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

Percentage Accurate: 99.6% → 99.8%

Time: 9.4s

Alternatives: 13

Speedup: 1.1×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.6% 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, 1.0× speedup

Math

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

FPCore

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

C

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

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) * z) * 6.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 99.8

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

4. Applied rewrites 99.8%

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

Alternative 2: 97.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -0.155:\\ \;\;\;\;\left(\left(y - x\right) \cdot z\right) \cdot 6\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{-23}:\\ \;\;\;\;x + z \cdot \left(y \cdot 6\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot -6\right) \cdot \left(x - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -0.155)
   (* (* (- y x) z) 6.0)
   (if (<= z 6.8e-23) (+ x (* z (* y 6.0))) (* (* z -6.0) (- x y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -0.155) {
		tmp = ((y - x) * z) * 6.0;
	} else if (z <= 6.8e-23) {
		tmp = x + (z * (y * 6.0));
	} else {
		tmp = (z * -6.0) * (x - y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-0.155d0)) then
        tmp = ((y - x) * z) * 6.0d0
    else if (z <= 6.8d-23) then
        tmp = x + (z * (y * 6.0d0))
    else
        tmp = (z * (-6.0d0)) * (x - y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -0.155) {
		tmp = ((y - x) * z) * 6.0;
	} else if (z <= 6.8e-23) {
		tmp = x + (z * (y * 6.0));
	} else {
		tmp = (z * -6.0) * (x - y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -0.155:
		tmp = ((y - x) * z) * 6.0
	elif z <= 6.8e-23:
		tmp = x + (z * (y * 6.0))
	else:
		tmp = (z * -6.0) * (x - y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -0.155)
		tmp = Float64(Float64(Float64(y - x) * z) * 6.0);
	elseif (z <= 6.8e-23)
		tmp = Float64(x + Float64(z * Float64(y * 6.0)));
	else
		tmp = Float64(Float64(z * -6.0) * Float64(x - y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -0.155)
		tmp = ((y - x) * z) * 6.0;
	elseif (z <= 6.8e-23)
		tmp = x + (z * (y * 6.0));
	else
		tmp = (z * -6.0) * (x - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -0.155], N[(N[(N[(y - x), $MachinePrecision] * z), $MachinePrecision] * 6.0), $MachinePrecision], If[LessEqual[z, 6.8e-23], N[(x + N[(z * N[(y * 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(z * -6.0), $MachinePrecision] * N[(x - y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -0.154999999999999999

   1. Initial program 99.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-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. lower-*.f64 99.6

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

   4. Applied rewrites 99.6%

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

   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. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 98.7

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

   7. Applied rewrites 98.7%

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

   if -0.154999999999999999 < z < 6.8000000000000001e-23

   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 99.3

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

   5. Applied rewrites 99.3%

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

   if 6.8000000000000001e-23 < z

   1. Initial program 98.6%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
   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. 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 98.1%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.155:\\ \;\;\;\;\left(\left(y - x\right) \cdot z\right) \cdot 6\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{-23}:\\ \;\;\;\;x + z \cdot \left(y \cdot 6\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot -6\right) \cdot \left(x - y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -0.155:\\ \;\;\;\;\left(\left(y - x\right) \cdot z\right) \cdot 6\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{-23}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot 6, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot -6\right) \cdot \left(x - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -0.155)
   (* (* (- y x) z) 6.0)
   (if (<= z 6.8e-23) (fma (* y 6.0) z x) (* (* z -6.0) (- x y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -0.155) {
		tmp = ((y - x) * z) * 6.0;
	} else if (z <= 6.8e-23) {
		tmp = fma((y * 6.0), z, x);
	} else {
		tmp = (z * -6.0) * (x - y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -0.155)
		tmp = Float64(Float64(Float64(y - x) * z) * 6.0);
	elseif (z <= 6.8e-23)
		tmp = fma(Float64(y * 6.0), z, x);
	else
		tmp = Float64(Float64(z * -6.0) * Float64(x - y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -0.155], N[(N[(N[(y - x), $MachinePrecision] * z), $MachinePrecision] * 6.0), $MachinePrecision], If[LessEqual[z, 6.8e-23], N[(N[(y * 6.0), $MachinePrecision] * z + x), $MachinePrecision], N[(N[(z * -6.0), $MachinePrecision] * N[(x - y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -0.154999999999999999

   1. Initial program 99.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-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. lower-*.f64 99.6

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

   4. Applied rewrites 99.6%

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

   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. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 98.7

