Data.Colour.RGBSpace.HSL:hsl from colour-2.3.3, E

Percentage Accurate: 99.7% → 98.7%

Time: 7.5s

Alternatives: 12

Speedup: 1.1×

• 20.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ x + \left(\left(y - x\right) \cdot 6\right) \cdot z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ x (* (* (- y x) 6.0) z)))

C

double code(double x, double y, double z) {
	return x + (((y - x) * 6.0) * z);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x + (((y - x) * 6.0d0) * z)
end function

Java

public static double code(double x, double y, double z) {
	return x + (((y - x) * 6.0) * z);
}

Python

def code(x, y, z):
	return x + (((y - x) * 6.0) * z)

Julia

function code(x, y, z)
	return Float64(x + Float64(Float64(Float64(y - x) * 6.0) * z))
end

MATLAB

function tmp = code(x, y, z)
	tmp = x + (((y - x) * 6.0) * z);
end

Wolfram

code[x_, y_, z_] := N[(x + N[(N[(N[(y - x), $MachinePrecision] * 6.0), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x + \left(\left(y - x\right) \cdot 6\right) \cdot z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ x (* (* (- y x) 6.0) z)))

C

double code(double x, double y, double z) {
	return x + (((y - x) * 6.0) * z);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x + (((y - x) * 6.0d0) * z)
end function

Java

public static double code(double x, double y, double z) {
	return x + (((y - x) * 6.0) * z);
}

Python

def code(x, y, z):
	return x + (((y - x) * 6.0) * z)

Julia

function code(x, y, z)
	return Float64(x + Float64(Float64(Float64(y - x) * 6.0) * z))
end

MATLAB

function tmp = code(x, y, z)
	tmp = x + (((y - x) * 6.0) * z);
end

Wolfram

code[x_, y_, z_] := N[(x + N[(N[(N[(y - x), $MachinePrecision] * 6.0), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]

Alternative 1: 98.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ x + \mathsf{fma}\left(6 \cdot y, z, \left(-6 \cdot x\right) \cdot z\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ x (fma (* 6.0 y) z (* (* -6.0 x) z))))

C

double code(double x, double y, double z) {
	return x + fma((6.0 * y), z, ((-6.0 * x) * z));
}

Julia

function code(x, y, z)
	return Float64(x + fma(Float64(6.0 * y), z, Float64(Float64(-6.0 * x) * z)))
end

Wolfram

code[x_, y_, z_] := N[(x + N[(N[(6.0 * y), $MachinePrecision] * z + N[(N[(-6.0 * x), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto x + \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot z} \]

   2. *-commutative N/A

      \[\leadsto x + \color{blue}{z \cdot \left(\left(y - x\right) \cdot 6\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto x + z \cdot \color{blue}{\left(\left(y - x\right) \cdot 6\right)} \]

   4. *-commutative N/A

      \[\leadsto x + z \cdot \color{blue}{\left(6 \cdot \left(y - x\right)\right)} \]

   5. lift--.f64 N/A

      \[\leadsto x + z \cdot \left(6 \cdot \color{blue}{\left(y - x\right)}\right) \]

   6. sub-neg N/A

      \[\leadsto x + z \cdot \left(6 \cdot \color{blue}{\left(y + \left(\mathsf{neg}\left(x\right)\right)\right)}\right) \]

   7. distribute-rgt-in N/A

      \[\leadsto x + z \cdot \color{blue}{\left(y \cdot 6 + \left(\mathsf{neg}\left(x\right)\right) \cdot 6\right)} \]

   8. distribute-rgt-in N/A

      \[\leadsto x + \color{blue}{\left(\left(y \cdot 6\right) \cdot z + \left(\left(\mathsf{neg}\left(x\right)\right) \cdot 6\right) \cdot z\right)} \]

   9. lower-fma.f64 N/A

      \[\leadsto x + \color{blue}{\mathsf{fma}\left(y \cdot 6, z, \left(\left(\mathsf{neg}\left(x\right)\right) \cdot 6\right) \cdot z\right)} \]

   10. *-commutative N/A

       \[\leadsto x + \mathsf{fma}\left(\color{blue}{6 \cdot y}, z, \left(\left(\mathsf{neg}\left(x\right)\right) \cdot 6\right) \cdot z\right) \]

   11. lower-*.f64 N/A

       \[\leadsto x + \mathsf{fma}\left(\color{blue}{6 \cdot y}, z, \left(\left(\mathsf{neg}\left(x\right)\right) \cdot 6\right) \cdot z\right) \]

   12. lower-*.f64 N/A

       \[\leadsto x + \mathsf{fma}\left(6 \cdot y, z, \color{blue}{\left(\left(\mathsf{neg}\left(x\right)\right) \cdot 6\right) \cdot z}\right) \]

   13. *-commutative N/A

       \[\leadsto x + \mathsf{fma}\left(6 \cdot y, z, \color{blue}{\left(6 \cdot \left(\mathsf{neg}\left(x\right)\right)\right)} \cdot z\right) \]

   14. neg-mul-1 N/A

       \[\leadsto x + \mathsf{fma}\left(6 \cdot y, z, \left(6 \cdot \color{blue}{\left(-1 \cdot x\right)}\right) \cdot z\right) \]

   15. associate-*r* N/A

       \[\leadsto x + \mathsf{fma}\left(6 \cdot y, z, \color{blue}{\left(\left(6 \cdot -1\right) \cdot x\right)} \cdot z\right) \]

   16. metadata-eval N/A

       \[\leadsto x + \mathsf{fma}\left(6 \cdot y, z, \left(\color{blue}{-6} \cdot x\right) \cdot z\right) \]

   17. metadata-eval N/A

       \[\leadsto x + \mathsf{fma}\left(6 \cdot y, z, \left(\color{blue}{\left(\mathsf{neg}\left(6\right)\right)} \cdot x\right) \cdot z\right) \]

   18. lower-*.f64 N/A

       \[\leadsto x + \mathsf{fma}\left(6 \cdot y, z, \color{blue}{\left(\left(\mathsf{neg}\left(6\right)\right) \cdot x\right)} \cdot z\right) \]

   19. metadata-eval 99.8

       \[\leadsto x + \mathsf{fma}\left(6 \cdot y, z, \left(\color{blue}{-6} \cdot x\right) \cdot z\right) \]

4. Applied rewrites 99.8%

   \[\leadsto x + \color{blue}{\mathsf{fma}\left(6 \cdot y, z, \left(-6 \cdot x\right) \cdot z\right)} \]
5. Add Preprocessing

