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

Percentage Accurate: 99.7% → 99.8%

Time: 8.4s

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

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

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(Float64(y - x) * z), 6.0, x)
end

Wolfram

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

Derivation

1. Initial program 99.5%

   \[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
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. Final simplification 99.8%

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

Alternative 2: 98.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -0.2:\\ \;\;\;\;\left(\left(y - x\right) \cdot z\right) \cdot 6\\ \mathbf{elif}\;z \leq 0.00015:\\ \;\;\;\;\left(6 \cdot y\right) \cdot z + x\\ \mathbf{else}:\\ \;\;\;\;\left(6 \cdot \left(y - x\right)\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -0.2)
   (* (* (- y x) z) 6.0)
   (if (<= z 0.00015) (+ (* (* 6.0 y) z) x) (* (* 6.0 (- y x)) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -0.2) {
		tmp = ((y - x) * z) * 6.0;
	} else if (z <= 0.00015) {
		tmp = ((6.0 * y) * z) + x;
	} else {
		tmp = (6.0 * (y - x)) * 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 <= (-0.2d0)) then
        tmp = ((y - x) * z) * 6.0d0
    else if (z <= 0.00015d0) then
        tmp = ((6.0d0 * y) * z) + x
    else
        tmp = (6.0d0 * (y - x)) * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -0.2) {
		tmp = ((y - x) * z) * 6.0;
	} else if (z <= 0.00015) {
		tmp = ((6.0 * y) * z) + x;
	} else {
		tmp = (6.0 * (y - x)) * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -0.2:
		tmp = ((y - x) * z) * 6.0
	elif z <= 0.00015:
		tmp = ((6.0 * y) * z) + x
	else:
		tmp = (6.0 * (y - x)) * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -0.2)
		tmp = Float64(Float64(Float64(y - x) * z) * 6.0);
	elseif (z <= 0.00015)
		tmp = Float64(Float64(Float64(6.0 * y) * z) + x);
	else
		tmp = Float64(Float64(6.0 * Float64(y - x)) * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -0.2)
		tmp = ((y - x) * z) * 6.0;
	elseif (z <= 0.00015)
		tmp = ((6.0 * y) * z) + x;
	else
		tmp = (6.0 * (y - x)) * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -0.2], N[(N[(N[(y - x), $MachinePrecision] * z), $MachinePrecision] * 6.0), $MachinePrecision], If[LessEqual[z, 0.00015], N[(N[(N[(6.0 * y), $MachinePrecision] * z), $MachinePrecision] + x), $MachinePrecision], N[(N[(6.0 * N[(y - x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -0.20000000000000001

   1. Initial program 99.6%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{6 \cdot \left(z \cdot \left(y - x\right)\right)} \]
   4. 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 99.7

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

   5. Applied rewrites 99.7%

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

   if -0.20000000000000001 < z < 1.49999999999999987e-4

   1. Initial program 99.2%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 98.7

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

   5. Applied rewrites 98.7%

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

   if 1.49999999999999987e-4 < 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. *-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 z around inf

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

      1. associate-*r* N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. neg-mul-1 N/A

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

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

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

      10. metadata-eval N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

   7. Applied rewrites 99.0%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.2:\\ \;\;\;\;\left(\left(y - x\right) \cdot z\right) \cdot 6\\ \mathbf{elif}\;z \leq 0.00015:\\ \;\;\;\;\left(6 \cdot y\right) \cdot z + x\\ \mathbf{else}:\\ \;\;\;\;\left(6 \cdot \left(y - x\right)\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -0.162:\\ \;\;\;\;\left(\left(y - x\right) \cdot z\right) \cdot 6\\ \mathbf{elif}\;z \leq 0.00015:\\ \;\;\;\;\mathsf{fma}\left(y \cdot z, 6, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(6 \cdot \left(y - x\right)\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -0.162)
   (* (* (- y x) z) 6.0)
   (if (<= z 0.00015) (fma (* y z) 6.0 x) (* (* 6.0 (- y x)) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -0.162) {
		tmp = ((y - x) * z) * 6.0;
	} else if (z <= 0.00015) {
		tmp = fma((y * z), 6.0, x);
	} else {
		tmp = (6.0 * (y - x)) * z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -0.162)
		tmp = Float64(Float64(Float64(y - x) * z) * 6.0);
	elseif (z <= 0.00015)
		tmp = fma(Float64(y * z), 6.0, x);
	else
		tmp = Float64(Float64(6.0 * Float64(y - x)) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -0.162], N[(N[(N[(y - x), $MachinePrecision] * z), $MachinePrecision] * 6.0), $MachinePrecision], If[LessEqual[z, 0.00015], N[(N[(y * z), $MachinePrecision] * 6.0 + x), $MachinePrecision], N[(N[(6.0 * N[(y - x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -0.162000000000000005

   1. Initial program 99.6%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{6 \cdot \left(z \cdot \left(y - x\right)\right)} \]
   4. 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 99.7

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

   5. Applied rewrites 99.7%

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

   if -0.162000000000000005 < z < 1.49999999999999987e-4

   1. Initial program 99.2%

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

      \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot z}, 6, x\right) \]
   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 1.49999999999999987e-4 < 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. *-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 z around inf

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

      1. associate-*r* N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. neg-mul-1 N/A

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

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

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

      10. metadata-eval N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

   7. Applied rewrites 99.0%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.162:\\ \;\;\;\;\left(\left(y - x\right) \cdot z\right) \cdot 6\\ \mathbf{elif}\;z \leq 0.00015:\\ \;\;\;\;\mathsf{fma}\left(y \cdot z, 6, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(6 \cdot \left(y - x\right)\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(y - x\right) \cdot z\right) \cdot 6\\ \mathbf{if}\;z \leq -0.162:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 0.00015:\\ \;\;\;\;\mathsf{fma}\left(y \cdot z, 6, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (* (- y x) z) 6.0)))
   (if (<= z -0.162) t_0 (if (<= z 0.00015) (fma (* y z) 6.0 x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = ((y - x) * z) * 6.0;
	double tmp;
	if (z <= -0.162) {
		tmp = t_0;
	} else if (z <= 0.00015) {
		tmp = fma((y * z), 6.0, x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(y - x) * z) * 6.0)
	tmp = 0.0
	if (z <= -0.162)
		tmp = t_0;
	elseif (z <= 0.00015)
		tmp = fma(Float64(y * z), 6.0, x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(y - x), $MachinePrecision] * z), $MachinePrecision] * 6.0), $MachinePrecision]}, If[LessEqual[z, -0.162], t$95$0, If[LessEqual[z, 0.00015], N[(N[(y * z), $MachinePrecision] * 6.0 + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -0.162000000000000005 or 1.49999999999999987e-4 < z

   1. Initial program 99.7%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{6 \cdot \left(z \cdot \left(y - x\right)\right)} \]
   4. 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 99.3

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

   5. Applied rewrites 99.3%

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

   if -0.162000000000000005 < z < 1.49999999999999987e-4

   1. Initial program 99.2%

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

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

Alternative 5: 85.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x \cdot z, -6, x\right)\\ \mathbf{if}\;x \leq -3.8 \cdot 10^{+148}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{+113}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot z, 6, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (* x z) -6.0 x)))
   (if (<= x -3.8e+148) t_0 (if (<= x 2.6e+113) (fma (* y z) 6.0 x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma((x * z), -6.0, x);
	double tmp;
	if (x <= -3.8e+148) {
		tmp = t_0;
	} else if (x <= 2.6e+113) {
		tmp = fma((y * z), 6.0, x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(Float64(x * z), -6.0, x)
	tmp = 0.0
	if (x <= -3.8e+148)
		tmp = t_0;
	elseif (x <= 2.6e+113)
		tmp = fma(Float64(y * z), 6.0, x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x * z), $MachinePrecision] * -6.0 + x), $MachinePrecision]}, If[LessEqual[x, -3.8e+148], t$95$0, If[LessEqual[x, 2.6e+113], N[(N[(y * z), $MachinePrecision] * 6.0 + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.7999999999999998e148 or 2.5999999999999999e113 < x

   1. Initial program 98.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 97.4

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

   5. Applied rewrites 97.4%

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

   if -3.7999999999999998e148 < x < 2.5999999999999999e113

   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 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in y around inf

