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

Percentage Accurate: 99.7% → 99.8%

Time: 8.0s

Alternatives: 10

Speedup: 1.1×

• 20.8% 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: 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 99.4%

   \[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
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. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r* N/A

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

   7. lower-fma.f64 N/A

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

   8. 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. Add Preprocessing

Alternative 2: 59.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.2 \cdot 10^{+146}:\\ \;\;\;\;\left(-6 \cdot x\right) \cdot z\\ \mathbf{elif}\;z \leq -9.2 \cdot 10^{-132} \lor \neg \left(z \leq 2.05 \cdot 10^{-44}\right):\\ \;\;\;\;\left(6 \cdot z\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.2e+146)
   (* (* -6.0 x) z)
   (if (or (<= z -9.2e-132) (not (<= z 2.05e-44))) (* (* 6.0 z) y) (* 1.0 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.2e+146) {
		tmp = (-6.0 * x) * z;
	} else if ((z <= -9.2e-132) || !(z <= 2.05e-44)) {
		tmp = (6.0 * z) * y;
	} else {
		tmp = 1.0 * x;
	}
	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 <= (-1.2d+146)) then
        tmp = ((-6.0d0) * x) * z
    else if ((z <= (-9.2d-132)) .or. (.not. (z <= 2.05d-44))) then
        tmp = (6.0d0 * z) * y
    else
        tmp = 1.0d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.2e+146) {
		tmp = (-6.0 * x) * z;
	} else if ((z <= -9.2e-132) || !(z <= 2.05e-44)) {
		tmp = (6.0 * z) * y;
	} else {
		tmp = 1.0 * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.2e+146:
		tmp = (-6.0 * x) * z
	elif (z <= -9.2e-132) or not (z <= 2.05e-44):
		tmp = (6.0 * z) * y
	else:
		tmp = 1.0 * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.2e+146)
		tmp = Float64(Float64(-6.0 * x) * z);
	elseif ((z <= -9.2e-132) || !(z <= 2.05e-44))
		tmp = Float64(Float64(6.0 * z) * y);
	else
		tmp = Float64(1.0 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.2e+146)
		tmp = (-6.0 * x) * z;
	elseif ((z <= -9.2e-132) || ~((z <= 2.05e-44)))
		tmp = (6.0 * z) * y;
	else
		tmp = 1.0 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.2e+146], N[(N[(-6.0 * x), $MachinePrecision] * z), $MachinePrecision], If[Or[LessEqual[z, -9.2e-132], N[Not[LessEqual[z, 2.05e-44]], $MachinePrecision]], N[(N[(6.0 * z), $MachinePrecision] * y), $MachinePrecision], N[(1.0 * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.2000000000000001e146

   1. Initial program 100.0%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 66.9

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

   5. Applied rewrites 66.9%

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

   6. Taylor expanded in z around 0

      \[\leadsto 1 \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 4.3%

      \[\leadsto 1 \cdot x \]

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 67.0%

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

   13. Applied rewrites 67.1%

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

   if -1.2000000000000001e146 < z < -9.20000000000000012e-132 or 2.04999999999999996e-44 < z

   1. Initial program 99.7%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
   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 56.5

