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

Percentage Accurate: 99.7% → 99.8%

Time: 7.0s

Alternatives: 12

Speedup: 1.1×

• 20.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.1× speedup

Math

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

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

Wolfram

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

Alternative 2: 98.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -21000000 \lor \neg \left(z \leq 0.17\right):\\ \;\;\;\;\left(\left(y - x\right) \cdot z\right) \cdot 6\\ \mathbf{else}:\\ \;\;\;\;x + \left(6 \cdot y\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -21000000.0) (not (<= z 0.17)))
   (* (* (- y x) z) 6.0)
   (+ x (* (* 6.0 y) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -21000000.0) || !(z <= 0.17)) {
		tmp = ((y - x) * z) * 6.0;
	} else {
		tmp = x + ((6.0 * y) * z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-21000000.0d0)) .or. (.not. (z <= 0.17d0))) then
        tmp = ((y - x) * z) * 6.0d0
    else
        tmp = x + ((6.0d0 * y) * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -21000000.0) || !(z <= 0.17)) {
		tmp = ((y - x) * z) * 6.0;
	} else {
		tmp = x + ((6.0 * y) * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -21000000.0) or not (z <= 0.17):
		tmp = ((y - x) * z) * 6.0
	else:
		tmp = x + ((6.0 * y) * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -21000000.0) || !(z <= 0.17))
		tmp = Float64(Float64(Float64(y - x) * z) * 6.0);
	else
		tmp = Float64(x + Float64(Float64(6.0 * y) * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -21000000.0) || ~((z <= 0.17)))
		tmp = ((y - x) * z) * 6.0;
	else
		tmp = x + ((6.0 * y) * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -21000000.0], N[Not[LessEqual[z, 0.17]], $MachinePrecision]], N[(N[(N[(y - x), $MachinePrecision] * z), $MachinePrecision] * 6.0), $MachinePrecision], N[(x + N[(N[(6.0 * y), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.1e7 or 0.170000000000000012 < 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 99.7

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

   4. Applied rewrites 99.7%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 97.3

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

   6. Applied rewrites 97.3%

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

   7. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. mul-1-neg N/A

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

      3. sub-neg N/A

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

      4. associate-*r* N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. distribute-lft-in N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. +-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. metadata-eval N/A

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

      14. associate-*r* N/A

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

      15. distribute-lft-in N/A

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

      16. mul-1-neg N/A

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

      17. sub-neg N/A

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

      18. lower-*.f64 N/A

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

      19. lower--.f64 99.5

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

   9. Applied rewrites 99.5%

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

   10. Taylor expanded in z around inf

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

       1. *-commutative N/A

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

       2. mul-1-neg N/A

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

       3. sub-neg N/A

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

       4. lower-*.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. lower--.f64 99.6

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

   12. Applied rewrites 99.6%

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

   if -2.1e7 < z < 0.170000000000000012

   1. Initial program 99.3%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 99.4

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

   5. Applied rewrites 99.4%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -21000000 \lor \neg \left(z \leq 0.17\right):\\ \;\;\;\;\left(\left(y - x\right) \cdot z\right) \cdot 6\\ \mathbf{else}:\\ \;\;\;\;x + \left(6 \cdot y\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -0.16 \lor \neg \left(z \leq 1200000\right):\\ \;\;\;\;\left(6 \cdot \left(y - x\right)\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot y, 6, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -0.16) (not (<= z 1200000.0)))
   (* (* 6.0 (- y x)) z)
   (fma (* z y) 6.0 x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -0.16) || !(z <= 1200000.0)) {
		tmp = (6.0 * (y - x)) * z;
	} else {
		tmp = fma((z * y), 6.0, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -0.16) || !(z <= 1200000.0))
		tmp = Float64(Float64(6.0 * Float64(y - x)) * z);
	else
		tmp = fma(Float64(z * y), 6.0, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -0.16], N[Not[LessEqual[z, 1200000.0]], $MachinePrecision]], N[(N[(6.0 * N[(y - x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], N[(N[(z * y), $MachinePrecision] * 6.0 + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -0.160000000000000003 or 1.2e6 < 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 99.7

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

   4. Applied rewrites 99.7%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 97.3

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

   6. Applied rewrites 97.3%

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

   7. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. mul-1-neg N/A

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

      3. sub-neg N/A

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

      4. associate-*r* N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. distribute-lft-in N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. +-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. metadata-eval N/A