Alternative 2: 98.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.7 \cdot 10^{-11} \lor \neg \left(z \leq 0.165\right):\\ \;\;\;\;\left(\left(y - x\right) \cdot z\right) \cdot 6\\ \mathbf{else}:\\ \;\;\;\;x + \left(6 \cdot y\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.7e-11) (not (<= z 0.165)))
   (* (* (- y x) z) 6.0)
   (+ x (* (* 6.0 y) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.7e-11) || !(z <= 0.165)) {
		tmp = ((y - x) * z) * 6.0;
	} else {
		tmp = x + ((6.0 * y) * z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-1.7d-11)) .or. (.not. (z <= 0.165d0))) then
        tmp = ((y - x) * z) * 6.0d0
    else
        tmp = x + ((6.0d0 * y) * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.7e-11) || !(z <= 0.165)) {
		tmp = ((y - x) * z) * 6.0;
	} else {
		tmp = x + ((6.0 * y) * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.7e-11) or not (z <= 0.165):
		tmp = ((y - x) * z) * 6.0
	else:
		tmp = x + ((6.0 * y) * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.7e-11) || !(z <= 0.165))
		tmp = Float64(Float64(Float64(y - x) * z) * 6.0);
	else
		tmp = Float64(x + Float64(Float64(6.0 * y) * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.7e-11) || ~((z <= 0.165)))
		tmp = ((y - x) * z) * 6.0;
	else
		tmp = x + ((6.0 * y) * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.7e-11], N[Not[LessEqual[z, 0.165]], $MachinePrecision]], N[(N[(N[(y - x), $MachinePrecision] * z), $MachinePrecision] * 6.0), $MachinePrecision], N[(x + N[(N[(6.0 * y), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.6999999999999999e-11 or 0.165000000000000008 < z

   1. Initial program 99.7%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(\left(y - x\right) \cdot 6\right) \cdot z} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot z + x} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot z} + x \]

      4. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right)} \cdot z + x \]

      5. associate-*l* N/A

         \[\leadsto \color{blue}{\left(y - x\right) \cdot \left(6 \cdot z\right)} + x \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 6 \cdot z, x\right)} \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y - x, \color{blue}{z \cdot 6}, x\right) \]

      8. lower-*.f64 99.8

         \[\leadsto \mathsf{fma}\left(y - x, \color{blue}{z \cdot 6}, x\right) \]

   4. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, z \cdot 6, x\right)} \]

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(z \cdot \left(y - x\right)\right) \cdot 6} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(z \cdot \left(y - x\right)\right) \cdot 6} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot z\right)} \cdot 6 \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot z\right)} \cdot 6 \]

      5. lower--.f64 98.2

         \[\leadsto \left(\color{blue}{\left(y - x\right)} \cdot z\right) \cdot 6 \]

   7. Applied rewrites 98.2%

      \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot z\right) \cdot 6} \]

   if -1.6999999999999999e-11 < z < 0.165000000000000008

   1. Initial program 99.9%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 99.3

         \[\leadsto x + \color{blue}{\left(6 \cdot y\right)} \cdot z \]

   5. Applied rewrites 99.3%

      \[\leadsto x + \color{blue}{\left(6 \cdot y\right)} \cdot z \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.7 \cdot 10^{-11} \lor \neg \left(z \leq 0.165\right):\\ \;\;\;\;\left(\left(y - x\right) \cdot z\right) \cdot 6\\ \mathbf{else}:\\ \;\;\;\;x + \left(6 \cdot y\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -0.17 \lor \neg \left(z \leq 0.165\right):\\ \;\;\;\;\left(\left(y - x\right) \cdot z\right) \cdot 6\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot y, 6, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -0.17) (not (<= z 0.165)))
   (* (* (- y x) z) 6.0)
   (fma (* z y) 6.0 x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -0.17) || !(z <= 0.165)) {
		tmp = ((y - x) * z) * 6.0;
	} else {
		tmp = fma((z * y), 6.0, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -0.17) || !(z <= 0.165))
		tmp = Float64(Float64(Float64(y - x) * z) * 6.0);
	else
		tmp = fma(Float64(z * y), 6.0, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -0.17], N[Not[LessEqual[z, 0.165]], $MachinePrecision]], N[(N[(N[(y - x), $MachinePrecision] * z), $MachinePrecision] * 6.0), $MachinePrecision], N[(N[(z * y), $MachinePrecision] * 6.0 + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -0.170000000000000012 or 0.165000000000000008 < z

   1. Initial program 99.7%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(\left(y - x\right) \cdot 6\right) \cdot z} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot z + x} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot z} + x \]

      4. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right)} \cdot z + x \]

      5. associate-*l* N/A

         \[\leadsto \color{blue}{\left(y - x\right) \cdot \left(6 \cdot z\right)} + x \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 6 \cdot z, x\right)} \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y - x, \color{blue}{z \cdot 6}, x\right) \]

      8. lower-*.f64 99.8

         \[\leadsto \mathsf{fma}\left(y - x, \color{blue}{z \cdot 6}, x\right) \]

   4. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, z \cdot 6, x\right)} \]

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(z \cdot \left(y - x\right)\right) \cdot 6} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(z \cdot \left(y - x\right)\right) \cdot 6} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot z\right)} \cdot 6 \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot z\right)} \cdot 6 \]

      5. lower--.f64 98.2

         \[\leadsto \left(\color{blue}{\left(y - x\right)} \cdot z\right) \cdot 6 \]

   7. Applied rewrites 98.2%

      \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot z\right) \cdot 6} \]

   if -0.170000000000000012 < z < 0.165000000000000008

   1. Initial program 99.9%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(\left(y - x\right) \cdot 6\right) \cdot z} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot z + x} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot z} + x \]

      4. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right)} \cdot z + x \]

      5. associate-*l* N/A

         \[\leadsto \color{blue}{\left(y - x\right) \cdot \left(6 \cdot z\right)} + x \]

      6. *-commutative N/A

         \[\leadsto \left(y - x\right) \cdot \color{blue}{\left(z \cdot 6\right)} + x \]

      7. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot z\right) \cdot 6} + x \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(y - x\right) \cdot z, 6, x\right)} \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot \left(y - x\right)}, 6, x\right) \]

      10. lower-*.f64 99.9

          \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot \left(y - x\right)}, 6, x\right) \]

   4. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z \cdot \left(y - x\right), 6, x\right)} \]

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot y}, 6, x\right) \]

      2. lower-*.f64 99.2

         \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot y}, 6, x\right) \]