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

      1. lower-*.f64 85.8

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

   7. Applied rewrites 85.8%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{+148}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot z, -6, x\right)\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{+113}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot z, 6, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot z, -6, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x \cdot z, -6, x\right)\\ \mathbf{if}\;x \leq -1.7 \cdot 10^{-153}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-113}:\\ \;\;\;\;\left(6 \cdot y\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (* x z) -6.0 x)))
   (if (<= x -1.7e-153) t_0 (if (<= x 3e-113) (* (* 6.0 y) z) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma((x * z), -6.0, x);
	double tmp;
	if (x <= -1.7e-153) {
		tmp = t_0;
	} else if (x <= 3e-113) {
		tmp = (6.0 * y) * z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(Float64(x * z), -6.0, x)
	tmp = 0.0
	if (x <= -1.7e-153)
		tmp = t_0;
	elseif (x <= 3e-113)
		tmp = Float64(Float64(6.0 * y) * z);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x * z), $MachinePrecision] * -6.0 + x), $MachinePrecision]}, If[LessEqual[x, -1.7e-153], t$95$0, If[LessEqual[x, 3e-113], N[(N[(6.0 * y), $MachinePrecision] * z), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.6999999999999999e-153 or 3.0000000000000001e-113 < x

   1. Initial program 99.4%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 79.2

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

   5. Applied rewrites 79.2%

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

   if -1.6999999999999999e-153 < x < 3.0000000000000001e-113

   1. Initial program 99.7%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{6 \cdot \left(y \cdot z\right)} \]
   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. lower-*.f64 79.8

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

   5. Applied rewrites 79.8%

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

   7. Applied rewrites 79.9%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{-153}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot z, -6, x\right)\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-113}:\\ \;\;\;\;\left(6 \cdot y\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot z, -6, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 51.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{+148}:\\ \;\;\;\;\left(-6 \cdot z\right) \cdot x\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{+113}:\\ \;\;\;\;\left(6 \cdot y\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\left(-6 \cdot x\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -3.8e+148)
   (* (* -6.0 z) x)
   (if (<= x 2.6e+113) (* (* 6.0 y) z) (* (* -6.0 x) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -3.8e+148) {
		tmp = (-6.0 * z) * x;
	} else if (x <= 2.6e+113) {
		tmp = (6.0 * y) * z;
	} else {
		tmp = (-6.0 * x) * 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 (x <= (-3.8d+148)) then
        tmp = ((-6.0d0) * z) * x
    else if (x <= 2.6d+113) then
        tmp = (6.0d0 * y) * z
    else
        tmp = ((-6.0d0) * x) * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -3.8e+148) {
		tmp = (-6.0 * z) * x;
	} else if (x <= 2.6e+113) {
		tmp = (6.0 * y) * z;
	} else {
		tmp = (-6.0 * x) * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -3.8e+148:
		tmp = (-6.0 * z) * x
	elif x <= 2.6e+113:
		tmp = (6.0 * y) * z
	else:
		tmp = (-6.0 * x) * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -3.8e+148)
		tmp = Float64(Float64(-6.0 * z) * x);
	elseif (x <= 2.6e+113)
		tmp = Float64(Float64(6.0 * y) * z);
	else
		tmp = Float64(Float64(-6.0 * x) * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -3.8e+148)
		tmp = (-6.0 * z) * x;
	elseif (x <= 2.6e+113)
		tmp = (6.0 * y) * z;
	else
		tmp = (-6.0 * x) * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -3.8e+148], N[(N[(-6.0 * z), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[x, 2.6e+113], N[(N[(6.0 * y), $MachinePrecision] * z), $MachinePrecision], N[(N[(-6.0 * x), $MachinePrecision] * z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.7999999999999998e148

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{6 \cdot \left(z \cdot \left(y - x\right)\right)} \]
   4. 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 55.5

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

   5. Applied rewrites 55.5%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 52.4%

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

   10. Applied rewrites 52.5%

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

   if -3.7999999999999998e148 < x < 2.5999999999999999e113

   1. Initial program 99.8%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{6 \cdot \left(y \cdot z\right)} \]
   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. lower-*.f64 58.2