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

   7. Applied rewrites 98.7%

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

   if -0.154999999999999999 < z < 6.8000000000000001e-23

   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 99.3

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

   5. Applied rewrites 99.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.3

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

   7. Applied rewrites 99.3%

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

   if 6.8000000000000001e-23 < z

   1. Initial program 98.6%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
   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. 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 98.1%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.155:\\ \;\;\;\;\left(\left(y - x\right) \cdot z\right) \cdot 6\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{-23}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot 6, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot -6\right) \cdot \left(x - y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(z \cdot -6\right) \cdot \left(x - y\right)\\ \mathbf{if}\;z \leq -0.17:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{-23}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot 6, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (* z -6.0) (- x y))))
   (if (<= z -0.17) t_0 (if (<= z 6.8e-23) (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 <= -0.17) {
		tmp = t_0;
	} else if (z <= 6.8e-23) {
		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 <= -0.17)
		tmp = t_0;
	elseif (z <= 6.8e-23)
		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, -0.17], t$95$0, If[LessEqual[z, 6.8e-23], N[(N[(y * 6.0), $MachinePrecision] * z + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -0.170000000000000012 or 6.8000000000000001e-23 < z

   1. Initial program 99.1%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
   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 98.3%

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

   if -0.170000000000000012 < z < 6.8000000000000001e-23

   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 99.3

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

   5. Applied rewrites 99.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.3

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

   7. Applied rewrites 99.3%

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

Alternative 5: 86.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{-79}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot z, 6, x\right)\\ \mathbf{elif}\;y \leq 2.06 \cdot 10^{-51}:\\ \;\;\;\;\mathsf{fma}\left(z, x \cdot -6, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot 6, z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -3.2e-79)
   (fma (* y z) 6.0 x)
   (if (<= y 2.06e-51) (fma z (* x -6.0) x) (fma (* y 6.0) z x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.2e-79) {
		tmp = fma((y * z), 6.0, x);
	} else if (y <= 2.06e-51) {
		tmp = fma(z, (x * -6.0), x);
	} else {
		tmp = fma((y * 6.0), z, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -3.2e-79)
		tmp = fma(Float64(y * z), 6.0, x);
	elseif (y <= 2.06e-51)
		tmp = fma(z, Float64(x * -6.0), x);
	else
		tmp = fma(Float64(y * 6.0), z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -3.2e-79], N[(N[(y * z), $MachinePrecision] * 6.0 + x), $MachinePrecision], If[LessEqual[y, 2.06e-51], N[(z * N[(x * -6.0), $MachinePrecision] + x), $MachinePrecision], N[(N[(y * 6.0), $MachinePrecision] * z + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.19999999999999988e-79

   1. Initial program 98.8%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
   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. lower-*.f64 99.7

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

   4. Applied rewrites 99.7%

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

   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 88.1

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

   7. Applied rewrites 88.1%

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

   if -3.19999999999999988e-79 < y < 2.0600000000000001e-51

   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 89.3

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

   5. Applied rewrites 89.3%

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

   if 2.0600000000000001e-51 < y

   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

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

      1. lower-*.f64 90.4

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

   5. Applied rewrites 90.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 90.3

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

   7. Applied rewrites 90.3%

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

Alternative 6: 86.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y \cdot 6, z, x\right)\\ \mathbf{if}\;y \leq -3.2 \cdot 10^{-79}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2.06 \cdot 10^{-51}:\\ \;\;\;\;\mathsf{fma}\left(z, x \cdot -6, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (* y 6.0) z x)))
   (if (<= y -3.2e-79) t_0 (if (<= y 2.06e-51) (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 <= -3.2e-79) {
		tmp = t_0;
	} else if (y <= 2.06e-51) {
		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 <= -3.2e-79)
		tmp = t_0;
	elseif (y <= 2.06e-51)
		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, -3.2e-79], t$95$0, If[LessEqual[y, 2.06e-51], N[(z * N[(x * -6.0), $MachinePrecision] + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.19999999999999988e-79 or 2.0600000000000001e-51 < y

