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

Percentage Accurate: 99.7% → 99.7%

Time: 8.0s

Alternatives: 14

Speedup: 1.1×

• 21.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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

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

   2. *-commutative N/A

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

   3. lift--.f64 N/A

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

   4. sub-neg N/A

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

   5. distribute-rgt-in N/A

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

   6. lower-fma.f64 N/A

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

   7. *-commutative N/A

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

   8. neg-mul-1 N/A

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

   9. associate-*r* N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. lower-*.f64 N/A

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

   13. metadata-eval 99.8

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

4. Applied rewrites 99.8%

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

   1. lift-fma.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 99.8

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

6. Applied rewrites 99.8%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 99.8

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

   5. lift-fma.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 99.8

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 99.8

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

8. Applied rewrites 99.8%

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

   1. lift-fma.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 99.8

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

10. Applied rewrites 99.8%

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

11. Final simplification 99.8%

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

Alternative 2: 98.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(y - x\right) \cdot 6\right) \cdot z\\ \mathbf{if}\;z \leq -0.115:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 0.165:\\ \;\;\;\;\mathsf{fma}\left(z \cdot y, 6, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (* (- y x) 6.0) z)))
   (if (<= z -0.115) t_0 (if (<= z 0.165) (fma (* z y) 6.0 x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = ((y - x) * 6.0) * z;
	double tmp;
	if (z <= -0.115) {
		tmp = t_0;
	} else if (z <= 0.165) {
		tmp = fma((z * y), 6.0, x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(y - x) * 6.0) * z)
	tmp = 0.0
	if (z <= -0.115)
		tmp = t_0;
	elseif (z <= 0.165)
		tmp = fma(Float64(z * y), 6.0, x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(y - x), $MachinePrecision] * 6.0), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -0.115], t$95$0, If[LessEqual[z, 0.165], N[(N[(z * y), $MachinePrecision] * 6.0 + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -0.115000000000000005 or 0.165000000000000008 < z

   1. Initial program 99.8%

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

   3. Taylor expanded in 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 61.4

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

   5. Applied rewrites 61.4%

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

   7. Applied rewrites 61.5%

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

   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. *-commutative N/A

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

      2. associate-*r* N/A

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

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

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

      4. unsub-neg N/A

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

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

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. metadata-eval N/A

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

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

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

      12. unsub-neg N/A

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

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

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

      14. lower-*.f64 N/A

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

      15. lower--.f64 98.7

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

   10. Applied rewrites 98.7%

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

   if -0.115000000000000005 < z < 0.165000000000000008

   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 99.9

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

   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 96.9

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

   7. Applied rewrites 96.9%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.115:\\ \;\;\;\;\left(\left(y - x\right) \cdot 6\right) \cdot z\\ \mathbf{elif}\;z \leq 0.165:\\ \;\;\;\;\mathsf{fma}\left(z \cdot y, 6, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y - x\right) \cdot 6\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-6, z, 1\right) \cdot x\\ \mathbf{if}\;x \leq -5.4 \cdot 10^{+17}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{-93}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot y, 6, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (fma -6.0 z 1.0) x)))
   (if (<= x -5.4e+17) t_0 (if (<= x 2.1e-93) (fma (* z y) 6.0 x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma(-6.0, z, 1.0) * x;
	double tmp;
	if (x <= -5.4e+17) {
		tmp = t_0;
	} else if (x <= 2.1e-93) {
		tmp = fma((z * y), 6.0, x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(fma(-6.0, z, 1.0) * x)
	tmp = 0.0
	if (x <= -5.4e+17)
		tmp = t_0;
	elseif (x <= 2.1e-93)
		tmp = fma(Float64(z * y), 6.0, x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(-6.0 * z + 1.0), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -5.4e+17], t$95$0, If[LessEqual[x, 2.1e-93], N[(N[(z * y), $MachinePrecision] * 6.0 + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -5.4e17 or 2.1000000000000001e-93 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      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 87.1

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

   5. Applied rewrites 87.1%

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

   if -5.4e17 < x < 2.1000000000000001e-93

   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 95.2

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

   7. Applied rewrites 95.2%

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

Alternative 4: 74.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.3 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot x, -6, x\right)\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-125}:\\ \;\;\;\;\left(z \cdot y\right) \cdot 6\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-6, z, 1\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -4.3e-5)
   (fma (* z x) -6.0 x)
   (if (<= x 1.9e-125) (* (* z y) 6.0) (* (fma -6.0 z 1.0) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -4.3e-5) {
		tmp = fma((z * x), -6.0, x);
	} else if (x <= 1.9e-125) {
		tmp = (z * y) * 6.0;
	} else {
		tmp = fma(-6.0, z, 1.0) * x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -4.3e-5)
		tmp = fma(Float64(z * x), -6.0, x);
	elseif (x <= 1.9e-125)
		tmp = Float64(Float64(z * y) * 6.0);
	else
		tmp = Float64(fma(-6.0, z, 1.0) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -4.3e-5], N[(N[(z * x), $MachinePrecision] * -6.0 + x), $MachinePrecision], If[LessEqual[x, 1.9e-125], N[(N[(z * y), $MachinePrecision] * 6.0), $MachinePrecision], N[(N[(-6.0 * z + 1.0), $MachinePrecision] * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.3000000000000002e-5

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-neg N/A

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

      5. distribute-rgt-in N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. neg-mul-1 N/A

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

      9. associate-*r* N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval 100.0

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

   4. Applied rewrites 100.0%

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

   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. *-rgt-identity N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      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 93.5

