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

Percentage Accurate: 99.7% → 99.8%

Time: 7.2s

Alternatives: 10

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

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

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 99.0%

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

Alternative 2: 98.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -0.165 \lor \neg \left(z \leq 0.00012\right):\\ \;\;\;\;\left(6 \cdot \left(y - x\right)\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot y, 6, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -0.165) (not (<= z 0.00012)))
   (* (* 6.0 (- y x)) z)
   (fma (* z y) 6.0 x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -0.165) || !(z <= 0.00012)) {
		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 <= -0.165) || !(z <= 0.00012))
		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, -0.165], N[Not[LessEqual[z, 0.00012]], $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 < -0.165000000000000008 or 1.20000000000000003e-4 < z

   1. Initial program 99.7%

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

   3. Taylor expanded in 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 51.8

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

   5. Applied rewrites 51.8%

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

   7. Applied rewrites 52.7%

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

   8. Taylor expanded in z around inf

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

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

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

      2. unsub-neg N/A

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

      3. mul-1-neg N/A

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

      4. distribute-lft-in N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

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

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

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

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

      9. metadata-eval N/A

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

      10. associate-*r* N/A

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

      11. distribute-rgt-in N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. associate-*r* N/A

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

      17. *-commutative N/A

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

      18. *-commutative N/A

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

      19. distribute-rgt-in N/A

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

      20. +-commutative N/A

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

      21. mul-1-neg N/A

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

      22. sub-neg N/A

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

      23. lower-*.f64 N/A

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

      24. lower--.f64 98.6

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

   10. Applied rewrites 98.6%

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

   if -0.165000000000000008 < 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. *-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 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.165 \lor \neg \left(z \leq 0.00012\right):\\ \;\;\;\;\left(6 \cdot \left(y - x\right)\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot y, 6, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{-79} \lor \neg \left(x \leq 2.15 \cdot 10^{-96}\right):\\ \;\;\;\;\mathsf{fma}\left(-6 \cdot x, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(6 \cdot z\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -2.5e-79) (not (<= x 2.15e-96)))
   (fma (* -6.0 x) z x)
   (* (* 6.0 z) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2.5e-79) || !(x <= 2.15e-96)) {
		tmp = fma((-6.0 * x), z, x);
	} else {
		tmp = (6.0 * z) * y;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -2.5e-79) || !(x <= 2.15e-96))
		tmp = fma(Float64(-6.0 * x), z, x);
	else
		tmp = Float64(Float64(6.0 * z) * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -2.5e-79], N[Not[LessEqual[x, 2.15e-96]], $MachinePrecision]], N[(N[(-6.0 * x), $MachinePrecision] * z + x), $MachinePrecision], N[(N[(6.0 * z), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.5e-79 or 2.1499999999999999e-96 < 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. 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 85.1

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

   5. Applied rewrites 85.1%

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

   if -2.5e-79 < x < 2.1499999999999999e-96

   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. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{-79} \lor \neg \left(x \leq 2.15 \cdot 10^{-96}\right):\\ \;\;\;\;\mathsf{fma}\left(-6 \cdot x, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(6 \cdot z\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{+24}:\\ \;\;\;\;\mathsf{fma}\left(-6 \cdot z, x, x\right)\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+46}:\\ \;\;\;\;\mathsf{fma}\left(6 \cdot y, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-6, z, 1\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.4e+24)
   (fma (* -6.0 z) x x)
   (if (<= x 4e+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 <= -1.4e+24) {
		tmp = fma((-6.0 * z), x, x);
	} else if (x <= 4e+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 <= -1.4e+24)
		tmp = fma(Float64(-6.0 * z), x, x);
	elseif (x <= 4e+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, -1.4e+24], N[(N[(-6.0 * z), $MachinePrecision] * x + x), $MachinePrecision], If[LessEqual[x, 4e+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 < -1.4000000000000001e24

   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 91.5

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

   5. Applied rewrites 91.5%

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

   7. Applied rewrites 91.5%

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

   if -1.4000000000000001e24 < x < 4e46

   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 4e46 < x

   1. Initial program 96.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. 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 98.6

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

   7. 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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{+24}:\\ \;\;\;\;\mathsf{fma}\left(-6 \cdot z, x, x\right)\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+46}:\\ \;\;\;\;\mathsf{fma}\left(6 \cdot y, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-6, z, 1\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{+24}:\\ \;\;\;\;\mathsf{fma}\left(-6 \cdot z, x, x\right)\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+46}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot y, 6, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-6, z, 1\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.4e+24)
   (fma (* -6.0 z) x x)
   (if (<= x 4e+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 <= -1.4e+24) {
		tmp = fma((-6.0 * z), x, x);
	} else if (x <= 4e+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 <= -1.4e+24)
		tmp = fma(Float64(-6.0 * z), x, x);
	elseif (x <= 4e+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, -1.4e+24], N[(N[(-6.0 * z), $MachinePrecision] * x + x), $MachinePrecision], If[LessEqual[x, 4e+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 < -1.4000000000000001e24