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

      14. associate-*r* N/A

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

      15. distribute-lft-in N/A

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

      16. mul-1-neg N/A

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

      17. sub-neg N/A

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

      18. lower-*.f64 N/A

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

      19. lower--.f64 99.5

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

   9. Applied rewrites 99.5%

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

   if -0.160000000000000003 < z < 1.2e6

   1. Initial program 99.3%

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

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

      2. lower-*.f64 99.3

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

   7. Applied rewrites 99.3%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.16 \lor \neg \left(z \leq 1200000\right):\\ \;\;\;\;\left(6 \cdot \left(y - x\right)\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot y, 6, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -21000000:\\ \;\;\;\;\left(\left(y - x\right) \cdot z\right) \cdot 6\\ \mathbf{elif}\;z \leq 1200000:\\ \;\;\;\;\mathsf{fma}\left(z \cdot y, 6, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(6 \cdot \left(y - x\right)\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -21000000.0)
   (* (* (- y x) z) 6.0)
   (if (<= z 1200000.0) (fma (* z y) 6.0 x) (* (* 6.0 (- y x)) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -21000000.0) {
		tmp = ((y - x) * z) * 6.0;
	} else if (z <= 1200000.0) {
		tmp = fma((z * y), 6.0, x);
	} else {
		tmp = (6.0 * (y - x)) * z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -21000000.0)
		tmp = Float64(Float64(Float64(y - x) * z) * 6.0);
	elseif (z <= 1200000.0)
		tmp = fma(Float64(z * y), 6.0, x);
	else
		tmp = Float64(Float64(6.0 * Float64(y - x)) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -21000000.0], N[(N[(N[(y - x), $MachinePrecision] * z), $MachinePrecision] * 6.0), $MachinePrecision], If[LessEqual[z, 1200000.0], N[(N[(z * y), $MachinePrecision] * 6.0 + x), $MachinePrecision], N[(N[(6.0 * N[(y - x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.1e7

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 98.0

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

   6. Applied rewrites 98.0%

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

   7. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. mul-1-neg N/A

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

      3. sub-neg N/A

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

      4. associate-*r* N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. distribute-lft-in N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. +-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. metadata-eval N/A

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

      14. associate-*r* N/A

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

      15. distribute-lft-in N/A

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

      16. mul-1-neg N/A

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

      17. sub-neg N/A

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

      18. lower-*.f64 N/A

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

      19. lower--.f64 99.3

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

   9. Applied rewrites 99.3%

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

   10. Taylor expanded in z around inf

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

       1. *-commutative N/A

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

       2. mul-1-neg N/A

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

       3. sub-neg N/A

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

       4. lower-*.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. lower--.f64 99.5

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

   12. Applied rewrites 99.5%

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

   if -2.1e7 < z < 1.2e6

   1. Initial program 99.3%

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

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

      2. lower-*.f64 99.3

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

   7. Applied rewrites 99.3%

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

   if 1.2e6 < 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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 96.6

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

   6. Applied rewrites 96.6%

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

   7. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. mul-1-neg N/A

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

      3. sub-neg N/A

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

      4. associate-*r* N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. distribute-lft-in N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. +-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. metadata-eval N/A

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

      14. associate-*r* N/A

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

      15. distribute-lft-in N/A

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

      16. mul-1-neg N/A

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

      17. sub-neg N/A

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

      18. lower-*.f64 N/A

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

      19. lower--.f64 99.7

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

   9. Applied rewrites 99.7%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -21000000:\\ \;\;\;\;\left(\left(y - x\right) \cdot z\right) \cdot 6\\ \mathbf{elif}\;z \leq 1200000:\\ \;\;\;\;\mathsf{fma}\left(z \cdot y, 6, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(6 \cdot \left(y - x\right)\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-72} \lor \neg \left(y \leq 9.6 \cdot 10^{-56}\right):\\ \;\;\;\;\mathsf{fma}\left(z \cdot y, 6, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot -6, x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -5e-72) (not (<= y 9.6e-56)))
   (fma (* z y) 6.0 x)
   (fma (* z -6.0) x x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -5e-72) || !(y <= 9.6e-56)) {
		tmp = fma((z * y), 6.0, x);
	} else {
		tmp = fma((z * -6.0), x, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -5e-72) || !(y <= 9.6e-56))
		tmp = fma(Float64(z * y), 6.0, x);
	else
		tmp = fma(Float64(z * -6.0), x, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -5e-72], N[Not[LessEqual[y, 9.6e-56]], $MachinePrecision]], N[(N[(z * y), $MachinePrecision] * 6.0 + x), $MachinePrecision], N[(N[(z * -6.0), $MachinePrecision] * x + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.9999999999999996e-72 or 9.60000000000000002e-56 < y

