.0% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

      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 99.6

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

   5. Applied rewrites 99.6%

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

   if -1 < 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 96.8%

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

      1. *-lft-identity 96.8

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

   7. Applied rewrites 96.8%

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

Alternative 7: 61.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{-74}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{-17}:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -4.8e-74) (+ x y) (if (<= y 2.6e-17) (- x) (+ x y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -4.8e-74) {
		tmp = x + y;
	} else if (y <= 2.6e-17) {
		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 <= (-4.8d-74)) then
        tmp = x + y
    else if (y <= 2.6d-17) 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 <= -4.8e-74) {
		tmp = x + y;
	} else if (y <= 2.6e-17) {
		tmp = -x;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -4.8e-74:
		tmp = x + y
	elif y <= 2.6e-17:
		tmp = -x
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -4.8e-74)
		tmp = Float64(x + y);
	elseif (y <= 2.6e-17)
		tmp = Float64(-x);
	else
		tmp = Float64(x + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -4.8e-74)
		tmp = x + y;
	elseif (y <= 2.6e-17)
		tmp = -x;
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -4.8e-74], N[(x + y), $MachinePrecision], If[LessEqual[y, 2.6e-17], (-x), N[(x + y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.7999999999999998e-74 or 2.60000000000000003e-17 < y

   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 61.4%

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

      1. *-lft-identity 61.4

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

   7. Applied rewrites 61.4%

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

   9. Applied rewrites 55.4%

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

   if -4.7999999999999998e-74 < y < 2.60000000000000003e-17

   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 87.4

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

   5. Applied rewrites 87.4%

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

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{-74}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{-17}:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 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 9: 74.5% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 77.5%

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

   1. *-lft-identity 77.5

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

7. Applied rewrites 77.5%

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

Alternative 10: 38.5% 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 41.6

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

5. Applied rewrites 41.6%

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

Reproduce

herbie shell --seed 2024214 
(FPCore (x y)
  :name "Data.Colour.SRGB:transferFunction from colour-2.3.3"
  :precision binary64
  (- (* (+ x 1.0) y) x))