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

   5. Applied rewrites 56.5%

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

   7. Applied rewrites 56.6%

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

   if -9.20000000000000012e-132 < z < 2.04999999999999996e-44

   1. Initial program 98.8%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 80.6

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

   5. Applied rewrites 80.6%

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

   6. Taylor expanded in z around 0

      \[\leadsto 1 \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 80.6%

      \[\leadsto 1 \cdot x \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.2 \cdot 10^{+146}:\\ \;\;\;\;\left(-6 \cdot x\right) \cdot z\\ \mathbf{elif}\;z \leq -9.2 \cdot 10^{-132} \lor \neg \left(z \leq 2.05 \cdot 10^{-44}\right):\\ \;\;\;\;\left(6 \cdot z\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -660 \lor \neg \left(z \leq 0.165\right):\\ \;\;\;\;\left(6 \cdot \left(y - x\right)\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot z, 6, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -660.0) (not (<= z 0.165)))
   (* (* 6.0 (- y x)) z)
   (fma (* y z) 6.0 x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -660.0) || !(z <= 0.165)) {
		tmp = (6.0 * (y - x)) * z;
	} else {
		tmp = fma((y * z), 6.0, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -660.0) || !(z <= 0.165))
		tmp = Float64(Float64(6.0 * Float64(y - x)) * z);
	else
		tmp = fma(Float64(y * z), 6.0, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -660.0], N[Not[LessEqual[z, 0.165]], $MachinePrecision]], N[(N[(6.0 * N[(y - x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], N[(N[(y * z), $MachinePrecision] * 6.0 + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -660 or 0.165000000000000008 < z

   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. 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

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 99.0

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

   7. Applied rewrites 99.0%

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

   if -660 < z < 0.165000000000000008

   1. Initial program 99.1%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
   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. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. 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. lower-*.f64 98.7

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

   7. Applied rewrites 98.7%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -660 \lor \neg \left(z \leq 0.165\right):\\ \;\;\;\;\left(6 \cdot \left(y - x\right)\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot z, 6, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -660:\\ \;\;\;\;\left(6 \cdot \left(y - x\right)\right) \cdot z\\ \mathbf{elif}\;z \leq 0.165:\\ \;\;\;\;\mathsf{fma}\left(y \cdot z, 6, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot \left(y - x\right)\right) \cdot 6\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -660.0)
   (* (* 6.0 (- y x)) z)
   (if (<= z 0.165) (fma (* y z) 6.0 x) (* (* z (- y x)) 6.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -660.0) {
		tmp = (6.0 * (y - x)) * z;
	} else if (z <= 0.165) {
		tmp = fma((y * z), 6.0, x);
	} else {
		tmp = (z * (y - x)) * 6.0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -660.0)
		tmp = Float64(Float64(6.0 * Float64(y - x)) * z);
	elseif (z <= 0.165)
		tmp = fma(Float64(y * z), 6.0, x);
	else
		tmp = Float64(Float64(z * Float64(y - x)) * 6.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -660.0], N[(N[(6.0 * N[(y - x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[z, 0.165], N[(N[(y * z), $MachinePrecision] * 6.0 + x), $MachinePrecision], N[(N[(z * N[(y - x), $MachinePrecision]), $MachinePrecision] * 6.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -660

   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. 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

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 99.4

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

   7. Applied rewrites 99.4%

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

   if -660 < z < 0.165000000000000008

   1. Initial program 99.1%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
   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. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. 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. lower-*.f64 98.7

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

   7. Applied rewrites 98.7%

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

   if 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 99.7

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

   4. Applied rewrites 99.7%

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

   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

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 98.4

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

   7. Applied rewrites 98.4%

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

   9. Applied rewrites 98.5%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -660:\\ \;\;\;\;\left(6 \cdot \left(y - x\right)\right) \cdot z\\ \mathbf{elif}\;z \leq 0.165:\\ \;\;\;\;\mathsf{fma}\left(y \cdot z, 6, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot \left(y - x\right)\right) \cdot 6\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{+134} \lor \neg \left(x \leq 1.52 \cdot 10^{+140}\right):\\ \;\;\;\;\mathsf{fma}\left(x \cdot z, -6, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot z, 6, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -2.1e+134) (not (<= x 1.52e+140)))
   (fma (* x z) -6.0 x)
   (fma (* y z) 6.0 x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2.1e+134) || !(x <= 1.52e+140)) {
		tmp = fma((x * z), -6.0, x);
	} else {
		tmp = fma((y * z), 6.0, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -2.1e+134) || !(x <= 1.52e+140))
		tmp = fma(Float64(x * z), -6.0, x);
	else
		tmp = fma(Float64(y * z), 6.0, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -2.1e+134], N[Not[LessEqual[x, 1.52e+140]], $MachinePrecision]], N[(N[(x * z), $MachinePrecision] * -6.0 + x), $MachinePrecision], N[(N[(y * z), $MachinePrecision] * 6.0 + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.1000000000000001e134 or 1.52e140 < x