   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 91.5

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

   5. Applied rewrites 91.5%

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

   7. Applied rewrites 91.5%

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

   if -1.4000000000000001e24 < x < 4e46

   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. *-commutative N/A

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

      2. lower-*.f64 89.1

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

   7. Applied rewrites 89.1%

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

   if 4e46 < x

   1. Initial program 96.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. 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 98.6

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

   7. 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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{+24}:\\ \;\;\;\;\mathsf{fma}\left(-6 \cdot z, x, x\right)\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+46}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot y, 6, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-6, z, 1\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{-79}:\\ \;\;\;\;\mathsf{fma}\left(-6 \cdot x, z, x\right)\\ \mathbf{elif}\;x \leq 2.15 \cdot 10^{-96}:\\ \;\;\;\;\left(6 \cdot z\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-6, z, 1\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -2.5e-79)
   (fma (* -6.0 x) z x)
   (if (<= x 2.15e-96) (* (* 6.0 z) y) (* (fma -6.0 z 1.0) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.5e-79) {
		tmp = fma((-6.0 * x), z, x);
	} else if (x <= 2.15e-96) {
		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 <= -2.5e-79)
		tmp = fma(Float64(-6.0 * x), z, x);
	elseif (x <= 2.15e-96)
		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, -2.5e-79], N[(N[(-6.0 * x), $MachinePrecision] * z + x), $MachinePrecision], If[LessEqual[x, 2.15e-96], 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 < -2.5e-79

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

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

      1. 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 84.4

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

   5. Applied rewrites 84.4%

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

   if -2.5e-79 < x < 2.1499999999999999e-96

   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 2.1499999999999999e-96 < x

   1. Initial program 97.4%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
   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. 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 88.2

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

   7. Applied rewrites 88.2%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{-79}:\\ \;\;\;\;\mathsf{fma}\left(-6 \cdot x, z, x\right)\\ \mathbf{elif}\;x \leq 2.15 \cdot 10^{-96}:\\ \;\;\;\;\left(6 \cdot z\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-6, z, 1\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{-79}:\\ \;\;\;\;\mathsf{fma}\left(-6 \cdot x, z, x\right)\\ \mathbf{elif}\;x \leq 2.15 \cdot 10^{-96}:\\ \;\;\;\;\left(6 \cdot z\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-6 \cdot z, x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -2.5e-79)
   (fma (* -6.0 x) z x)
   (if (<= x 2.15e-96) (* (* 6.0 z) y) (fma (* -6.0 z) x x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.5e-79) {
		tmp = fma((-6.0 * x), z, x);
	} else if (x <= 2.15e-96) {
		tmp = (6.0 * z) * y;
	} else {
		tmp = fma((-6.0 * z), x, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -2.5e-79)
		tmp = fma(Float64(-6.0 * x), z, x);
	elseif (x <= 2.15e-96)
		tmp = Float64(Float64(6.0 * z) * y);
	else
		tmp = fma(Float64(-6.0 * z), x, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -2.5e-79], N[(N[(-6.0 * x), $MachinePrecision] * z + x), $MachinePrecision], If[LessEqual[x, 2.15e-96], N[(N[(6.0 * z), $MachinePrecision] * y), $MachinePrecision], N[(N[(-6.0 * z), $MachinePrecision] * x + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.5e-79

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

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

      1. 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 84.4

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

   5. Applied rewrites 84.4%

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

   if -2.5e-79 < x < 2.1499999999999999e-96

   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 2.1499999999999999e-96 < 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. 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 85.8

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

   5. Applied rewrites 85.8%

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

   7. Applied rewrites 88.2%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{-79}:\\ \;\;\;\;\mathsf{fma}\left(-6 \cdot x, z, x\right)\\ \mathbf{elif}\;x \leq 2.15 \cdot 10^{-96}:\\ \;\;\;\;\left(6 \cdot z\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-6 \cdot z, x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 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 9: 42.0% 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 99.0%

   \[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
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 40.2

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

5. Applied rewrites 40.2%

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

7. Applied rewrites 40.7%

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

8. Final simplification 40.7%

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

Alternative 10: 42.0% 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 99.0%

   \[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
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 40.2

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

5. Applied rewrites 40.2%

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

7. Applied rewrites 40.3%

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

8. Final simplification 40.3%

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