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

   7. Applied rewrites 93.5%

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

   if -4.3000000000000002e-5 < x < 1.9000000000000001e-125

   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 78.1

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

   5. Applied rewrites 78.1%

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

   if 1.9000000000000001e-125 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      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 78.5

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

   5. Applied rewrites 78.5%

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

Alternative 5: 74.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-6, z, 1\right) \cdot x\\ \mathbf{if}\;x \leq -4.3 \cdot 10^{-5}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-125}:\\ \;\;\;\;\left(z \cdot y\right) \cdot 6\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (fma -6.0 z 1.0) x)))
   (if (<= x -4.3e-5) t_0 (if (<= x 1.9e-125) (* (* z y) 6.0) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma(-6.0, z, 1.0) * x;
	double tmp;
	if (x <= -4.3e-5) {
		tmp = t_0;
	} else if (x <= 1.9e-125) {
		tmp = (z * y) * 6.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(fma(-6.0, z, 1.0) * x)
	tmp = 0.0
	if (x <= -4.3e-5)
		tmp = t_0;
	elseif (x <= 1.9e-125)
		tmp = Float64(Float64(z * y) * 6.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(-6.0 * z + 1.0), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -4.3e-5], t$95$0, If[LessEqual[x, 1.9e-125], N[(N[(z * y), $MachinePrecision] * 6.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.3000000000000002e-5 or 1.9000000000000001e-125 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 84.4

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

   5. Applied rewrites 84.4%

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

   if -4.3000000000000002e-5 < x < 1.9000000000000001e-125

   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 78.1

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

   5. Applied rewrites 78.1%

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

Alternative 6: 60.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -0.00042:\\ \;\;\;\;\left(6 \cdot y\right) \cdot z\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{-79}:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot y\right) \cdot 6\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -0.00042)
   (* (* 6.0 y) z)
   (if (<= z 3.1e-79) (* 1.0 x) (* (* z y) 6.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -0.00042) {
		tmp = (6.0 * y) * z;
	} else if (z <= 3.1e-79) {
		tmp = 1.0 * x;
	} 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 (z <= (-0.00042d0)) then
        tmp = (6.0d0 * y) * z
    else if (z <= 3.1d-79) then
        tmp = 1.0d0 * x
    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 (z <= -0.00042) {
		tmp = (6.0 * y) * z;
	} else if (z <= 3.1e-79) {
		tmp = 1.0 * x;
	} else {
		tmp = (z * y) * 6.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -0.00042:
		tmp = (6.0 * y) * z
	elif z <= 3.1e-79:
		tmp = 1.0 * x
	else:
		tmp = (z * y) * 6.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -0.00042)
		tmp = Float64(Float64(6.0 * y) * z);
	elseif (z <= 3.1e-79)
		tmp = Float64(1.0 * x);
	else
		tmp = Float64(Float64(z * y) * 6.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -0.00042)
		tmp = (6.0 * y) * z;
	elseif (z <= 3.1e-79)
		tmp = 1.0 * x;
	else
		tmp = (z * y) * 6.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -0.00042], N[(N[(6.0 * y), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[z, 3.1e-79], N[(1.0 * x), $MachinePrecision], N[(N[(z * y), $MachinePrecision] * 6.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.2000000000000002e-4

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. *-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 65.1

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

   5. Applied rewrites 65.1%

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

   7. Applied rewrites 65.2%

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

   if -4.2000000000000002e-4 < z < 3.0999999999999999e-79

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      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 76.9

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

   5. Applied rewrites 76.9%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 75.4%

      \[\leadsto 1 \cdot x \]

   if 3.0999999999999999e-79 < 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 55.7

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

   5. Applied rewrites 55.7%

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

Alternative 7: 60.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -0.00042:\\ \;\;\;\;\left(6 \cdot y\right) \cdot z\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{-79}:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot 6\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -0.00042)
   (* (* 6.0 y) z)
   (if (<= z 3.1e-79) (* 1.0 x) (* (* z 6.0) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -0.00042) {
		tmp = (6.0 * y) * z;
	} else if (z <= 3.1e-79) {
		tmp = 1.0 * x;
	} else {
		tmp = (z * 6.0) * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-0.00042d0)) then
        tmp = (6.0d0 * y) * z
    else if (z <= 3.1d-79) then
        tmp = 1.0d0 * x
    else
        tmp = (z * 6.0d0) * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -0.00042) {
		tmp = (6.0 * y) * z;
	} else if (z <= 3.1e-79) {
		tmp = 1.0 * x;
	} else {
		tmp = (z * 6.0) * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -0.00042:
		tmp = (6.0 * y) * z
	elif z <= 3.1e-79:
		tmp = 1.0 * x
	else:
		tmp = (z * 6.0) * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -0.00042)
		tmp = Float64(Float64(6.0 * y) * z);
	elseif (z <= 3.1e-79)
		tmp = Float64(1.0 * x);
	else
		tmp = Float64(Float64(z * 6.0) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -0.00042)
		tmp = (6.0 * y) * z;
	elseif (z <= 3.1e-79)
		tmp = 1.0 * x;
	else
		tmp = (z * 6.0) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -0.00042], N[(N[(6.0 * y), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[z, 3.1e-79], N[(1.0 * x), $MachinePrecision], N[(N[(z * 6.0), $MachinePrecision] * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.2000000000000002e-4

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. *-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 65.1

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

   5. Applied rewrites 65.1%

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

   7. Applied rewrites 65.2%

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

   if -4.2000000000000002e-4 < z < 3.0999999999999999e-79

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      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 76.9

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

   5. