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

Percentage Accurate: 99.7% → 99.7%

Time: 8.9s

Alternatives: 11

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: 99.7% 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]

Derivation

1. Initial program 99.9%

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

Alternative 2: 98.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(z \cdot -6\right) \cdot \left(x - y\right)\\ \mathbf{if}\;z \leq -1.32 \cdot 10^{+19}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 0.165:\\ \;\;\;\;x + z \cdot \left(y \cdot 6\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (* z -6.0) (- x y))))
   (if (<= z -1.32e+19) t_0 (if (<= z 0.165) (+ x (* z (* y 6.0))) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (z * -6.0) * (x - y);
	double tmp;
	if (z <= -1.32e+19) {
		tmp = t_0;
	} else if (z <= 0.165) {
		tmp = x + (z * (y * 6.0));
	} else {
		tmp = t_0;
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = (z * (-6.0d0)) * (x - y)
    if (z <= (-1.32d+19)) then
        tmp = t_0
    else if (z <= 0.165d0) then
        tmp = x + (z * (y * 6.0d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (z * -6.0) * (x - y);
	double tmp;
	if (z <= -1.32e+19) {
		tmp = t_0;
	} else if (z <= 0.165) {
		tmp = x + (z * (y * 6.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (z * -6.0) * (x - y)
	tmp = 0
	if z <= -1.32e+19:
		tmp = t_0
	elif z <= 0.165:
		tmp = x + (z * (y * 6.0))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(z * -6.0) * Float64(x - y))
	tmp = 0.0
	if (z <= -1.32e+19)
		tmp = t_0;
	elseif (z <= 0.165)
		tmp = Float64(x + Float64(z * Float64(y * 6.0)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (z * -6.0) * (x - y);
	tmp = 0.0;
	if (z <= -1.32e+19)
		tmp = t_0;
	elseif (z <= 0.165)
		tmp = x + (z * (y * 6.0));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(z * -6.0), $MachinePrecision] * N[(x - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.32e+19], t$95$0, If[LessEqual[z, 0.165], N[(x + N[(z * N[(y * 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.32e19 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 z around inf

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

      1. associate-*r* N/A

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

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

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

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

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

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

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

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

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

      16. mul-1-neg N/A

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

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

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

      18. distribute-lft-out-- N/A

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

      19. neg-mul-1 N/A

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

      20. neg-sub0 N/A

          \[\leadsto \left(-6 \cdot z\right) \cdot \color{blue}{\left(0 - \left(y - x\right)\right)} \]

      21. associate-+l- N/A

          \[\leadsto \left(-6 \cdot z\right) \cdot \color{blue}{\left(\left(0 - y\right) + x\right)} \]

      22. neg-sub0 N/A

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

      23. mul-1-neg N/A

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

      24. *-lft-identity N/A

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

      25. *-inverses N/A

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

      26. associate-*l/ N/A

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

      27. associate-*r/ N/A

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

      28. associate-*r/ N/A

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

      29. *-rgt-identity N/A

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

   5. Applied rewrites 99.4%

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

   if -1.32e19 < z < 0.165000000000000008

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 99.2

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

   5. Applied rewrites 99.2%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.32 \cdot 10^{+19}:\\ \;\;\;\;\left(z \cdot -6\right) \cdot \left(x - y\right)\\ \mathbf{elif}\;z \leq 0.165:\\ \;\;\;\;x + z \cdot \left(y \cdot 6\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot -6\right) \cdot \left(x - y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(z \cdot -6\right) \cdot \left(x - y\right)\\ \mathbf{if}\;z \leq -1.32 \cdot 10^{+19}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 0.165:\\ \;\;\;\;\mathsf{fma}\left(y \cdot 6, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (* z -6.0) (- x y))))
   (if (<= z -1.32e+19) t_0 (if (<= z 0.165) (fma (* y 6.0) z x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (z * -6.0) * (x - y);
	double tmp;
	if (z <= -1.32e+19) {
		tmp = t_0;
	} else if (z <= 0.165) {
		tmp = fma((y * 6.0), z, x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(z * -6.0) * Float64(x - y))
	tmp = 0.0
	if (z <= -1.32e+19)
		tmp = t_0;
	elseif (z <= 0.165)
		tmp = fma(Float64(y * 6.0), z, x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(z * -6.0), $MachinePrecision] * N[(x - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.32e+19], t$95$0, If[LessEqual[z, 0.165], N[(N[(y * 6.0), $MachinePrecision] * z + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.32e19 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 z around inf

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

      1. associate-*r* N/A

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

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

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

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

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

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

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

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

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

      16. mul-1-neg N/A

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

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

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

      18. distribute-lft-out-- N/A

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

      19. neg-mul-1 N/A

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

      20. neg-sub0 N/A

          \[\leadsto \left(-6 \cdot z\right) \cdot \color{blue}{\left(0 - \left(y - x\right)\right)} \]