   7. Applied rewrites 99.2%

      \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot y}, 6, x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.17 \lor \neg \left(z \leq 0.165\right):\\ \;\;\;\;\left(\left(y - x\right) \cdot z\right) \cdot 6\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot y, 6, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.3 \cdot 10^{-25} \lor \neg \left(y \leq 3.1 \cdot 10^{-118}\right):\\ \;\;\;\;\mathsf{fma}\left(z \cdot y, 6, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot -6, x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -3.3e-25) (not (<= y 3.1e-118)))
   (fma (* z y) 6.0 x)
   (fma (* z -6.0) x x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -3.3e-25) || !(y <= 3.1e-118)) {
		tmp = fma((z * y), 6.0, x);
	} else {
		tmp = fma((z * -6.0), x, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -3.3e-25) || !(y <= 3.1e-118))
		tmp = fma(Float64(z * y), 6.0, x);
	else
		tmp = fma(Float64(z * -6.0), x, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -3.3e-25], N[Not[LessEqual[y, 3.1e-118]], $MachinePrecision]], N[(N[(z * y), $MachinePrecision] * 6.0 + x), $MachinePrecision], N[(N[(z * -6.0), $MachinePrecision] * x + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.2999999999999998e-25 or 3.1000000000000001e-118 < y

   1. Initial program 99.8%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(\left(y - x\right) \cdot 6\right) \cdot z} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot z + x} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot z} + x \]

      4. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right)} \cdot z + x \]

      5. associate-*l* N/A

         \[\leadsto \color{blue}{\left(y - x\right) \cdot \left(6 \cdot z\right)} + x \]

      6. *-commutative N/A

         \[\leadsto \left(y - x\right) \cdot \color{blue}{\left(z \cdot 6\right)} + x \]

      7. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot z\right) \cdot 6} + x \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(y - x\right) \cdot z, 6, x\right)} \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot \left(y - x\right)}, 6, x\right) \]

      10. lower-*.f64 99.8

          \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot \left(y - x\right)}, 6, x\right) \]

   4. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z \cdot \left(y - x\right), 6, x\right)} \]

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot y}, 6, x\right) \]

      2. lower-*.f64 91.0

         \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot y}, 6, x\right) \]

   7. Applied rewrites 91.0%

      \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot y}, 6, x\right) \]

   if -3.2999999999999998e-25 < y < 3.1000000000000001e-118

   1. Initial program 99.7%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto x + \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot z} \]

      2. *-commutative N/A

         \[\leadsto x + \color{blue}{z \cdot \left(\left(y - x\right) \cdot 6\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto x + z \cdot \color{blue}{\left(\left(y - x\right) \cdot 6\right)} \]

      4. *-commutative N/A

         \[\leadsto x + z \cdot \color{blue}{\left(6 \cdot \left(y - x\right)\right)} \]

      5. lift--.f64 N/A

         \[\leadsto x + z \cdot \left(6 \cdot \color{blue}{\left(y - x\right)}\right) \]

      6. sub-neg N/A

         \[\leadsto x + z \cdot \left(6 \cdot \color{blue}{\left(y + \left(\mathsf{neg}\left(x\right)\right)\right)}\right) \]

      7. distribute-rgt-in N/A

         \[\leadsto x + z \cdot \color{blue}{\left(y \cdot 6 + \left(\mathsf{neg}\left(x\right)\right) \cdot 6\right)} \]

      8. distribute-rgt-in N/A

         \[\leadsto x + \color{blue}{\left(\left(y \cdot 6\right) \cdot z + \left(\left(\mathsf{neg}\left(x\right)\right) \cdot 6\right) \cdot z\right)} \]

      9. lower-fma.f64 N/A

         \[\leadsto x + \color{blue}{\mathsf{fma}\left(y \cdot 6, z, \left(\left(\mathsf{neg}\left(x\right)\right) \cdot 6\right) \cdot z\right)} \]

      10. *-commutative N/A

          \[\leadsto x + \mathsf{fma}\left(\color{blue}{6 \cdot y}, z, \left(\left(\mathsf{neg}\left(x\right)\right) \cdot 6\right) \cdot z\right) \]

      11. lower-*.f64 N/A

          \[\leadsto x + \mathsf{fma}\left(\color{blue}{6 \cdot y}, z, \left(\left(\mathsf{neg}\left(x\right)\right) \cdot 6\right) \cdot z\right) \]

      12. lower-*.f64 N/A

          \[\leadsto x + \mathsf{fma}\left(6 \cdot y, z, \color{blue}{\left(\left(\mathsf{neg}\left(x\right)\right) \cdot 6\right) \cdot z}\right) \]

      13. *-commutative N/A

          \[\leadsto x + \mathsf{fma}\left(6 \cdot y, z, \color{blue}{\left(6 \cdot \left(\mathsf{neg}\left(x\right)\right)\right)} \cdot z\right) \]

      14. neg-mul-1 N/A

          \[\leadsto x + \mathsf{fma}\left(6 \cdot y, z, \left(6 \cdot \color{blue}{\left(-1 \cdot x\right)}\right) \cdot z\right) \]

      15. associate-*r* N/A

          \[\leadsto x + \mathsf{fma}\left(6 \cdot y, z, \color{blue}{\left(\left(6 \cdot -1\right) \cdot x\right)} \cdot z\right) \]

      16. metadata-eval N/A

          \[\leadsto x + \mathsf{fma}\left(6 \cdot y, z, \left(\color{blue}{-6} \cdot x\right) \cdot z\right) \]

      17. metadata-eval N/A

          \[\leadsto x + \mathsf{fma}\left(6 \cdot y, z, \left(\color{blue}{\left(\mathsf{neg}\left(6\right)\right)} \cdot x\right) \cdot z\right) \]

      18. lower-*.f64 N/A

          \[\leadsto x + \mathsf{fma}\left(6 \cdot y, z, \color{blue}{\left(\left(\mathsf{neg}\left(6\right)\right) \cdot x\right)} \cdot z\right) \]

      19. metadata-eval 99.7

          \[\leadsto x + \mathsf{fma}\left(6 \cdot y, z, \left(\color{blue}{-6} \cdot x\right) \cdot z\right) \]

   4. Applied rewrites 99.7%

      \[\leadsto x + \color{blue}{\mathsf{fma}\left(6 \cdot y, z, \left(-6 \cdot x\right) \cdot z\right)} \]

   5. Taylor expanded in x around inf

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

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(-6 \cdot z + 1\right)} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(-6 \cdot z\right) \cdot x + 1 \cdot x} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{-6 \cdot \left(z \cdot x\right)} + 1 \cdot x \]

      4. *-commutative N/A

         \[\leadsto -6 \cdot \color{blue}{\left(x \cdot z\right)} + 1 \cdot x \]

      5. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-6 \cdot x\right) \cdot z} + 1 \cdot x \]

      6. *-lft-identity N/A

         \[\leadsto \left(-6 \cdot x\right) \cdot z + \color{blue}{x} \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-6 \cdot x, z, x\right)} \]

      8. lower-*.f64 89.8

         \[\leadsto \mathsf{fma}\left(\color{blue}{-6 \cdot x}, z, x\right) \]