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 91.7

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

   7. Applied rewrites 91.7%

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

   if -4.9999999999999996e-72 < y < 9.60000000000000002e-56

   1. Initial program 99.1%

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

   3. Taylor expanded in x around inf

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 88.8

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

   5. Applied rewrites 88.8%

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

   7. Applied rewrites 89.5%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-72} \lor \neg \left(y \leq 9.6 \cdot 10^{-56}\right):\\ \;\;\;\;\mathsf{fma}\left(z \cdot y, 6, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot -6, x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{-41} \lor \neg \left(x \leq 1.7 \cdot 10^{-148}\right):\\ \;\;\;\;\mathsf{fma}\left(-6 \cdot x, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(6 \cdot y\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -9.5e-41) (not (<= x 1.7e-148)))
   (fma (* -6.0 x) z x)
   (* (* 6.0 y) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -9.5e-41) || !(x <= 1.7e-148)) {
		tmp = fma((-6.0 * x), z, x);
	} else {
		tmp = (6.0 * y) * z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -9.5e-41) || !(x <= 1.7e-148))
		tmp = fma(Float64(-6.0 * x), z, x);
	else
		tmp = Float64(Float64(6.0 * y) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -9.5e-41], N[Not[LessEqual[x, 1.7e-148]], $MachinePrecision]], N[(N[(-6.0 * x), $MachinePrecision] * z + x), $MachinePrecision], N[(N[(6.0 * y), $MachinePrecision] * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -9.4999999999999997e-41 or 1.7000000000000001e-148 < x

   1. Initial program 99.4%

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

   3. Taylor expanded in x around inf

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 83.3

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

   5. Applied rewrites 83.3%

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

   if -9.4999999999999997e-41 < x < 1.7000000000000001e-148

   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 74.3

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

   5. Applied rewrites 74.3%

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

   7. Applied rewrites 74.4%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{-41} \lor \neg \left(x \leq 1.7 \cdot 10^{-148}\right):\\ \;\;\;\;\mathsf{fma}\left(-6 \cdot x, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(6 \cdot y\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{-41}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot x, -6, x\right)\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{-148}:\\ \;\;\;\;\left(6 \cdot y\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-6 \cdot x, z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -9.5e-41)
   (fma (* z x) -6.0 x)
   (if (<= x 1.7e-148) (* (* 6.0 y) z) (fma (* -6.0 x) z x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -9.5e-41) {
		tmp = fma((z * x), -6.0, x);
	} else if (x <= 1.7e-148) {
		tmp = (6.0 * y) * z;
	} else {
		tmp = fma((-6.0 * x), z, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -9.5e-41)
		tmp = fma(Float64(z * x), -6.0, x);
	elseif (x <= 1.7e-148)
		tmp = Float64(Float64(6.0 * y) * z);
	else
		tmp = fma(Float64(-6.0 * x), z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -9.5e-41], N[(N[(z * x), $MachinePrecision] * -6.0 + x), $MachinePrecision], If[LessEqual[x, 1.7e-148], N[(N[(6.0 * y), $MachinePrecision] * z), $MachinePrecision], N[(N[(-6.0 * x), $MachinePrecision] * z + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -9.4999999999999997e-41

   1. Initial program 98.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 99.9

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

   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

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 85.9

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

   7. Applied rewrites 85.9%

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

   if -9.4999999999999997e-41 < x < 1.7000000000000001e-148

   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 74.3

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

   5. Applied rewrites 74.3%

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

   7. Applied rewrites 74.4%

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

   if 1.7000000000000001e-148 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 82.2

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

   5. Applied rewrites 82.2%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{-41}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot x, -6, x\right)\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{-148}:\\ \;\;\;\;\left(6 \cdot y\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-6 \cdot x, z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{-38}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot -6, x, x\right)\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{-148}:\\ \;\;\;\;\left(6 \cdot y\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-6 \cdot x, z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.2e-38)
   (fma (* z -6.0) x x)
   (if (<= x 1.7e-148) (* (* 6.0 y) z) (fma (* -6.0 x) z x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.2e-38) {
		tmp = fma((z * -6.0), x, x);
	} else if (x <= 1.7e-148) {
		tmp = (6.0 * y) * z;
	} else {
		tmp = fma((-6.0 * x), z, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.2e-38)
		tmp = fma(Float64(z * -6.0), x, x);
	elseif (x <= 1.7e-148)
		tmp = Float64(Float64(6.0 * y) * z);
	else
		tmp = fma(Float64(-6.0 * x), z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.2e-38], N[(N[(z * -6.0), $MachinePrecision] * x + x), $MachinePrecision], If[LessEqual[x, 1.7e-148], N[(N[(6.0 * y), $MachinePrecision] * z), $MachinePrecision], N[(N[(-6.0 * x), $MachinePrecision] * z + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.20000000000000011e-38

