Data.Colour.SRGB:transferFunction from colour-2.3.3

Percentage Accurate: 100.0% → 100.0%

Time: 6.0s

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: 86.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.53 \lor \neg \left(x \leq 3.6 \cdot 10^{-68}\right):\\ \;\;\;\;\left(y - 1\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -0.53) (not (<= x 3.6e-68))) (* (- y 1.0) x) (fma y x y)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -0.53) || !(x <= 3.6e-68)) {
		tmp = (y - 1.0) * x;
	} else {
		tmp = fma(y, x, y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -0.53) || !(x <= 3.6e-68))
		tmp = Float64(Float64(y - 1.0) * x);
	else
		tmp = fma(y, x, y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -0.53], N[Not[LessEqual[x, 3.6e-68]], $MachinePrecision]], N[(N[(y - 1.0), $MachinePrecision] * x), $MachinePrecision], N[(y * x + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.53000000000000003 or 3.60000000000000007e-68 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 96.8

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

   5. Applied rewrites 96.8%

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

   if -0.53000000000000003 < x < 3.60000000000000007e-68

   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. associate-*r* N/A

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

      2. distribute-rgt-out-- N/A

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

      3. mul-1-neg N/A

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

      4. distribute-lft-neg-in N/A

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

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

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. *-commutative N/A

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

      9. mul-1-neg N/A

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

      10. fp-cancel-sign-sub-inv N/A

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

      11. *-commutative N/A

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

      12. *-rgt-identity N/A

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

      13. lower-fma.f64 81.5

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

   7. Applied rewrites 81.5%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.53 \lor \neg \left(x \leq 3.6 \cdot 10^{-68}\right):\\ \;\;\;\;\left(y - 1\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -54:\\ \;\;\;\;\left(y - 1\right) \cdot x\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;1 \cdot y - x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, -x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -54.0)
   (* (- y 1.0) x)
   (if (<= x 1.0) (- (* 1.0 y) x) (fma y x (- x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -54.0) {
		tmp = (y - 1.0) * x;
	} else if (x <= 1.0) {
		tmp = (1.0 * y) - x;
	} else {
		tmp = fma(y, x, -x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -54.0)
		tmp = Float64(Float64(y - 1.0) * x);
	elseif (x <= 1.0)
		tmp = Float64(Float64(1.0 * y) - x);
	else
		tmp = fma(y, x, Float64(-x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -54.0], N[(N[(y - 1.0), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[x, 1.0], N[(N[(1.0 * y), $MachinePrecision] - x), $MachinePrecision], N[(y * x + (-x)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -54

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 99.7

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

   5. Applied rewrites 99.7%

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

   if -54 < x < 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 98.4%

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

   if 1 < x

   1. Initial program 99.9%

      \[\left(x + 1\right) \cdot y - x \]
   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 49.7

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

   5. Applied rewrites 49.7%

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

   6. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

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

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

      3. +-commutative N/A

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

      4. distribute-neg-in N/A

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

      5. mul-1-neg N/A

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

      6. remove-double-neg N/A

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

      7. metadata-eval N/A

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

      8. distribute-rgt-out N/A

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

      9. lower-fma.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 98.9

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

   8. Applied rewrites 98.9%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -54:\\ \;\;\;\;\left(y - 1\right) \cdot x\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;1 \cdot y - x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, -x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.53:\\ \;\;\;\;\left(y - 1\right) \cdot x\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-68}:\\ \;\;\;\;\mathsf{fma}\left(y, x, y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, -x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.53)
   (* (- y 1.0) x)
   (if (<= x 3.6e-68) (fma y x y) (fma y x (- x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.53) {
		tmp = (y - 1.0) * x;
	} else if (x <= 3.6e-68) {
		tmp = fma(y, x, y);
	} else {
		tmp = fma(y, x, -x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.53)
		tmp = Float64(Float64(y - 1.0) * x);
	elseif (x <= 3.6e-68)
		tmp = fma(y, x, y);
	else
		tmp = fma(y, x, Float64(-x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.53], N[(N[(y - 1.0), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[x, 3.6e-68], N[(y * x + y), $MachinePrecision], N[(y * x + (-x)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.53000000000000003

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 99.7

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

   5. Applied rewrites 99.7%

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

   if -0.53000000000000003 < x < 3.60000000000000007e-68

   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. associate-*r* N/A

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

      2. distribute-rgt-out-- N/A

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

      3. mul-1-neg N/A

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

      4. distribute-lft-neg-in N/A

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

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

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. *-commutative N/A

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

      9. mul-1-neg N/A

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

      10. fp-cancel-sign-sub-inv N/A

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

      11. *-commutative N/A

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

      12. *-rgt-identity N/A

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

      13. lower-fma.f64 81.5

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

   7. Applied rewrites 81.5%

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

   if 3.60000000000000007e-68 < x

   1. Initial program 99.9%

      \[\left(x + 1\right) \cdot y - x \]
   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 51.0

