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

Percentage Accurate: 99.7% → 99.8%

Time: 8.4s

Alternatives: 12

Speedup: 1.1×

• 21.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.1× speedup

Math

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

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

Wolfram

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

Derivation

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. lower-fma.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 60.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(6 \cdot z\right) \cdot y\\ \mathbf{if}\;z \leq -3.1 \cdot 10^{+268}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq -0.165:\\ \;\;\;\;\left(-6 \cdot x\right) \cdot z\\ \mathbf{elif}\;z \leq 5.3 \cdot 10^{-52}:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (* 6.0 z) y)))
   (if (<= z -3.1e+268)
     t_0
     (if (<= z -0.165) (* (* -6.0 x) z) (if (<= z 5.3e-52) (* 1.0 x) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = (6.0 * z) * y;
	double tmp;
	if (z <= -3.1e+268) {
		tmp = t_0;
	} else if (z <= -0.165) {
		tmp = (-6.0 * x) * z;
	} else if (z <= 5.3e-52) {
		tmp = 1.0 * 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 * z) * y
    if (z <= (-3.1d+268)) then
        tmp = t_0
    else if (z <= (-0.165d0)) then
        tmp = ((-6.0d0) * x) * z
    else if (z <= 5.3d-52) then
        tmp = 1.0d0 * 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 * z) * y;
	double tmp;
	if (z <= -3.1e+268) {
		tmp = t_0;
	} else if (z <= -0.165) {
		tmp = (-6.0 * x) * z;
	} else if (z <= 5.3e-52) {
		tmp = 1.0 * x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (6.0 * z) * y
	tmp = 0
	if z <= -3.1e+268:
		tmp = t_0
	elif z <= -0.165:
		tmp = (-6.0 * x) * z
	elif z <= 5.3e-52:
		tmp = 1.0 * x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(6.0 * z) * y)
	tmp = 0.0
	if (z <= -3.1e+268)
		tmp = t_0;
	elseif (z <= -0.165)
		tmp = Float64(Float64(-6.0 * x) * z);
	elseif (z <= 5.3e-52)
		tmp = Float64(1.0 * x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (6.0 * z) * y;
	tmp = 0.0;
	if (z <= -3.1e+268)
		tmp = t_0;
	elseif (z <= -0.165)
		tmp = (-6.0 * x) * z;
	elseif (z <= 5.3e-52)
		tmp = 1.0 * x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(6.0 * z), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[z, -3.1e+268], t$95$0, If[LessEqual[z, -0.165], N[(N[(-6.0 * x), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[z, 5.3e-52], N[(1.0 * x), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.1000000000000001e268 or 5.3000000000000003e-52 < z

   1. Initial program 98.5%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 56.1

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

   5. Applied rewrites 56.1%

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

   7. Applied rewrites 57.4%

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

   if -3.1000000000000001e268 < z < -0.165000000000000008

   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. *-commutative N/A

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

      10. lower-*.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

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

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

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

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

      8. metadata-eval N/A

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

      9. cancel-sign-sub-inv N/A

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

      10. +-commutative N/A

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

      11. cancel-sign-sub-inv N/A

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

      12. metadata-eval N/A

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

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

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower--.f64 98.1

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

   7. Applied rewrites 98.1%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 65.7%

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

   if -0.165000000000000008 < z < 5.3000000000000003e-52

   1. Initial program 99.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 75.2

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

   7. Applied rewrites 75.2%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 74.7%

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

Alternative 3: 98.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -0.17:\\ \;\;\;\;\left(\left(x - y\right) \cdot z\right) \cdot -6\\ \mathbf{elif}\;z \leq 0.00012:\\ \;\;\;\;\mathsf{fma}\left(z \cdot y, 6, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x - y\right) \cdot -6\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -0.17)
   (* (* (- x y) z) -6.0)
   (if (<= z 0.00012) (fma (* z y) 6.0 x) (* (* (- x y) -6.0) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -0.17) {
		tmp = ((x - y) * z) * -6.0;
	} else if (z <= 0.00012) {
		tmp = fma((z * y), 6.0, x);
	} else {
		tmp = ((x - y) * -6.0) * z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -0.17)
		tmp = Float64(Float64(Float64(x - y) * z) * -6.0);
	elseif (z <= 0.00012)
		tmp = fma(Float64(z * y), 6.0, x);
	else
		tmp = Float64(Float64(Float64(x - y) * -6.0) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -0.17], N[(N[(N[(x - y), $MachinePrecision] * z), $MachinePrecision] * -6.0), $MachinePrecision], If[LessEqual[z, 0.00012], N[(N[(z * y), $MachinePrecision] * 6.0 + x), $MachinePrecision], N[(N[(N[(x - y), $MachinePrecision] * -6.0), $MachinePrecision] * z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -0.170000000000000012