   1. Initial program 98.7%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 97.3

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

   5. Applied rewrites 97.3%

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

   7. Applied rewrites 97.3%

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

   if -2.1000000000000001e134 < x < 1.52e140

   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. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. 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. lower-*.f64 84.6

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

   7. Applied rewrites 84.6%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{+134} \lor \neg \left(x \leq 1.52 \cdot 10^{+140}\right):\\ \;\;\;\;\mathsf{fma}\left(x \cdot z, -6, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot z, 6, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{-67} \lor \neg \left(x \leq 7 \cdot 10^{-97}\right):\\ \;\;\;\;\mathsf{fma}\left(x \cdot z, -6, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(6 \cdot z\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -2.9e-67) (not (<= x 7e-97)))
   (fma (* x z) -6.0 x)
   (* (* 6.0 z) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2.9e-67) || !(x <= 7e-97)) {
		tmp = fma((x * z), -6.0, x);
	} else {
		tmp = (6.0 * z) * y;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -2.9e-67) || !(x <= 7e-97))
		tmp = fma(Float64(x * z), -6.0, x);
	else
		tmp = Float64(Float64(6.0 * z) * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -2.9e-67], N[Not[LessEqual[x, 7e-97]], $MachinePrecision]], N[(N[(x * z), $MachinePrecision] * -6.0 + x), $MachinePrecision], N[(N[(6.0 * z), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.90000000000000005e-67 or 7.00000000000000038e-97 < x

   1. Initial program 99.3%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 81.0

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

   5. Applied rewrites 81.0%

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

   7. Applied rewrites 81.0%

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

   if -2.90000000000000005e-67 < x < 7.00000000000000038e-97

   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 74.7

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

   5. Applied rewrites 74.7%

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

   7. Applied rewrites 74.8%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{-67} \lor \neg \left(x \leq 7 \cdot 10^{-97}\right):\\ \;\;\;\;\mathsf{fma}\left(x \cdot z, -6, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(6 \cdot z\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{-67} \lor \neg \left(x \leq 7 \cdot 10^{-97}\right):\\ \;\;\;\;\mathsf{fma}\left(-6, z, 1\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(6 \cdot z\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -2.9e-67) (not (<= x 7e-97)))
   (* (fma -6.0 z 1.0) x)
   (* (* 6.0 z) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2.9e-67) || !(x <= 7e-97)) {
		tmp = fma(-6.0, z, 1.0) * x;
	} else {
		tmp = (6.0 * z) * y;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -2.9e-67) || !(x <= 7e-97))
		tmp = Float64(fma(-6.0, z, 1.0) * x);
	else
		tmp = Float64(Float64(6.0 * z) * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -2.9e-67], N[Not[LessEqual[x, 7e-97]], $MachinePrecision]], N[(N[(-6.0 * z + 1.0), $MachinePrecision] * x), $MachinePrecision], N[(N[(6.0 * z), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.90000000000000005e-67 or 7.00000000000000038e-97 < x