   7. Applied rewrites 89.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-6 \cdot x, z, x\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 89.9%

      \[\leadsto \mathsf{fma}\left(z \cdot -6, \color{blue}{x}, x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.3 \cdot 10^{-25} \lor \neg \left(y \leq 3.1 \cdot 10^{-118}\right):\\ \;\;\;\;\mathsf{fma}\left(z \cdot y, 6, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot -6, x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{-10} \lor \neg \left(x \leq 3.2 \cdot 10^{-208}\right):\\ \;\;\;\;\mathsf{fma}\left(-6 \cdot x, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot 6\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.3e-10) (not (<= x 3.2e-208)))
   (fma (* -6.0 x) z x)
   (* (* z 6.0) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.3e-10) || !(x <= 3.2e-208)) {
		tmp = fma((-6.0 * x), z, x);
	} else {
		tmp = (z * 6.0) * y;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.3e-10) || !(x <= 3.2e-208))
		tmp = fma(Float64(-6.0 * x), z, x);
	else
		tmp = Float64(Float64(z * 6.0) * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.3e-10], N[Not[LessEqual[x, 3.2e-208]], $MachinePrecision]], N[(N[(-6.0 * x), $MachinePrecision] * z + x), $MachinePrecision], N[(N[(z * 6.0), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.29999999999999991e-10 or 3.2000000000000001e-208 < x

   1. Initial program 99.9%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. distribute-rgt-in N/A

         \[\leadsto \color{blue}{1 \cdot x + \left(-6 \cdot z\right) \cdot x} \]

      2. *-lft-identity N/A

         \[\leadsto \color{blue}{x} + \left(-6 \cdot z\right) \cdot x \]

      3. associate-*r* N/A

         \[\leadsto x + \color{blue}{-6 \cdot \left(z \cdot x\right)} \]

      4. *-commutative N/A

         \[\leadsto x + -6 \cdot \color{blue}{\left(x \cdot z\right)} \]

      5. +-commutative N/A

         \[\leadsto \color{blue}{-6 \cdot \left(x \cdot z\right) + x} \]

      6. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-6 \cdot x\right) \cdot z} + x \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-6 \cdot x, z, x\right)} \]

      8. lower-*.f64 79.0

         \[\leadsto \mathsf{fma}\left(\color{blue}{-6 \cdot x}, z, x\right) \]

   5. Applied rewrites 79.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-6 \cdot x, z, x\right)} \]

   if -1.29999999999999991e-10 < x < 3.2000000000000001e-208

   1. Initial program 99.6%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(y \cdot z\right) \cdot 6} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(y \cdot z\right) \cdot 6} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(z \cdot y\right)} \cdot 6 \]

      4. lower-*.f64 83.5

         \[\leadsto \color{blue}{\left(z \cdot y\right)} \cdot 6 \]

   5. Applied rewrites 83.5%

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

   7. Applied rewrites 83.6%

      \[\leadsto \left(z \cdot 6\right) \cdot \color{blue}{y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{-10} \lor \neg \left(x \leq 3.2 \cdot 10^{-208}\right):\\ \;\;\;\;\mathsf{fma}\left(-6 \cdot x, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot 6\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot -6, x, x\right)\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{-208}:\\ \;\;\;\;\left(z \cdot 6\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-6 \cdot x, z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.3e-10)
   (fma (* z -6.0) x x)
   (if (<= x 3.2e-208) (* (* z 6.0) y) (fma (* -6.0 x) z x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.3e-10) {
		tmp = fma((z * -6.0), x, x);
	} else if (x <= 3.2e-208) {
		tmp = (z * 6.0) * y;
	} else {
		tmp = fma((-6.0 * x), z, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.3e-10)
		tmp = fma(Float64(z * -6.0), x, x);
	elseif (x <= 3.2e-208)
		tmp = Float64(Float64(z * 6.0) * y);
	else
		tmp = fma(Float64(-6.0 * x), z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.3e-10], N[(N[(z * -6.0), $MachinePrecision] * x + x), $MachinePrecision], If[LessEqual[x, 3.2e-208], N[(N[(z * 6.0), $MachinePrecision] * y), $MachinePrecision], N[(N[(-6.0 * x), $MachinePrecision] * z + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.29999999999999991e-10

   1. Initial program 99.9%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto x + \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot z} \]

      2. *-commutative N/A

         \[\leadsto x + \color{blue}{z \cdot \left(\left(y - x\right) \cdot 6\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto x + z \cdot \color{blue}{\left(\left(y - x\right) \cdot 6\right)} \]

      4. *-commutative N/A

         \[\leadsto x + z \cdot \color{blue}{\left(6 \cdot \left(y - x\right)\right)} \]

      5. lift--.f64 N/A

         \[\leadsto x + z \cdot \left(6 \cdot \color{blue}{\left(y - x\right)}\right) \]

      6. sub-neg N/A

         \[\leadsto x + z \cdot \left(6 \cdot \color{blue}{\left(y + \left(\mathsf{neg}\left(x\right)\right)\right)}\right) \]

      7. distribute-rgt-in N/A

         \[\leadsto x + z \cdot \color{blue}{\left(y \cdot 6 + \left(\mathsf{neg}\left(x\right)\right) \cdot 6\right)} \]

      8. distribute-rgt-in N/A

         \[\leadsto x + \color{blue}{\left(\left(y \cdot 6\right) \cdot z + \left(\left(\mathsf{neg}\left(x\right)\right) \cdot 6\right) \cdot z\right)} \]

      9. lower-fma.f64 N/A

         \[\leadsto x + \color{blue}{\mathsf{fma}\left(y \cdot 6, z, \left(\left(\mathsf{neg}\left(x\right)\right) \cdot 6\right) \cdot z\right)} \]

      10. *-commutative N/A

          \[\leadsto x + \mathsf{fma}\left(\color{blue}{6 \cdot y}, z, \left(\left(\mathsf{neg}\left(x\right)\right) \cdot 6\right) \cdot z\right) \]

      11. lower-*.f64 N/A

          \[\leadsto x + \mathsf{fma}\left(\color{blue}{6 \cdot y}, z, \left(\left(\mathsf{neg}\left(x\right)\right) \cdot 6\right) \cdot z\right) \]

      12. lower-*.f64 N/A

          \[\leadsto x + \mathsf{fma}\left(6 \cdot y, z, \color{blue}{\left(\left(\mathsf{neg}\left(x\right)\right) \cdot 6\right) \cdot z}\right) \]

      13. *-commutative N/A

          \[\leadsto x + \mathsf{fma}\left(6 \cdot y, z, \color{blue}{\left(6 \cdot \left(\mathsf{neg}\left(x\right)\right)\right)} \cdot z\right) \]