   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. *-commutative N/A

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

      10. lower-*.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

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

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

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

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

      8. metadata-eval N/A

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

      9. cancel-sign-sub-inv N/A

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

      10. +-commutative N/A

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

      11. cancel-sign-sub-inv N/A

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

      12. metadata-eval N/A

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

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

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower--.f64 98.3

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

   7. Applied rewrites 98.3%

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

   9. Applied rewrites 98.4%

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

   if -0.170000000000000012 < z < 1.20000000000000003e-4

   1. Initial program 98.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 99.3

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

   7. Applied rewrites 99.3%

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

   if 1.20000000000000003e-4 < z

   1. Initial program 99.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

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

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

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

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

      8. metadata-eval N/A

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

      9. cancel-sign-sub-inv N/A

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

      10. +-commutative N/A

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

      11. cancel-sign-sub-inv N/A

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

      12. metadata-eval N/A

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

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

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower--.f64 98.8

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

   7. Applied rewrites 98.8%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.17:\\ \;\;\;\;\left(\left(x - y\right) \cdot z\right) \cdot -6\\ \mathbf{elif}\;z \leq 0.00012:\\ \;\;\;\;\mathsf{fma}\left(z \cdot y, 6, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x - y\right) \cdot -6\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(x - y\right) \cdot z\right) \cdot -6\\ \mathbf{if}\;z \leq -0.17:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(6 \cdot y, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (* (- x y) z) -6.0)))
   (if (<= z -0.17) t_0 (if (<= z 1.7e-11) (fma (* 6.0 y) z x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = ((x - y) * z) * -6.0;
	double tmp;
	if (z <= -0.17) {
		tmp = t_0;
	} else if (z <= 1.7e-11) {
		tmp = fma((6.0 * y), z, x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(x - y) * z) * -6.0)
	tmp = 0.0
	if (z <= -0.17)
		tmp = t_0;
	elseif (z <= 1.7e-11)
		tmp = fma(Float64(6.0 * y), z, x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(x - y), $MachinePrecision] * z), $MachinePrecision] * -6.0), $MachinePrecision]}, If[LessEqual[z, -0.17], t$95$0, If[LessEqual[z, 1.7e-11], N[(N[(6.0 * y), $MachinePrecision] * z + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -0.170000000000000012 or 1.6999999999999999e-11 < z

   1. Initial program 99.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

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

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

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

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

      8. metadata-eval N/A

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

      9. cancel-sign-sub-inv N/A

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

      10. +-commutative N/A

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

      11. cancel-sign-sub-inv N/A

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

      12. metadata-eval N/A

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

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

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower--.f64 98.6

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

   7. Applied rewrites 98.6%

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

   9. Applied rewrites 98.6%

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

   if -0.170000000000000012 < z < 1.6999999999999999e-11

   1. Initial program 98.4%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 99.3

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

   5. Applied rewrites 99.3%

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

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.17:\\ \;\;\;\;\left(\left(x - y\right) \cdot z\right) \cdot -6\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(6 \cdot y, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x - y\right) \cdot z\right) \cdot -6\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{+24}:\\ \;\;\;\;\mathsf{fma}\left(-6 \cdot z, x, x\right)\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+46}:\\ \;\;\;\;\mathsf{fma}\left(6 \cdot y, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-6, z, 1\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -2.3e+24)
   (fma (* -6.0 z) x x)
   (if (<= x 1.1e+46) (fma (* 6.0 y) z x) (* (fma -6.0 z 1.0) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.3e+24) {
		tmp = fma((-6.0 * z), x, x);
	} else if (x <= 1.1e+46) {
		tmp = fma((6.0 * y), z, x);
	} else {
		tmp = fma(-6.0, z, 1.0) * x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -2.3e+24)
		tmp = fma(Float64(-6.0 * z), x, x);
	elseif (x <= 1.1e+46)
		tmp = fma(Float64(6.0 * y), z, x);
	else
		tmp = Float64(fma(-6.0, z, 1.0) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -2.3e+24], N[(N[(-6.0 * z), $MachinePrecision] * x + x), $MachinePrecision], If[LessEqual[x, 1.1e+46], N[(N[(6.0 * y), $MachinePrecision] * z + x), $MachinePrecision], N[(N[(-6.0 * z + 1.0), $MachinePrecision] * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.2999999999999999e24

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 91.5

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

   7. Applied rewrites 91.5%

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

   9. Applied rewrites 91.5%

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

   if -2.2999999999999999e24 < x < 1.1e46

   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

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

      1. lower-*.f64 89.1

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

   5. Applied rewrites 89.1%

      \[\leadsto x + \color{blue}{\left(6 \cdot y\right)} \cdot z \]
   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 89.1