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

   5. Applied rewrites 51.0%

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

   6. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

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

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

      3. +-commutative N/A

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

      4. distribute-neg-in N/A

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

      5. mul-1-neg N/A

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

      6. remove-double-neg N/A

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

      7. metadata-eval N/A

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

      8. distribute-rgt-out N/A

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

      9. lower-fma.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 94.7

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

   8. Applied rewrites 94.7%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.53:\\ \;\;\;\;\left(y - 1\right) \cdot x\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-68}:\\ \;\;\;\;\mathsf{fma}\left(y, x, y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, -x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.48 \cdot 10^{-114} \lor \neg \left(y \leq 2.8 \cdot 10^{-17}\right):\\ \;\;\;\;\mathsf{fma}\left(y, x, y\right)\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.48e-114) (not (<= y 2.8e-17))) (fma y x y) (- x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.48e-114) || !(y <= 2.8e-17)) {
		tmp = fma(y, x, y);
	} else {
		tmp = -x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.48e-114) || !(y <= 2.8e-17))
		tmp = fma(y, x, y);
	else
		tmp = Float64(-x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.48e-114], N[Not[LessEqual[y, 2.8e-17]], $MachinePrecision]], N[(y * x + y), $MachinePrecision], (-x)]

Derivation

1. Split input into 2 regimes

2. if y < -1.48e-114 or 2.7999999999999999e-17 < 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. associate-*r* N/A

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

      2. distribute-rgt-out-- N/A

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

      3. mul-1-neg N/A

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

      4. distribute-lft-neg-in N/A

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

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

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. *-commutative N/A

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

      9. mul-1-neg N/A

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

      10. fp-cancel-sign-sub-inv N/A

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

      11. *-commutative N/A

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

      12. *-rgt-identity N/A

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

      13. lower-fma.f64 93.8

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

   7. Applied rewrites 93.8%

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

   if -1.48e-114 < y < 2.7999999999999999e-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 83.3

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

   5. Applied rewrites 83.3%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.48 \cdot 10^{-114} \lor \neg \left(y \leq 2.8 \cdot 10^{-17}\right):\\ \;\;\;\;\mathsf{fma}\left(y, x, y\right)\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 61.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{+18} \lor \neg \left(y \leq 1\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -2.1e+18) (not (<= y 1.0))) (* x y) (- x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -2.1e+18) || !(y <= 1.0)) {
		tmp = x * y;
	} else {
		tmp = -x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-2.1d+18)) .or. (.not. (y <= 1.0d0))) then
        tmp = x * y
    else
        tmp = -x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -2.1e+18) || !(y <= 1.0)) {
		tmp = x * y;
	} else {
		tmp = -x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -2.1e+18) or not (y <= 1.0):
		tmp = x * y
	else:
		tmp = -x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -2.1e+18) || !(y <= 1.0))
		tmp = Float64(x * y);
	else
		tmp = Float64(-x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -2.1e+18) || ~((y <= 1.0)))
		tmp = x * y;
	else
		tmp = -x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -2.1e+18], N[Not[LessEqual[y, 1.0]], $MachinePrecision]], N[(x * y), $MachinePrecision], (-x)]

Derivation

1. Split input into 2 regimes

2. if y < -2.1e18 or 1 < 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. associate-*r* N/A

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

      2. distribute-rgt-out-- N/A

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

      3. mul-1-neg N/A

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

      4. distribute-lft-neg-in N/A

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

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

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. *-commutative N/A

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

      9. mul-1-neg N/A

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

      10. fp-cancel-sign-sub-inv N/A

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

      11. *-commutative N/A

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

      12. *-rgt-identity N/A

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

      13. lower-fma.f64 98.3

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

   7. Applied rewrites 98.3%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 53.0%

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

   if -2.1e18 < 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 73.2

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

   5. Applied rewrites 73.2%

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

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{+18} \lor \neg \left(y \leq 1\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 38.7% 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 37.4

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

5. Applied rewrites 37.4%

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

6. Final simplification 37.4%

   \[\leadsto -x \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024320 
(FPCore (x y)
  :name "Data.Colour.SRGB:transferFunction from colour-2.3.3"
  :precision binary64
  (- (* (+ x 1.0) y) x))