      14. neg-mul-1 N/A

          \[\leadsto x + \mathsf{fma}\left(6 \cdot y, z, \left(6 \cdot \color{blue}{\left(-1 \cdot x\right)}\right) \cdot z\right) \]

      15. associate-*r* N/A

          \[\leadsto x + \mathsf{fma}\left(6 \cdot y, z, \color{blue}{\left(\left(6 \cdot -1\right) \cdot x\right)} \cdot z\right) \]

      16. metadata-eval N/A

          \[\leadsto x + \mathsf{fma}\left(6 \cdot y, z, \left(\color{blue}{-6} \cdot x\right) \cdot z\right) \]

      17. metadata-eval N/A

          \[\leadsto x + \mathsf{fma}\left(6 \cdot y, z, \left(\color{blue}{\left(\mathsf{neg}\left(6\right)\right)} \cdot x\right) \cdot z\right) \]

      18. lower-*.f64 N/A

          \[\leadsto x + \mathsf{fma}\left(6 \cdot y, z, \color{blue}{\left(\left(\mathsf{neg}\left(6\right)\right) \cdot x\right)} \cdot z\right) \]

      19. metadata-eval 99.9

          \[\leadsto x + \mathsf{fma}\left(6 \cdot y, z, \left(\color{blue}{-6} \cdot x\right) \cdot z\right) \]

   4. Applied rewrites 99.9%

      \[\leadsto x + \color{blue}{\mathsf{fma}\left(6 \cdot y, z, \left(-6 \cdot x\right) \cdot z\right)} \]

   5. Taylor expanded in x around inf

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

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(-6 \cdot z + 1\right)} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(-6 \cdot z\right) \cdot x + 1 \cdot x} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{-6 \cdot \left(z \cdot x\right)} + 1 \cdot x \]

      4. *-commutative N/A

         \[\leadsto -6 \cdot \color{blue}{\left(x \cdot z\right)} + 1 \cdot x \]

      5. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-6 \cdot x\right) \cdot z} + 1 \cdot x \]

      6. *-lft-identity N/A

         \[\leadsto \left(-6 \cdot x\right) \cdot z + \color{blue}{x} \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-6 \cdot x, z, x\right)} \]

      8. lower-*.f64 86.7

         \[\leadsto \mathsf{fma}\left(\color{blue}{-6 \cdot x}, z, x\right) \]

   7. Applied rewrites 86.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-6 \cdot x, z, x\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 86.7%

      \[\leadsto \mathsf{fma}\left(z \cdot -6, \color{blue}{x}, x\right) \]

   if -1.29999999999999991e-10 < x < 3.2000000000000001e-208

   1. Initial program 99.6%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(y \cdot z\right) \cdot 6} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(y \cdot z\right) \cdot 6} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(z \cdot y\right)} \cdot 6 \]

      4. lower-*.f64 83.5

         \[\leadsto \color{blue}{\left(z \cdot y\right)} \cdot 6 \]

   5. Applied rewrites 83.5%

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

   7. Applied rewrites 83.6%

      \[\leadsto \left(z \cdot 6\right) \cdot \color{blue}{y} \]

   if 3.2000000000000001e-208 < x

   1. Initial program 99.9%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. distribute-rgt-in N/A

         \[\leadsto \color{blue}{1 \cdot x + \left(-6 \cdot z\right) \cdot x} \]

      2. *-lft-identity N/A

         \[\leadsto \color{blue}{x} + \left(-6 \cdot z\right) \cdot x \]

      3. associate-*r* N/A

         \[\leadsto x + \color{blue}{-6 \cdot \left(z \cdot x\right)} \]

      4. *-commutative N/A

         \[\leadsto x + -6 \cdot \color{blue}{\left(x \cdot z\right)} \]

      5. +-commutative N/A

         \[\leadsto \color{blue}{-6 \cdot \left(x \cdot z\right) + x} \]

      6. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-6 \cdot x\right) \cdot z} + x \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-6 \cdot x, z, x\right)} \]

      8. lower-*.f64 74.9

         \[\leadsto \mathsf{fma}\left(\color{blue}{-6 \cdot x}, z, x\right) \]

   5. Applied rewrites 74.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-6 \cdot x, z, x\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot -6, x, x\right)\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{-208}:\\ \;\;\;\;\left(z \cdot 6\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-6 \cdot x, z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 52.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.3 \cdot 10^{-25}:\\ \;\;\;\;\left(y \cdot 6\right) \cdot z\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{-115}:\\ \;\;\;\;\left(z \cdot -6\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot 6\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -3.3e-25)
   (* (* y 6.0) z)
   (if (<= y 8.2e-115) (* (* z -6.0) x) (* (* z 6.0) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.3e-25) {
		tmp = (y * 6.0) * z;
	} else if (y <= 8.2e-115) {
		tmp = (z * -6.0) * x;
	} else {
		tmp = (z * 6.0) * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-3.3d-25)) then
        tmp = (y * 6.0d0) * z
    else if (y <= 8.2d-115) then
        tmp = (z * (-6.0d0)) * x
    else
        tmp = (z * 6.0d0) * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.3e-25) {
		tmp = (y * 6.0) * z;
	} else if (y <= 8.2e-115) {
		tmp = (z * -6.0) * x;
	} else {
		tmp = (z * 6.0) * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -3.3e-25:
		tmp = (y * 6.0) * z
	elif y <= 8.2e-115:
		tmp = (z * -6.0) * x
	else:
		tmp = (z * 6.0) * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -3.3e-25)
		tmp = Float64(Float64(y * 6.0) * z);
	elseif (y <= 8.2e-115)
		tmp = Float64(Float64(z * -6.0) * x);
	else
		tmp = Float64(Float64(z * 6.0) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -3.3e-25)
		tmp = (y * 6.0) * z;
	elseif (y <= 8.2e-115)
		tmp = (z * -6.0) * x;
	else
		tmp = (z * 6.0) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -3.3e-25], N[(N[(y * 6.0), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[y, 8.2e-115], N[(N[(z * -6.0), $MachinePrecision] * x), $MachinePrecision], N[(N[(z * 6.0), $MachinePrecision] * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.2999999999999998e-25

   1. Initial program 99.9%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(y \cdot z\right) \cdot 6} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(y \cdot z\right) \cdot 6} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(z \cdot y\right)} \cdot 6 \]

      4. lower-*.f64 66.7

         \[\leadsto \color{blue}{\left(z \cdot y\right)} \cdot 6 \]

   5. Applied rewrites 66.7%

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

   7. Applied rewrites 66.8%

      \[\leadsto \left(y \cdot 6\right) \cdot \color{blue}{z} \]

   if -3.2999999999999998e-25 < y < 8.1999999999999993e-115

   1. Initial program 99.7%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto x + \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot z} \]