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

   7. Applied rewrites 89.1%

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

   if 1.1e46 < x

   1. Initial program 96.2%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 98.6

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

   5. Applied rewrites 98.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-6, z, 1\right) \cdot x} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 86.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{+24}:\\ \;\;\;\;\mathsf{fma}\left(-6 \cdot z, x, x\right)\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+46}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot y, 6, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-6, z, 1\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -2.3e+24)
   (fma (* -6.0 z) x x)
   (if (<= x 1.1e+46) (fma (* z y) 6.0 x) (* (fma -6.0 z 1.0) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.3e+24) {
		tmp = fma((-6.0 * z), x, x);
	} else if (x <= 1.1e+46) {
		tmp = fma((z * y), 6.0, x);
	} else {
		tmp = fma(-6.0, z, 1.0) * x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -2.3e+24)
		tmp = fma(Float64(-6.0 * z), x, x);
	elseif (x <= 1.1e+46)
		tmp = fma(Float64(z * y), 6.0, x);
	else
		tmp = Float64(fma(-6.0, z, 1.0) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -2.3e+24], N[(N[(-6.0 * z), $MachinePrecision] * x + x), $MachinePrecision], If[LessEqual[x, 1.1e+46], N[(N[(z * y), $MachinePrecision] * 6.0 + x), $MachinePrecision], N[(N[(-6.0 * z + 1.0), $MachinePrecision] * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.2999999999999999e24

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 91.5

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

   7. Applied rewrites 91.5%

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

   9. Applied rewrites 91.5%

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

   if -2.2999999999999999e24 < x < 1.1e46

   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. *-commutative N/A

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

      10. lower-*.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 89.1

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

   7. Applied rewrites 89.1%

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

   if 1.1e46 < x

   1. Initial program 96.2%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 98.6

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

   5. Applied rewrites 98.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-6, z, 1\right) \cdot x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{+24}:\\ \;\;\;\;\mathsf{fma}\left(-6 \cdot z, x, x\right)\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+46}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot y, 6, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-6, z, 1\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.08 \cdot 10^{-79}:\\ \;\;\;\;\mathsf{fma}\left(-6 \cdot z, x, x\right)\\ \mathbf{elif}\;x \leq 1.52 \cdot 10^{-93}:\\ \;\;\;\;\left(6 \cdot z\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-6, z, 1\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.08e-79)
   (fma (* -6.0 z) x x)
   (if (<= x 1.52e-93) (* (* 6.0 z) y) (* (fma -6.0 z 1.0) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.08e-79) {
		tmp = fma((-6.0 * z), x, x);
	} else if (x <= 1.52e-93) {
		tmp = (6.0 * z) * y;
	} else {
		tmp = fma(-6.0, z, 1.0) * x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.08e-79)
		tmp = fma(Float64(-6.0 * z), x, x);
	elseif (x <= 1.52e-93)
		tmp = Float64(Float64(6.0 * z) * y);
	else
		tmp = Float64(fma(-6.0, z, 1.0) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.08e-79], N[(N[(-6.0 * z), $MachinePrecision] * x + x), $MachinePrecision], If[LessEqual[x, 1.52e-93], N[(N[(6.0 * z), $MachinePrecision] * y), $MachinePrecision], N[(N[(-6.0 * z + 1.0), $MachinePrecision] * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.0800000000000001e-79

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 84.3

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

   7. Applied rewrites 84.3%

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

   9. Applied rewrites 84.4%

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

   if -1.0800000000000001e-79 < x < 1.52e-93

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 73.3

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

   5. Applied rewrites 73.3%

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

   7. Applied rewrites 73.5%

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

   if 1.52e-93 < x

   1. Initial program 97.4%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 88.2

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

   5. Applied rewrites 88.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-6, z, 1\right) \cdot x} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 75.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-6, z, 1\right) \cdot x\\ \mathbf{if}\;x \leq -1.08 \cdot 10^{-79}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.52 \cdot 10^{-93}:\\ \;\;\;\;\left(6 \cdot z\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (fma -6.0 z 1.0) x)))
   (if (<= x -1.08e-79) t_0 (if (<= x 1.52e-93) (* (* 6.0 z) y) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma(-6.0, z, 1.0) * x;
	double tmp;
	if (x <= -1.08e-79) {
		tmp = t_0;
	} else if (x <= 1.52e-93) {
		tmp = (6.0 * z) * y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(fma(-6.0, z, 1.0) * x)
	tmp = 0.0
	if (x <= -1.08e-79)
		tmp = t_0;
	elseif (x <= 1.52e-93)
		tmp = Float64(Float64(6.0 * z) * y);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(-6.0 * z + 1.0), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -1.08e-79], t$95$0, If[LessEqual[x, 1.52e-93], N[(N[(6.0 * z), $MachinePrecision] * y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.0800000000000001e-79 or 1.52e-93 < x

