Data.Colour.SRGB:transferFunction from colour-2.3.3

Percentage Accurate: 100.0% → 100.0%

Time: 5.0s

Alternatives: 7

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: 98.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.6 \cdot 10^{+28}:\\ \;\;\;\;\mathsf{fma}\left(x, y, y\right)\\ \mathbf{elif}\;y \leq 0.042:\\ \;\;\;\;\mathsf{fma}\left(x, -1, y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, y, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.6e+28) (fma x y y) (if (<= y 0.042) (fma x -1.0 y) (fma x y y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.6e+28) {
		tmp = fma(x, y, y);
	} else if (y <= 0.042) {
		tmp = fma(x, -1.0, y);
	} else {
		tmp = fma(x, y, y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.6e+28)
		tmp = fma(x, y, y);
	elseif (y <= 0.042)
		tmp = fma(x, -1.0, y);
	else
		tmp = fma(x, y, y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.6e+28], N[(x * y + y), $MachinePrecision], If[LessEqual[y, 0.042], N[(x * -1.0 + y), $MachinePrecision], N[(x * y + y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.6000000000000002e28 or 0.0420000000000000026 < y

   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. Taylor expanded in y around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-in N/A

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

      6. +-commutative N/A

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

      7. remove-double-neg N/A

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

      8. +-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. *-lft-identity N/A

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

      11. lower-fma.f64 100.0

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

   7. Applied rewrites 100.0%

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

   if -2.6000000000000002e28 < y < 0.0420000000000000026

   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

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. lift-*.f64 N/A

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

      9. lift-neg.f64 N/A

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

      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. Taylor expanded in y around 0

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

   9. Applied rewrites 99.3%

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

Alternative 3: 86.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.65 \cdot 10^{-68}:\\ \;\;\;\;\mathsf{fma}\left(x, y, y\right)\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{-14}:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, y, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.65e-68) (fma x y y) (if (<= y 3.5e-14) (- x) (fma x y y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.65e-68) {
		tmp = fma(x, y, y);
	} else if (y <= 3.5e-14) {
		tmp = -x;
	} else {
		tmp = fma(x, y, y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.65e-68)
		tmp = fma(x, y, y);
	elseif (y <= 3.5e-14)
		tmp = Float64(-x);
	else
		tmp = fma(x, y, y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.65e-68], N[(x * y + y), $MachinePrecision], If[LessEqual[y, 3.5e-14], (-x), N[(x * y + y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.6499999999999999e-68 or 3.5000000000000002e-14 < y

   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. Taylor expanded in y around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-in N/A

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

      6. +-commutative N/A

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

      7. remove-double-neg N/A

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

      8. +-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. *-lft-identity N/A

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

      11. lower-fma.f64 96.0

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

   7. Applied rewrites 96.0%

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

   if -1.6499999999999999e-68 < y < 3.5000000000000002e-14

   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 82.5

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

   5. Applied rewrites 82.5%

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

Alternative 4: 61.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.185:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 0.042:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -0.185) (* x y) (if (<= y 0.042) (- x) (* x y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -0.185) {
		tmp = x * y;
	} else if (y <= 0.042) {
		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 <= (-0.185d0)) then
        tmp = x * y
    else if (y <= 0.042d0) 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 <= -0.185) {
		tmp = x * y;
	} else if (y <= 0.042) {
		tmp = -x;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -0.185:
		tmp = x * y
	elif y <= 0.042:
		tmp = -x
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -0.185)
		tmp = Float64(x * y);
	elseif (y <= 0.042)
		tmp = Float64(-x);
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -0.185)
		tmp = x * y;
	elseif (y <= 0.042)
		tmp = -x;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -0.185], N[(x * y), $MachinePrecision], If[LessEqual[y, 0.042], (-x), N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.185 or 0.0420000000000000026 < y

   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. Taylor expanded in y around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-in N/A

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

      6. +-commutative N/A

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

      7. remove-double-neg N/A

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

      8. +-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. *-lft-identity N/A

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

      11. lower-fma.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 49.0%

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

   if -0.185 < y < 0.0420000000000000026

   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 79.0

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

   5. Applied rewrites 79.0%

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

Alternative 5: 100.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   6. +-commutative N/A

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

   7. associate-+l+ N/A

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

   8. lift-*.f64 N/A

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

   9. lift-neg.f64 N/A

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

   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. Final simplification 100.0%

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

Alternative 6: 37.6% 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 41.0

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

5. Applied rewrites 41.0%

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

Alternative 7: 2.7% accurate, 12.0× 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 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 41.0

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

5. Applied rewrites 41.0%

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

7. Applied rewrites 2.7%

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

Reproduce

herbie shell --seed 2024294 
(FPCore (x y)
  :name "Data.Colour.SRGB:transferFunction from colour-2.3.3"
  :precision binary64
  (- (* (+ x 1.0) y) x))