      21. associate-+l- N/A

          \[\leadsto \left(-6 \cdot z\right) \cdot \color{blue}{\left(\left(0 - y\right) + x\right)} \]

      22. neg-sub0 N/A

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

      23. mul-1-neg N/A

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

      24. *-lft-identity N/A

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

      25. *-inverses N/A

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

      26. associate-*l/ N/A

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

      27. associate-*r/ N/A

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

      28. associate-*r/ N/A

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

      29. *-rgt-identity N/A

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

   5. Applied rewrites 99.4%

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

   if -1.32e19 < z < 0.165000000000000008

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 99.2

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

   5. Applied rewrites 99.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lower-fma.f64 99.1

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 99.1

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

   7. Applied rewrites 99.1%

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

Alternative 4: 86.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y \cdot 6, z, x\right)\\ \mathbf{if}\;y \leq -2.15 \cdot 10^{-58}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{-80}:\\ \;\;\;\;\mathsf{fma}\left(z, x \cdot -6, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (* y 6.0) z x)))
   (if (<= y -2.15e-58) t_0 (if (<= y 4.6e-80) (fma z (* x -6.0) x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma((y * 6.0), z, x);
	double tmp;
	if (y <= -2.15e-58) {
		tmp = t_0;
	} else if (y <= 4.6e-80) {
		tmp = fma(z, (x * -6.0), x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(Float64(y * 6.0), z, x)
	tmp = 0.0
	if (y <= -2.15e-58)
		tmp = t_0;
	elseif (y <= 4.6e-80)
		tmp = fma(z, Float64(x * -6.0), x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y * 6.0), $MachinePrecision] * z + x), $MachinePrecision]}, If[LessEqual[y, -2.15e-58], t$95$0, If[LessEqual[y, 4.6e-80], N[(z * N[(x * -6.0), $MachinePrecision] + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.15e-58 or 4.5999999999999996e-80 < y

   1. Initial program 99.9%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
   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. lower-*.f64 91.2

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

   5. Applied rewrites 91.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lower-fma.f64 91.1

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 91.1

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

   7. Applied rewrites 91.1%

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

   if -2.15e-58 < y < 4.5999999999999996e-80

   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

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 89.6

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

   5. Applied rewrites 89.6%

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

Alternative 5: 73.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(z, x \cdot -6, x\right)\\ \mathbf{if}\;x \leq -1.85 \cdot 10^{-37}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 4.3 \cdot 10^{-191}:\\ \;\;\;\;y \cdot \left(6 \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma z (* x -6.0) x)))
   (if (<= x -1.85e-37) t_0 (if (<= x 4.3e-191) (* y (* 6.0 z)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma(z, (x * -6.0), x);
	double tmp;
	if (x <= -1.85e-37) {
		tmp = t_0;
	} else if (x <= 4.3e-191) {
		tmp = y * (6.0 * z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(z, Float64(x * -6.0), x)
	tmp = 0.0
	if (x <= -1.85e-37)
		tmp = t_0;
	elseif (x <= 4.3e-191)
		tmp = Float64(y * Float64(6.0 * z));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(x * -6.0), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[x, -1.85e-37], t$95$0, If[LessEqual[x, 4.3e-191], N[(y * N[(6.0 * z), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.85e-37 or 4.29999999999999983e-191 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 83.9

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

   5. Applied rewrites 83.9%

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

   if -1.85e-37 < x < 4.29999999999999983e-191

   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. lower-*.f64 N/A

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

      2. lower-*.f64 76.7

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

   5. Applied rewrites 76.7%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 76.9

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

   7. Applied rewrites 76.9%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.85 \cdot 10^{-37}:\\ \;\;\;\;\mathsf{fma}\left(z, x \cdot -6, x\right)\\ \mathbf{elif}\;x \leq 4.3 \cdot 10^{-191}:\\ \;\;\;\;y \cdot \left(6 \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, x \cdot -6, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \mathsf{fma}\left(z, -6, 1\right)\\ \mathbf{if}\;x \leq -1.85 \cdot 10^{-37}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 4.3 \cdot 10^{-191}:\\ \;\;\;\;y \cdot \left(6 \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (fma z -6.0 1.0))))
   (if (<= x -1.85e-37) t_0 (if (<= x 4.3e-191) (* y (* 6.0 z)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x * fma(z, -6.0, 1.0);
	double tmp;
	if (x <= -1.85e-37) {
		tmp = t_0;
	} else if (x <= 4.3e-191) {
		tmp = y * (6.0 * z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x * fma(z, -6.0, 1.0))
	tmp = 0.0
	if (x <= -1.85e-37)
		tmp = t_0;
	elseif (x <= 4.3e-191)
		tmp = Float64(y * Float64(6.0 * z));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[(z * -6.0 + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.85e-37], t$95$0, If[LessEqual[x, 4.3e-191], N[(y * N[(6.0 * z), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.85e-37 or 4.29999999999999983e-191 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 83.9