   1. Initial program 99.3%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 81.0

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

   5. Applied rewrites 81.0%

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

   if -2.90000000000000005e-67 < x < 7.00000000000000038e-97

   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 74.7

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

   5. Applied rewrites 74.7%

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

   7. Applied rewrites 74.8%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{-67} \lor \neg \left(x \leq 7 \cdot 10^{-97}\right):\\ \;\;\;\;\mathsf{fma}\left(-6, z, 1\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(6 \cdot z\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 84.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{+134}:\\ \;\;\;\;\mathsf{fma}\left(-6 \cdot x, z, x\right)\\ \mathbf{elif}\;x \leq 1.52 \cdot 10^{+140}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot z, 6, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot z, -6, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -2.1e+134)
   (fma (* -6.0 x) z x)
   (if (<= x 1.52e+140) (fma (* y z) 6.0 x) (fma (* x z) -6.0 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.1e+134) {
		tmp = fma((-6.0 * x), z, x);
	} else if (x <= 1.52e+140) {
		tmp = fma((y * z), 6.0, x);
	} else {
		tmp = fma((x * z), -6.0, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -2.1e+134)
		tmp = fma(Float64(-6.0 * x), z, x);
	elseif (x <= 1.52e+140)
		tmp = fma(Float64(y * z), 6.0, x);
	else
		tmp = fma(Float64(x * z), -6.0, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -2.1e+134], N[(N[(-6.0 * x), $MachinePrecision] * z + x), $MachinePrecision], If[LessEqual[x, 1.52e+140], N[(N[(y * z), $MachinePrecision] * 6.0 + x), $MachinePrecision], N[(N[(x * z), $MachinePrecision] * -6.0 + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.1000000000000001e134

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 100.0

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

   7. Applied rewrites 100.0%

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

   if -2.1000000000000001e134 < x < 1.52e140

   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. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. 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. lower-*.f64 84.6

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

   7. Applied rewrites 84.6%

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

   if 1.52e140 < x

   1. Initial program 97.6%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 95.1

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

   5. Applied rewrites 95.1%

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

   7. Applied rewrites 95.2%

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

Alternative 9: 61.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -660 \lor \neg \left(z \leq 0.165\right):\\ \;\;\;\;\left(-6 \cdot x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -660.0) (not (<= z 0.165))) (* (* -6.0 x) z) (* 1.0 x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -660.0) || !(z <= 0.165)) {
		tmp = (-6.0 * x) * z;
	} else {
		tmp = 1.0 * x;
	}
	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 <= (-660.0d0)) .or. (.not. (z <= 0.165d0))) then
        tmp = ((-6.0d0) * x) * z
    else
        tmp = 1.0d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -660.0) || !(z <= 0.165)) {
		tmp = (-6.0 * x) * z;
	} else {
		tmp = 1.0 * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -660.0) or not (z <= 0.165):
		tmp = (-6.0 * x) * z
	else:
		tmp = 1.0 * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -660.0) || !(z <= 0.165))
		tmp = Float64(Float64(-6.0 * x) * z);
	else
		tmp = Float64(1.0 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -660.0) || ~((z <= 0.165)))
		tmp = (-6.0 * x) * z;
	else
		tmp = 1.0 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -660.0], N[Not[LessEqual[z, 0.165]], $MachinePrecision]], N[(N[(-6.0 * x), $MachinePrecision] * z), $MachinePrecision], N[(1.0 * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -660 or 0.165000000000000008 < z

   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 inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 53.3

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

   5. Applied rewrites 53.3%

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

   6. Taylor expanded in z around 0

      \[\leadsto 1 \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 3.3%

      \[\leadsto 1 \cdot x \]

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 52.6%

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

   13. Applied rewrites 52.6%

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

   if -660 < z < 0.165000000000000008

   1. Initial program 99.1%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 68.8

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

   5. Applied rewrites 68.8%

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

   6. Taylor expanded in z around 0

      \[\leadsto 1 \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 67.6%

      \[\leadsto 1 \cdot x \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -660 \lor \neg \left(z \leq 0.165\right):\\ \;\;\;\;\left(-6 \cdot x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 36.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ 1 \cdot x \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* 1.0 x))

C

double code(double x, double y, double z) {
	return 1.0 * x;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 1.0d0 * x
end function

Java

public static double code(double x, double y, double z) {
	return 1.0 * x;
}

Python

def code(x, y, z):
	return 1.0 * x

Julia

function code(x, y, z)
	return Float64(1.0 * x)
end

MATLAB

function tmp = code(x, y, z)
	tmp = 1.0 * x;
end

Wolfram

code[x_, y_, z_] := N[(1.0 * x), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
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. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f64 61.3

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

5. Applied rewrites 61.3%

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

6. Taylor expanded in z around 0

   \[\leadsto 1 \cdot x \]
7. Step-by-step derivation

8. Applied rewrites 36.5%

   \[\leadsto 1 \cdot x \]
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 2024337 
(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)))