   1. Initial program 98.6%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 86.3

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

   5. Applied rewrites 86.3%

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

   if -1.0800000000000001e-79 < x < 1.52e-93

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 73.3

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

   5. Applied rewrites 73.3%

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

   7. Applied rewrites 73.5%

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

Alternative 9: 60.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(6 \cdot z\right) \cdot y\\ \mathbf{if}\;z \leq -1 \cdot 10^{-64}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 5.3 \cdot 10^{-52}:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (* 6.0 z) y)))
   (if (<= z -1e-64) t_0 (if (<= z 5.3e-52) (* 1.0 x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (6.0 * z) * y;
	double tmp;
	if (z <= -1e-64) {
		tmp = t_0;
	} else if (z <= 5.3e-52) {
		tmp = 1.0 * 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 * z) * y
    if (z <= (-1d-64)) then
        tmp = t_0
    else if (z <= 5.3d-52) then
        tmp = 1.0d0 * 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 * z) * y;
	double tmp;
	if (z <= -1e-64) {
		tmp = t_0;
	} else if (z <= 5.3e-52) {
		tmp = 1.0 * x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (6.0 * z) * y
	tmp = 0
	if z <= -1e-64:
		tmp = t_0
	elif z <= 5.3e-52:
		tmp = 1.0 * x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(6.0 * z) * y)
	tmp = 0.0
	if (z <= -1e-64)
		tmp = t_0;
	elseif (z <= 5.3e-52)
		tmp = Float64(1.0 * x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (6.0 * z) * y;
	tmp = 0.0;
	if (z <= -1e-64)
		tmp = t_0;
	elseif (z <= 5.3e-52)
		tmp = 1.0 * x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(6.0 * z), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[z, -1e-64], t$95$0, If[LessEqual[z, 5.3e-52], N[(1.0 * x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -9.99999999999999965e-65 or 5.3000000000000003e-52 < z

   1. Initial program 99.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 52.5

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

   5. Applied rewrites 52.5%

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

   7. Applied rewrites 53.2%

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

   if -9.99999999999999965e-65 < z < 5.3000000000000003e-52

   1. Initial program 99.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 77.5

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

   7. Applied rewrites 77.5%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 77.5%

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

Alternative 10: 60.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(6 \cdot y\right) \cdot z\\ \mathbf{if}\;z \leq -1 \cdot 10^{-64}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 5.3 \cdot 10^{-52}:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (* 6.0 y) z)))
   (if (<= z -1e-64) t_0 (if (<= z 5.3e-52) (* 1.0 x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (6.0 * y) * z;
	double tmp;
	if (z <= -1e-64) {
		tmp = t_0;
	} else if (z <= 5.3e-52) {
		tmp = 1.0 * 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 * y) * z
    if (z <= (-1d-64)) then
        tmp = t_0
    else if (z <= 5.3d-52) then
        tmp = 1.0d0 * 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 * y) * z;
	double tmp;
	if (z <= -1e-64) {
		tmp = t_0;
	} else if (z <= 5.3e-52) {
		tmp = 1.0 * x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (6.0 * y) * z
	tmp = 0
	if z <= -1e-64:
		tmp = t_0
	elif z <= 5.3e-52:
		tmp = 1.0 * x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(6.0 * y) * z)
	tmp = 0.0
	if (z <= -1e-64)
		tmp = t_0;
	elseif (z <= 5.3e-52)
		tmp = Float64(1.0 * x);
	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 (z <= -1e-64)
		tmp = t_0;
	elseif (z <= 5.3e-52)
		tmp = 1.0 * x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(6.0 * y), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -1e-64], t$95$0, If[LessEqual[z, 5.3e-52], N[(1.0 * x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -9.99999999999999965e-65 or 5.3000000000000003e-52 < z

   1. Initial program 99.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 52.5

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

   5. Applied rewrites 52.5%

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

   7. Applied rewrites 52.5%

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

   if -9.99999999999999965e-65 < z < 5.3000000000000003e-52

   1. Initial program 99.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 77.5

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

   7. Applied rewrites 77.5%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 77.5%

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

Alternative 11: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.0%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. associate-*r* N/A

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

   8. lower-fma.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 99.8

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

4. Applied rewrites 99.8%

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

Alternative 12: 35.8% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.0%

   \[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
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. lower-fma.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f64 65.3

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

7. Applied rewrites 65.3%

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

8. Taylor expanded in z around 0

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

10. Applied rewrites 40.0%

    \[\leadsto 1 \cdot x \]
11. Add Preprocessing

Developer Target 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024298 
(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)))