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

   5. Applied rewrites 83.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. distribute-lft1-in N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 83.8

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

   7. Applied rewrites 83.8%

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

   if -1.85e-37 < x < 4.29999999999999983e-191

   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. lower-*.f64 N/A

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

      2. lower-*.f64 76.7

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

   5. Applied rewrites 76.7%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 76.9

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

   7. Applied rewrites 76.9%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.85 \cdot 10^{-37}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(z, -6, 1\right)\\ \mathbf{elif}\;x \leq 4.3 \cdot 10^{-191}:\\ \;\;\;\;y \cdot \left(6 \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(z, -6, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 61.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -0.165:\\ \;\;\;\;x \cdot \left(z \cdot -6\right)\\ \mathbf{elif}\;z \leq 0.16:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(x \cdot -6\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -0.165) (* x (* z -6.0)) (if (<= z 0.16) x (* z (* x -6.0)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -0.165) {
		tmp = x * (z * -6.0);
	} else if (z <= 0.16) {
		tmp = x;
	} else {
		tmp = z * (x * -6.0);
	}
	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 <= (-0.165d0)) then
        tmp = x * (z * (-6.0d0))
    else if (z <= 0.16d0) then
        tmp = x
    else
        tmp = z * (x * (-6.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -0.165) {
		tmp = x * (z * -6.0);
	} else if (z <= 0.16) {
		tmp = x;
	} else {
		tmp = z * (x * -6.0);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -0.165:
		tmp = x * (z * -6.0)
	elif z <= 0.16:
		tmp = x
	else:
		tmp = z * (x * -6.0)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -0.165)
		tmp = Float64(x * Float64(z * -6.0));
	elseif (z <= 0.16)
		tmp = x;
	else
		tmp = Float64(z * Float64(x * -6.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -0.165)
		tmp = x * (z * -6.0);
	elseif (z <= 0.16)
		tmp = x;
	else
		tmp = z * (x * -6.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -0.165], N[(x * N[(z * -6.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 0.16], x, N[(z * N[(x * -6.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -0.165000000000000008

   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 inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 60.4

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

   5. Applied rewrites 60.4%

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

   6. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 60.4

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

   8. Applied rewrites 60.4%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 60.5

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

   10. Applied rewrites 60.5%

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

   if -0.165000000000000008 < z < 0.160000000000000003

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 72.1

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

   5. Applied rewrites 72.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. distribute-lft1-in N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 72.1

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

   7. Applied rewrites 72.1%

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

   8. Taylor expanded in z around 0

      \[\leadsto \color{blue}{1} \cdot x \]
   9. Step-by-step derivation

   10. Applied rewrites 71.3%

       \[\leadsto \color{blue}{1} \cdot x \]
   11. Step-by-step derivation

       1. *-lft-identity 71.3

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

   12. Applied rewrites 71.3%

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

   if 0.160000000000000003 < 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

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 56.8

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

   5. Applied rewrites 56.8%

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

   6. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 56.1

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

   8. Applied rewrites 56.1%

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 56.2

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

   10. Applied rewrites 56.2%

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

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.165:\\ \;\;\;\;x \cdot \left(z \cdot -6\right)\\ \mathbf{elif}\;z \leq 0.16:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(x \cdot -6\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 61.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -0.165:\\ \;\;\;\;-6 \cdot \left(x \cdot z\right)\\ \mathbf{elif}\;z \leq 0.16:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(x \cdot -6\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -0.165) (* -6.0 (* x z)) (if (<= z 0.16) x (* z (* x -6.0)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -0.165) {
		tmp = -6.0 * (x * z);
	} else if (z <= 0.16) {
		tmp = x;
	} else {
		tmp = z * (x * -6.0);
	}
	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 <= (-0.165d0)) then
        tmp = (-6.0d0) * (x * z)
    else if (z <= 0.16d0) then
        tmp = x
    else
        tmp = z * (x * (-6.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -0.165) {
		tmp = -6.0 * (x * z);
	} else if (z <= 0.16) {
		tmp = x;
	} else {
		tmp = z * (x * -6.0);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -0.165:
		tmp = -6.0 * (x * z)
	elif z <= 0.16:
		tmp = x
	else:
		tmp = z * (x * -6.0)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -0.165)
		tmp = Float64(-6.0 * Float64(x * z));
	elseif (z <= 0.16)
		tmp = x;
	else
		tmp = Float64(z * Float64(x * -6.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -0.165)
		tmp = -6.0 * (x * z);
	elseif (z <= 0.16)
		tmp = x;
	else
		tmp = z * (x * -6.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -0.165], N[(-6.0 * N[(x * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 0.16], x, N[(z * N[(x * -6.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -0.165000000000000008