   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. lower-*.f64 88.4

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

   5. Applied rewrites 88.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 88.4

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

   7. Applied rewrites 88.4%

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

   if -3.19999999999999988e-79 < y < 2.0600000000000001e-51

   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 89.3

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

   5. Applied rewrites 89.3%

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

Alternative 7: 74.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(z \cdot 6\right)\\ \mathbf{if}\;y \leq -8.5 \cdot 10^{+23}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{+112}:\\ \;\;\;\;\mathsf{fma}\left(z, x \cdot -6, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (* z 6.0))))
   (if (<= y -8.5e+23) t_0 (if (<= y 6.5e+112) (fma z (* x -6.0) x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = y * (z * 6.0);
	double tmp;
	if (y <= -8.5e+23) {
		tmp = t_0;
	} else if (y <= 6.5e+112) {
		tmp = fma(z, (x * -6.0), x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(y * Float64(z * 6.0))
	tmp = 0.0
	if (y <= -8.5e+23)
		tmp = t_0;
	elseif (y <= 6.5e+112)
		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[(y * N[(z * 6.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -8.5e+23], t$95$0, If[LessEqual[y, 6.5e+112], N[(z * N[(x * -6.0), $MachinePrecision] + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -8.5000000000000001e23 or 6.4999999999999998e112 < y

   1. Initial program 99.0%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
   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. lower-*.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in y around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 69.5

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

   7. Applied rewrites 69.5%

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

   if -8.5000000000000001e23 < y < 6.4999999999999998e112

   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 82.3

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

   5. Applied rewrites 82.3%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{+23}:\\ \;\;\;\;y \cdot \left(z \cdot 6\right)\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{+112}:\\ \;\;\;\;\mathsf{fma}\left(z, x \cdot -6, x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(z \cdot 6\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 53.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.25 \cdot 10^{-82}:\\ \;\;\;\;y \cdot \left(z \cdot 6\right)\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{-40}:\\ \;\;\;\;z \cdot \left(x \cdot -6\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot 6\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.25e-82)
   (* y (* z 6.0))
   (if (<= y 3.2e-40) (* z (* x -6.0)) (* z (* y 6.0)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.25e-82) {
		tmp = y * (z * 6.0);
	} else if (y <= 3.2e-40) {
		tmp = z * (x * -6.0);
	} else {
		tmp = z * (y * 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 (y <= (-1.25d-82)) then
        tmp = y * (z * 6.0d0)
    else if (y <= 3.2d-40) then
        tmp = z * (x * (-6.0d0))
    else
        tmp = z * (y * 6.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.25e-82) {
		tmp = y * (z * 6.0);
	} else if (y <= 3.2e-40) {
		tmp = z * (x * -6.0);
	} else {
		tmp = z * (y * 6.0);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.25e-82:
		tmp = y * (z * 6.0)
	elif y <= 3.2e-40:
		tmp = z * (x * -6.0)
	else:
		tmp = z * (y * 6.0)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.25e-82)
		tmp = Float64(y * Float64(z * 6.0));
	elseif (y <= 3.2e-40)
		tmp = Float64(z * Float64(x * -6.0));
	else
		tmp = Float64(z * Float64(y * 6.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1.25e-82)
		tmp = y * (z * 6.0);
	elseif (y <= 3.2e-40)
		tmp = z * (x * -6.0);
	else
		tmp = z * (y * 6.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.25e-82], N[(y * N[(z * 6.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.2e-40], N[(z * N[(x * -6.0), $MachinePrecision]), $MachinePrecision], N[(z * N[(y * 6.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.25e-82

   1. Initial program 98.8%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
   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. lower-*.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in y around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 58.4

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

   7. Applied rewrites 58.4%

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

   if -1.25e-82 < y < 3.20000000000000002e-40

   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 55.7%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 46.1%