      2. *-commutative N/A

         \[\leadsto x + \color{blue}{z \cdot \left(\left(y - x\right) \cdot 6\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto x + z \cdot \color{blue}{\left(\left(y - x\right) \cdot 6\right)} \]

      4. *-commutative N/A

         \[\leadsto x + z \cdot \color{blue}{\left(6 \cdot \left(y - x\right)\right)} \]

      5. lift--.f64 N/A

         \[\leadsto x + z \cdot \left(6 \cdot \color{blue}{\left(y - x\right)}\right) \]

      6. sub-neg N/A

         \[\leadsto x + z \cdot \left(6 \cdot \color{blue}{\left(y + \left(\mathsf{neg}\left(x\right)\right)\right)}\right) \]

      7. distribute-rgt-in N/A

         \[\leadsto x + z \cdot \color{blue}{\left(y \cdot 6 + \left(\mathsf{neg}\left(x\right)\right) \cdot 6\right)} \]

      8. distribute-rgt-in N/A

         \[\leadsto x + \color{blue}{\left(\left(y \cdot 6\right) \cdot z + \left(\left(\mathsf{neg}\left(x\right)\right) \cdot 6\right) \cdot z\right)} \]

      9. lower-fma.f64 N/A

         \[\leadsto x + \color{blue}{\mathsf{fma}\left(y \cdot 6, z, \left(\left(\mathsf{neg}\left(x\right)\right) \cdot 6\right) \cdot z\right)} \]

      10. *-commutative N/A

          \[\leadsto x + \mathsf{fma}\left(\color{blue}{6 \cdot y}, z, \left(\left(\mathsf{neg}\left(x\right)\right) \cdot 6\right) \cdot z\right) \]

      11. lower-*.f64 N/A

          \[\leadsto x + \mathsf{fma}\left(\color{blue}{6 \cdot y}, z, \left(\left(\mathsf{neg}\left(x\right)\right) \cdot 6\right) \cdot z\right) \]

      12. lower-*.f64 N/A

          \[\leadsto x + \mathsf{fma}\left(6 \cdot y, z, \color{blue}{\left(\left(\mathsf{neg}\left(x\right)\right) \cdot 6\right) \cdot z}\right) \]

      13. *-commutative N/A

          \[\leadsto x + \mathsf{fma}\left(6 \cdot y, z, \color{blue}{\left(6 \cdot \left(\mathsf{neg}\left(x\right)\right)\right)} \cdot z\right) \]

      14. neg-mul-1 N/A

          \[\leadsto x + \mathsf{fma}\left(6 \cdot y, z, \left(6 \cdot \color{blue}{\left(-1 \cdot x\right)}\right) \cdot z\right) \]

      15. associate-*r* N/A

          \[\leadsto x + \mathsf{fma}\left(6 \cdot y, z, \color{blue}{\left(\left(6 \cdot -1\right) \cdot x\right)} \cdot z\right) \]

      16. metadata-eval N/A

          \[\leadsto x + \mathsf{fma}\left(6 \cdot y, z, \left(\color{blue}{-6} \cdot x\right) \cdot z\right) \]

      17. metadata-eval N/A

          \[\leadsto x + \mathsf{fma}\left(6 \cdot y, z, \left(\color{blue}{\left(\mathsf{neg}\left(6\right)\right)} \cdot x\right) \cdot z\right) \]

      18. lower-*.f64 N/A

          \[\leadsto x + \mathsf{fma}\left(6 \cdot y, z, \color{blue}{\left(\left(\mathsf{neg}\left(6\right)\right) \cdot x\right)} \cdot z\right) \]

      19. metadata-eval 99.8

          \[\leadsto x + \mathsf{fma}\left(6 \cdot y, z, \left(\color{blue}{-6} \cdot x\right) \cdot z\right) \]

   4. Applied rewrites 99.8%

      \[\leadsto x + \color{blue}{\mathsf{fma}\left(6 \cdot y, z, \left(-6 \cdot x\right) \cdot z\right)} \]

   5. Taylor expanded in x around inf

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

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(-6 \cdot z + 1\right)} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(-6 \cdot z\right) \cdot x + 1 \cdot x} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{-6 \cdot \left(z \cdot x\right)} + 1 \cdot x \]

      4. *-commutative N/A

         \[\leadsto -6 \cdot \color{blue}{\left(x \cdot z\right)} + 1 \cdot x \]

      5. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-6 \cdot x\right) \cdot z} + 1 \cdot x \]

      6. *-lft-identity N/A

         \[\leadsto \left(-6 \cdot x\right) \cdot z + \color{blue}{x} \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-6 \cdot x, z, x\right)} \]

      8. lower-*.f64 89.0

         \[\leadsto \mathsf{fma}\left(\color{blue}{-6 \cdot x}, z, x\right) \]

   7. Applied rewrites 89.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-6 \cdot x, z, x\right)} \]

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 44.9%

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

   12. Applied rewrites 45.0%

       \[\leadsto \left(z \cdot -6\right) \cdot x \]

   if 8.1999999999999993e-115 < y

   1. Initial program 99.8%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(y \cdot z\right) \cdot 6} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(y \cdot z\right) \cdot 6} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(z \cdot y\right)} \cdot 6 \]

      4. lower-*.f64 60.1

         \[\leadsto \color{blue}{\left(z \cdot y\right)} \cdot 6 \]

   5. Applied rewrites 60.1%

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

   7. Applied rewrites 60.2%

      \[\leadsto \left(z \cdot 6\right) \cdot \color{blue}{y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 56.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.3 \cdot 10^{-25}:\\ \;\;\;\;\left(y \cdot 6\right) \cdot z\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{-115}:\\ \;\;\;\;\left(z \cdot -6\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot 6\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 52.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.3 \cdot 10^{-25}:\\ \;\;\;\;\left(y \cdot 6\right) \cdot z\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{-115}:\\ \;\;\;\;\left(-6 \cdot x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot 6\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -3.3e-25)
   (* (* y 6.0) z)
   (if (<= y 8.2e-115) (* (* -6.0 x) z) (* (* z 6.0) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.3e-25) {
		tmp = (y * 6.0) * z;
	} else if (y <= 8.2e-115) {
		tmp = (-6.0 * x) * z;
	} else {
		tmp = (z * 6.0) * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-3.3d-25)) then
        tmp = (y * 6.0d0) * z
    else if (y <= 8.2d-115) then
        tmp = ((-6.0d0) * x) * z
    else
        tmp = (z * 6.0d0) * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.3e-25) {
		tmp = (y * 6.0) * z;
	} else if (y <= 8.2e-115) {
		tmp = (-6.0 * x) * z;
	} else {
		tmp = (z * 6.0) * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -3.3e-25:
		tmp = (y * 6.0) * z
	elif y <= 8.2e-115:
		tmp = (-6.0 * x) * z
	else:
		tmp = (z * 6.0) * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -3.3e-25)
		tmp = Float64(Float64(y * 6.0) * z);
	elseif (y <= 8.2e-115)
		tmp = Float64(Float64(-6.0 * x) * z);
	else
		tmp = Float64(Float64(z * 6.0) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -3.3e-25)
		tmp = (y * 6.0) * z;
	elseif (y <= 8.2e-115)
		tmp = (-6.0 * x) * z;
	else
		tmp = (z * 6.0) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -3.3e-25], N[(N[(y * 6.0), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[y, 8.2e-115], N[(N[(-6.0 * x), $MachinePrecision] * z), $MachinePrecision], N[(N[(z * 6.0), $MachinePrecision] * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.2999999999999998e-25