   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 inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 60.4

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

   5. Applied rewrites 60.4%

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

   6. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 60.4

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

   8. Applied rewrites 60.4%

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

   if -0.165000000000000008 < z < 0.160000000000000003

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 72.1

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

   5. Applied rewrites 72.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. distribute-lft1-in N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 72.1

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

   7. Applied rewrites 72.1%

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

   8. Taylor expanded in z around 0

      \[\leadsto \color{blue}{1} \cdot x \]
   9. Step-by-step derivation

   10. Applied rewrites 71.3%

       \[\leadsto \color{blue}{1} \cdot x \]
   11. Step-by-step derivation

       1. *-lft-identity 71.3

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

   12. Applied rewrites 71.3%

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

   if 0.160000000000000003 < 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

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 56.8

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

   5. Applied rewrites 56.8%

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

   6. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 56.1

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

   8. Applied rewrites 56.1%

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 56.2

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

   10. Applied rewrites 56.2%

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

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.165:\\ \;\;\;\;-6 \cdot \left(x \cdot z\right)\\ \mathbf{elif}\;z \leq 0.16:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(x \cdot -6\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 61.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -6 \cdot \left(x \cdot z\right)\\ \mathbf{if}\;z \leq -0.165:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 0.16:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* -6.0 (* x z)))) (if (<= z -0.165) t_0 (if (<= z 0.16) x t_0))))

C

double code(double x, double y, double z) {
	double t_0 = -6.0 * (x * z);
	double tmp;
	if (z <= -0.165) {
		tmp = t_0;
	} else if (z <= 0.16) {
		tmp = x;
	} else {
		tmp = t_0;
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = (-6.0d0) * (x * z)
    if (z <= (-0.165d0)) then
        tmp = t_0
    else if (z <= 0.16d0) then
        tmp = x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = -6.0 * (x * z);
	double tmp;
	if (z <= -0.165) {
		tmp = t_0;
	} else if (z <= 0.16) {
		tmp = x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = -6.0 * (x * z)
	tmp = 0
	if z <= -0.165:
		tmp = t_0
	elif z <= 0.16:
		tmp = x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(-6.0 * Float64(x * z))
	tmp = 0.0
	if (z <= -0.165)
		tmp = t_0;
	elseif (z <= 0.16)
		tmp = x;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = -6.0 * (x * z);
	tmp = 0.0;
	if (z <= -0.165)
		tmp = t_0;
	elseif (z <= 0.16)
		tmp = x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(-6.0 * N[(x * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -0.165], t$95$0, If[LessEqual[z, 0.16], x, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -0.165000000000000008 or 0.160000000000000003 < 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

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 58.4

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

   5. Applied rewrites 58.4%

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

   6. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 58.0

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

   8. Applied rewrites 58.0%

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

   if -0.165000000000000008 < z < 0.160000000000000003

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 72.1

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

   5. Applied rewrites 72.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. distribute-lft1-in N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 72.1

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

   7. Applied rewrites 72.1%

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

   8. Taylor expanded in z around 0

      \[\leadsto \color{blue}{1} \cdot x \]
   9. Step-by-step derivation

   10. Applied rewrites 71.3%

       \[\leadsto \color{blue}{1} \cdot x \]
   11. Step-by-step derivation

       1. *-lft-identity 71.3

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

   12. Applied rewrites 71.3%

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

Alternative 10: 99.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. +-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. lower-fma.f64 99.9

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

4. Applied rewrites 99.9%

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

Alternative 11: 36.4% accurate, 17.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return 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 = x
end function

Java

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

Python

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

Julia

function code(x, y, z)
	return x
end

MATLAB

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

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around inf

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. *-lft-identity N/A

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

   6. lower-fma.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 65.1

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

5. Applied rewrites 65.1%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. lift-*.f64 N/A

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

   6. distribute-lft1-in N/A

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

   7. lower-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lower-fma.f64 65.1

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

7. Applied rewrites 65.1%

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

8. Taylor expanded in z around 0

   \[\leadsto \color{blue}{1} \cdot x \]
9. Step-by-step derivation

10. Applied rewrites 36.3%

    \[\leadsto \color{blue}{1} \cdot x \]
11. Step-by-step derivation

    1. *-lft-identity 36.3

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

12. Applied rewrites 36.3%

    \[\leadsto \color{blue}{x} \]
13. 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 2024219 
(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)))