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

   10. Applied rewrites 46.2%

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

   if 3.20000000000000002e-40 < y

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 59.7

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

   5. Applied rewrites 59.7%

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

   7. Applied rewrites 59.8%

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

4. Final simplification 53.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.25 \cdot 10^{-82}:\\ \;\;\;\;y \cdot \left(z \cdot 6\right)\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{-40}:\\ \;\;\;\;z \cdot \left(x \cdot -6\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot 6\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 53.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.25 \cdot 10^{-82}:\\ \;\;\;\;6 \cdot \left(y \cdot z\right)\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{-40}:\\ \;\;\;\;z \cdot \left(x \cdot -6\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot 6\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.25e-82)
   (* 6.0 (* y z))
   (if (<= y 3.2e-40) (* z (* x -6.0)) (* z (* y 6.0)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.25e-82) {
		tmp = 6.0 * (y * z);
	} else if (y <= 3.2e-40) {
		tmp = z * (x * -6.0);
	} else {
		tmp = z * (y * 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 (y <= (-1.25d-82)) then
        tmp = 6.0d0 * (y * z)
    else if (y <= 3.2d-40) then
        tmp = z * (x * (-6.0d0))
    else
        tmp = z * (y * 6.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.25e-82) {
		tmp = 6.0 * (y * z);
	} else if (y <= 3.2e-40) {
		tmp = z * (x * -6.0);
	} else {
		tmp = z * (y * 6.0);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.25e-82:
		tmp = 6.0 * (y * z)
	elif y <= 3.2e-40:
		tmp = z * (x * -6.0)
	else:
		tmp = z * (y * 6.0)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.25e-82)
		tmp = Float64(6.0 * Float64(y * z));
	elseif (y <= 3.2e-40)
		tmp = Float64(z * Float64(x * -6.0));
	else
		tmp = Float64(z * Float64(y * 6.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1.25e-82)
		tmp = 6.0 * (y * z);
	elseif (y <= 3.2e-40)
		tmp = z * (x * -6.0);
	else
		tmp = z * (y * 6.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.25e-82], N[(6.0 * N[(y * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.2e-40], N[(z * N[(x * -6.0), $MachinePrecision]), $MachinePrecision], N[(z * N[(y * 6.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.25e-82

   1. Initial program 98.8%

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

   3. Taylor expanded in x around 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 58.3

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

   5. Applied rewrites 58.3%

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

   if -1.25e-82 < y < 3.20000000000000002e-40

   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 55.7%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 46.1%

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

   10. Applied rewrites 46.2%

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

   if 3.20000000000000002e-40 < y

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 59.7

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

   5. Applied rewrites 59.7%

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

   7. Applied rewrites 59.8%

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

4. Final simplification 53.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.25 \cdot 10^{-82}:\\ \;\;\;\;6 \cdot \left(y \cdot z\right)\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{-40}:\\ \;\;\;\;z \cdot \left(x \cdot -6\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot 6\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 53.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 6 \cdot \left(y \cdot z\right)\\ \mathbf{if}\;y \leq -1.25 \cdot 10^{-82}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{-40}:\\ \;\;\;\;z \cdot \left(x \cdot -6\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* 6.0 (* y z))))
   (if (<= y -1.25e-82) t_0 (if (<= y 3.2e-40) (* z (* x -6.0)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = 6.0 * (y * z);
	double tmp;
	if (y <= -1.25e-82) {
		tmp = t_0;
	} else if (y <= 3.2e-40) {
		tmp = z * (x * -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 = 6.0d0 * (y * z)
    if (y <= (-1.25d-82)) then
        tmp = t_0
    else if (y <= 3.2d-40) then
        tmp = z * (x * (-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 = 6.0 * (y * z);
	double tmp;
	if (y <= -1.25e-82) {
		tmp = t_0;
	} else if (y <= 3.2e-40) {
		tmp = z * (x * -6.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 6.0 * (y * z)
	tmp = 0
	if y <= -1.25e-82:
		tmp = t_0
	elif y <= 3.2e-40:
		tmp = z * (x * -6.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(6.0 * Float64(y * z))
	tmp = 0.0
	if (y <= -1.25e-82)
		tmp = t_0;
	elseif (y <= 3.2e-40)
		tmp = Float64(z * Float64(x * -6.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 6.0 * (y * z);
	tmp = 0.0;
	if (y <= -1.25e-82)
		tmp = t_0;
	elseif (y <= 3.2e-40)
		tmp = z * (x * -6.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(6.0 * N[(y * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.25e-82], t$95$0, If[LessEqual[y, 3.2e-40], N[(z * N[(x * -6.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.25e-82 or 3.20000000000000002e-40 < y

   1. Initial program 99.2%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
   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 58.9

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

   5. Applied rewrites 58.9%

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

   if -1.25e-82 < y < 3.20000000000000002e-40

   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 55.7%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 46.1%

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

   10. Applied rewrites 46.2%

       \[\leadsto \left(-6 \cdot x\right) \cdot z \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.25 \cdot 10^{-82}:\\ \;\;\;\;6 \cdot \left(y \cdot z\right)\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{-40}:\\ \;\;\;\;z \cdot \left(x \cdot -6\right)\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \left(y \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 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. lower-*.f64 99.8

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

4. Applied rewrites 99.8%

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

Alternative 12: 28.2% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(z * N[(x * -6.0), $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. 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 63.6%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 26.9%

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

10. Applied rewrites 27.0%

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

11. Final simplification 27.0%

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

Alternative 13: 28.2% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

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 * (z * (-6.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(x * N[(z * -6.0), $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. 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 63.6%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 26.9%

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

Developer Target 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024221 
(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)))