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

   5. Applied rewrites 58.2%

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

   7. Applied rewrites 58.2%

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

   if 2.5999999999999999e113 < x

   1. Initial program 98.0%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
   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 100.0

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

   4. Applied rewrites 100.0%

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

   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. associate-*r* N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. neg-mul-1 N/A

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

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

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

      10. metadata-eval N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

   7. Applied rewrites 58.3%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 56.1%

       \[\leadsto \left(-6 \cdot x\right) \cdot z \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 51.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{+148}:\\ \;\;\;\;\left(-6 \cdot z\right) \cdot x\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{+113}:\\ \;\;\;\;\left(6 \cdot y\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;-6 \cdot \left(x \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -3.8e+148)
   (* (* -6.0 z) x)
   (if (<= x 2.6e+113) (* (* 6.0 y) z) (* -6.0 (* x z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -3.8e+148) {
		tmp = (-6.0 * z) * x;
	} else if (x <= 2.6e+113) {
		tmp = (6.0 * y) * z;
	} else {
		tmp = -6.0 * (x * 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 (x <= (-3.8d+148)) then
        tmp = ((-6.0d0) * z) * x
    else if (x <= 2.6d+113) then
        tmp = (6.0d0 * y) * z
    else
        tmp = (-6.0d0) * (x * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -3.8e+148) {
		tmp = (-6.0 * z) * x;
	} else if (x <= 2.6e+113) {
		tmp = (6.0 * y) * z;
	} else {
		tmp = -6.0 * (x * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -3.8e+148:
		tmp = (-6.0 * z) * x
	elif x <= 2.6e+113:
		tmp = (6.0 * y) * z
	else:
		tmp = -6.0 * (x * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -3.8e+148)
		tmp = Float64(Float64(-6.0 * z) * x);
	elseif (x <= 2.6e+113)
		tmp = Float64(Float64(6.0 * y) * z);
	else
		tmp = Float64(-6.0 * Float64(x * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -3.8e+148)
		tmp = (-6.0 * z) * x;
	elseif (x <= 2.6e+113)
		tmp = (6.0 * y) * z;
	else
		tmp = -6.0 * (x * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -3.8e+148], N[(N[(-6.0 * z), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[x, 2.6e+113], N[(N[(6.0 * y), $MachinePrecision] * z), $MachinePrecision], N[(-6.0 * N[(x * z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.7999999999999998e148

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{6 \cdot \left(z \cdot \left(y - x\right)\right)} \]
   4. 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 55.5

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

   5. Applied rewrites 55.5%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 52.4%

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

   10. Applied rewrites 52.5%

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

   if -3.7999999999999998e148 < x < 2.5999999999999999e113

   1. Initial program 99.8%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{6 \cdot \left(y \cdot z\right)} \]
   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. lower-*.f64 58.2

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

   5. Applied rewrites 58.2%

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

   7. Applied rewrites 58.2%

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

   if 2.5999999999999999e113 < x

   1. Initial program 98.0%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{6 \cdot \left(z \cdot \left(y - x\right)\right)} \]
   4. 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 58.1

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

   5. Applied rewrites 58.1%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 55.9%

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

4. Final simplification 57.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{+148}:\\ \;\;\;\;\left(-6 \cdot z\right) \cdot x\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{+113}:\\ \;\;\;\;\left(6 \cdot y\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;-6 \cdot \left(x \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 28.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return -6.0 * (x * 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 = (-6.0d0) * (x * z)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in z around inf

   \[\leadsto \color{blue}{6 \cdot \left(z \cdot \left(y - x\right)\right)} \]
4. 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 67.4

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

5. Applied rewrites 67.4%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 29.0%

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

9. Final simplification 29.0%

   \[\leadsto -6 \cdot \left(x \cdot z\right) \]
10. Add Preprocessing

Alternative 10: 28.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return (-6.0 * z) * 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 = ((-6.0d0) * z) * x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in z around inf

   \[\leadsto \color{blue}{6 \cdot \left(z \cdot \left(y - x\right)\right)} \]
4. 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 67.4

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

5. Applied rewrites 67.4%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 29.0%

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

10. Applied rewrites 29.0%

    \[\leadsto \left(-6 \cdot z\right) \cdot x \]
11. 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 2024235 
(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)))