   1. Initial program 98.7%

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

   3. Taylor expanded in x around inf

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 84.6

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

   5. Applied rewrites 84.6%

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

   7. Applied rewrites 85.8%

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

   if -1.20000000000000011e-38 < x < 1.7000000000000001e-148

   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 74.3

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

   5. Applied rewrites 74.3%

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

   7. Applied rewrites 74.4%

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

   if 1.7000000000000001e-148 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 82.2

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

   5. Applied rewrites 82.2%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{-38}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot -6, x, x\right)\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{-148}:\\ \;\;\;\;\left(6 \cdot y\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-6 \cdot x, z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 52.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.3 \cdot 10^{-115}:\\ \;\;\;\;\left(6 \cdot y\right) \cdot z\\ \mathbf{elif}\;y \leq 9.6 \cdot 10^{-56}:\\ \;\;\;\;\left(-6 \cdot x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot y\right) \cdot 6\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -3.3e-115)
   (* (* 6.0 y) z)
   (if (<= y 9.6e-56) (* (* -6.0 x) z) (* (* z y) 6.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.3e-115) {
		tmp = (6.0 * y) * z;
	} else if (y <= 9.6e-56) {
		tmp = (-6.0 * x) * z;
	} 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 <= (-3.3d-115)) then
        tmp = (6.0d0 * y) * z
    else if (y <= 9.6d-56) then
        tmp = ((-6.0d0) * x) * z
    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 <= -3.3e-115) {
		tmp = (6.0 * y) * z;
	} else if (y <= 9.6e-56) {
		tmp = (-6.0 * x) * z;
	} else {
		tmp = (z * y) * 6.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -3.3e-115:
		tmp = (6.0 * y) * z
	elif y <= 9.6e-56:
		tmp = (-6.0 * x) * z
	else:
		tmp = (z * y) * 6.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -3.3e-115)
		tmp = Float64(Float64(6.0 * y) * z);
	elseif (y <= 9.6e-56)
		tmp = Float64(Float64(-6.0 * x) * z);
	else
		tmp = Float64(Float64(z * y) * 6.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -3.3e-115)
		tmp = (6.0 * y) * z;
	elseif (y <= 9.6e-56)
		tmp = (-6.0 * x) * z;
	else
		tmp = (z * y) * 6.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -3.3e-115], N[(N[(6.0 * y), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[y, 9.6e-56], N[(N[(-6.0 * x), $MachinePrecision] * z), $MachinePrecision], N[(N[(z * y), $MachinePrecision] * 6.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.2999999999999999e-115

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 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 63.5

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

   5. Applied rewrites 63.5%

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

   7. Applied rewrites 63.7%

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

   if -3.2999999999999999e-115 < y < 9.60000000000000002e-56

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 99.9

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. mul-1-neg N/A

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

      3. sub-neg N/A

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

      4. associate-*r* N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. distribute-lft-in N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. +-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. metadata-eval N/A

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

      14. associate-*r* N/A

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

      15. distribute-lft-in N/A

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

      16. mul-1-neg N/A

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

      17. sub-neg N/A

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

      18. lower-*.f64 N/A

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

      19. lower--.f64 54.6

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

   9. Applied rewrites 54.6%

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

   10. Taylor expanded in x around inf

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

   12. Applied rewrites 44.9%

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

   if 9.60000000000000002e-56 < y

   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 57.6

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

   5. Applied rewrites 57.6%

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

4. Final simplification 54.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.3 \cdot 10^{-115}:\\ \;\;\;\;\left(6 \cdot y\right) \cdot z\\ \mathbf{elif}\;y \leq 9.6 \cdot 10^{-56}:\\ \;\;\;\;\left(-6 \cdot x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot y\right) \cdot 6\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 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 99.5%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. associate-*r* N/A

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

   8. lower-fma.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 99.8

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

4. Applied rewrites 99.8%

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

Alternative 11: 41.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(N[(6.0 * z), $MachinePrecision] * y), $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 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 40.6

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

5. Applied rewrites 40.6%

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

7. Applied rewrites 41.0%

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

8. Final simplification 41.0%

   \[\leadsto \left(6 \cdot z\right) \cdot y \]
9. Add Preprocessing

Alternative 12: 41.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(N[(6.0 * y), $MachinePrecision] * z), $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 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 40.6

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

5. Applied rewrites 40.6%

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

7. Applied rewrites 40.7%

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

8. Final simplification 40.7%

   \[\leadsto \left(6 \cdot y\right) \cdot z \]
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 2024296 
(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)))