   1. Initial program 99.9%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(y \cdot z\right) \cdot 6} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(y \cdot z\right) \cdot 6} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(z \cdot y\right)} \cdot 6 \]

      4. lower-*.f64 66.7

         \[\leadsto \color{blue}{\left(z \cdot y\right)} \cdot 6 \]

   5. Applied rewrites 66.7%

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

   7. Applied rewrites 66.8%

      \[\leadsto \left(y \cdot 6\right) \cdot \color{blue}{z} \]

   if -3.2999999999999998e-25 < y < 8.1999999999999993e-115

   1. Initial program 99.7%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto x + \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot z} \]

      2. *-commutative N/A

         \[\leadsto x + \color{blue}{z \cdot \left(\left(y - x\right) \cdot 6\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto x + z \cdot \color{blue}{\left(\left(y - x\right) \cdot 6\right)} \]

      4. *-commutative N/A

         \[\leadsto x + z \cdot \color{blue}{\left(6 \cdot \left(y - x\right)\right)} \]

      5. lift--.f64 N/A

         \[\leadsto x + z \cdot \left(6 \cdot \color{blue}{\left(y - x\right)}\right) \]

      6. sub-neg N/A

         \[\leadsto x + z \cdot \left(6 \cdot \color{blue}{\left(y + \left(\mathsf{neg}\left(x\right)\right)\right)}\right) \]

      7. distribute-rgt-in N/A

         \[\leadsto x + z \cdot \color{blue}{\left(y \cdot 6 + \left(\mathsf{neg}\left(x\right)\right) \cdot 6\right)} \]

      8. distribute-rgt-in N/A

         \[\leadsto x + \color{blue}{\left(\left(y \cdot 6\right) \cdot z + \left(\left(\mathsf{neg}\left(x\right)\right) \cdot 6\right) \cdot z\right)} \]

      9. lower-fma.f64 N/A

         \[\leadsto x + \color{blue}{\mathsf{fma}\left(y \cdot 6, z, \left(\left(\mathsf{neg}\left(x\right)\right) \cdot 6\right) \cdot z\right)} \]

      10. *-commutative N/A

          \[\leadsto x + \mathsf{fma}\left(\color{blue}{6 \cdot y}, z, \left(\left(\mathsf{neg}\left(x\right)\right) \cdot 6\right) \cdot z\right) \]

      11. lower-*.f64 N/A

          \[\leadsto x + \mathsf{fma}\left(\color{blue}{6 \cdot y}, z, \left(\left(\mathsf{neg}\left(x\right)\right) \cdot 6\right) \cdot z\right) \]

      12. lower-*.f64 N/A

          \[\leadsto x + \mathsf{fma}\left(6 \cdot y, z, \color{blue}{\left(\left(\mathsf{neg}\left(x\right)\right) \cdot 6\right) \cdot z}\right) \]

      13. *-commutative N/A

          \[\leadsto x + \mathsf{fma}\left(6 \cdot y, z, \color{blue}{\left(6 \cdot \left(\mathsf{neg}\left(x\right)\right)\right)} \cdot z\right) \]

      14. neg-mul-1 N/A

          \[\leadsto x + \mathsf{fma}\left(6 \cdot y, z, \left(6 \cdot \color{blue}{\left(-1 \cdot x\right)}\right) \cdot z\right) \]

      15. associate-*r* N/A

          \[\leadsto x + \mathsf{fma}\left(6 \cdot y, z, \color{blue}{\left(\left(6 \cdot -1\right) \cdot x\right)} \cdot z\right) \]

      16. metadata-eval N/A

          \[\leadsto x + \mathsf{fma}\left(6 \cdot y, z, \left(\color{blue}{-6} \cdot x\right) \cdot z\right) \]

      17. metadata-eval N/A

          \[\leadsto x + \mathsf{fma}\left(6 \cdot y, z, \left(\color{blue}{\left(\mathsf{neg}\left(6\right)\right)} \cdot x\right) \cdot z\right) \]

      18. lower-*.f64 N/A

          \[\leadsto x + \mathsf{fma}\left(6 \cdot y, z, \color{blue}{\left(\left(\mathsf{neg}\left(6\right)\right) \cdot x\right)} \cdot z\right) \]

      19. metadata-eval 99.8

          \[\leadsto x + \mathsf{fma}\left(6 \cdot y, z, \left(\color{blue}{-6} \cdot x\right) \cdot z\right) \]

   4. Applied rewrites 99.8%

      \[\leadsto x + \color{blue}{\mathsf{fma}\left(6 \cdot y, z, \left(-6 \cdot x\right) \cdot z\right)} \]

   5. Taylor expanded in x around inf

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

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(-6 \cdot z + 1\right)} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(-6 \cdot z\right) \cdot x + 1 \cdot x} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{-6 \cdot \left(z \cdot x\right)} + 1 \cdot x \]

      4. *-commutative N/A

         \[\leadsto -6 \cdot \color{blue}{\left(x \cdot z\right)} + 1 \cdot x \]

      5. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-6 \cdot x\right) \cdot z} + 1 \cdot x \]

      6. *-lft-identity N/A

         \[\leadsto \left(-6 \cdot x\right) \cdot z + \color{blue}{x} \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-6 \cdot x, z, x\right)} \]

      8. lower-*.f64 89.0

         \[\leadsto \mathsf{fma}\left(\color{blue}{-6 \cdot x}, z, x\right) \]

   7. Applied rewrites 89.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-6 \cdot x, z, x\right)} \]

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 44.9%

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

   12. Applied rewrites 44.9%

       \[\leadsto \left(-6 \cdot x\right) \cdot z \]

   if 8.1999999999999993e-115 < y

   1. Initial program 99.8%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(y \cdot z\right) \cdot 6} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(y \cdot z\right) \cdot 6} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(z \cdot y\right)} \cdot 6 \]

      4. lower-*.f64 60.1

         \[\leadsto \color{blue}{\left(z \cdot y\right)} \cdot 6 \]

   5. Applied rewrites 60.1%

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

   7. Applied rewrites 60.2%

      \[\leadsto \left(z \cdot 6\right) \cdot \color{blue}{y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 56.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.3 \cdot 10^{-25}:\\ \;\;\;\;\left(y \cdot 6\right) \cdot z\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{-115}:\\ \;\;\;\;\left(-6 \cdot x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot 6\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(y - x, z \cdot 6, x\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fma (- y x) (* z 6.0) x))

C

double code(double x, double y, double z) {
	return fma((y - x), (z * 6.0), x);
}

Julia

function code(x, y, z)
	return fma(Float64(y - x), Float64(z * 6.0), x)
end

Wolfram

code[x_, y_, z_] := N[(N[(y - x), $MachinePrecision] * N[(z * 6.0), $MachinePrecision] + x), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \color{blue}{x + \left(\left(y - x\right) \cdot 6\right) \cdot z} \]

   2. +-commutative N/A

      \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot z + x} \]

   3. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot z} + x \]

   4. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right)} \cdot z + x \]

   5. associate-*l* N/A

      \[\leadsto \color{blue}{\left(y - x\right) \cdot \left(6 \cdot z\right)} + x \]

   6. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 6 \cdot z, x\right)} \]

   7. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(y - x, \color{blue}{z \cdot 6}, x\right) \]

   8. lower-*.f64 99.8

      \[\leadsto \mathsf{fma}\left(y - x, \color{blue}{z \cdot 6}, x\right) \]

4. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, z \cdot 6, x\right)} \]
5. Add Preprocessing

Alternative 10: 99.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(z \cdot \left(y - x\right), 6, x\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fma (* z (- y x)) 6.0 x))

C

double code(double x, double y, double z) {
	return fma((z * (y - x)), 6.0, x);
}

Julia

function code(x, y, z)
	return fma(Float64(z * Float64(y - x)), 6.0, x)
end

Wolfram

code[x_, y_, z_] := N[(N[(z * N[(y - x), $MachinePrecision]), $MachinePrecision] * 6.0 + x), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \color{blue}{x + \left(\left(y - x\right) \cdot 6\right) \cdot z} \]

   2. +-commutative N/A

      \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot z + x} \]

   3. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot z} + x \]

   4. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right)} \cdot z + x \]

   5. associate-*l* N/A

      \[\leadsto \color{blue}{\left(y - x\right) \cdot \left(6 \cdot z\right)} + x \]

   6. *-commutative N/A

      \[\leadsto \left(y - x\right) \cdot \color{blue}{\left(z \cdot 6\right)} + x \]

   7. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot z\right) \cdot 6} + x \]

   8. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(y - x\right) \cdot z, 6, x\right)} \]

   9. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot \left(y - x\right)}, 6, x\right) \]

   10. lower-*.f64 99.8

       \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot \left(y - x\right)}, 6, x\right) \]

4. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(z \cdot \left(y - x\right), 6, x\right)} \]
5. Add Preprocessing

Alternative 11: 42.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \left(z \cdot 6\right) \cdot y \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* (* z 6.0) y))

C

double code(double x, double y, double z) {
	return (z * 6.0) * y;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (z * 6.0d0) * y
end function

Java

public static double code(double x, double y, double z) {
	return (z * 6.0) * y;
}

Python

def code(x, y, z):
	return (z * 6.0) * y

Julia

function code(x, y, z)
	return Float64(Float64(z * 6.0) * y)
end

MATLAB

function tmp = code(x, y, z)
	tmp = (z * 6.0) * y;
end

Wolfram

code[x_, y_, z_] := N[(N[(z * 6.0), $MachinePrecision] * y), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(y \cdot z\right) \cdot 6} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(y \cdot z\right) \cdot 6} \]

   3. *-commutative N/A

      \[\leadsto \color{blue}{\left(z \cdot y\right)} \cdot 6 \]

   4. lower-*.f64 45.6

      \[\leadsto \color{blue}{\left(z \cdot y\right)} \cdot 6 \]

5. Applied rewrites 45.6%

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

7. Applied rewrites 45.6%

   \[\leadsto \left(z \cdot 6\right) \cdot \color{blue}{y} \]

8. Final simplification 45.6%

   \[\leadsto \left(z \cdot 6\right) \cdot y \]
9. Add Preprocessing

Alternative 12: 42.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \left(y \cdot 6\right) \cdot z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* (* y 6.0) z))

C

double code(double x, double y, double z) {
	return (y * 6.0) * z;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (y * 6.0d0) * z
end function

Java

public static double code(double x, double y, double z) {
	return (y * 6.0) * z;
}

Python

def code(x, y, z):
	return (y * 6.0) * z

Julia

function code(x, y, z)
	return Float64(Float64(y * 6.0) * z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = (y * 6.0) * z;
end

Wolfram

code[x_, y_, z_] := N[(N[(y * 6.0), $MachinePrecision] * z), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(y \cdot z\right) \cdot 6} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(y \cdot z\right) \cdot 6} \]

   3. *-commutative N/A

      \[\leadsto \color{blue}{\left(z \cdot y\right)} \cdot 6 \]

   4. lower-*.f64 45.6

      \[\leadsto \color{blue}{\left(z \cdot y\right)} \cdot 6 \]

5. Applied rewrites 45.6%

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

7. Applied rewrites 45.6%

   \[\leadsto \left(y \cdot 6\right) \cdot \color{blue}{z} \]

8. Final simplification 45.6%

   \[\leadsto \left(y \cdot 6\right) \cdot z \]
9. Add Preprocessing

Developer Target 1: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x - \left(6 \cdot z\right) \cdot \left(x - y\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- x (* (* 6.0 z) (- x y))))

C

double code(double x, double y, double z) {
	return x - ((6.0 * z) * (x - y));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x - ((6.0d0 * z) * (x - y))
end function

Java

public static double code(double x, double y, double z) {
	return x - ((6.0 * z) * (x - y));
}

Python

def code(x, y, z):
	return x - ((6.0 * z) * (x - y))

Julia

function code(x, y, z)
	return Float64(x - Float64(Float64(6.0 * z) * Float64(x - y)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = x - ((6.0 * z) * (x - y));
end

Wolfram

code[x_, y_, z_] := N[(x - N[(N[(6.0 * z), $MachinePrecision] * N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024318 
(FPCore (x y z)
  :name "Data.Colour.RGBSpace.HSL:hsl from colour-2.3.3, E"
  :precision binary64

  :alt
  (! :herbie-platform default (- x (* (* 6 z) (- x y))))

  (+ x (* (* (- y x) 6.0) z)))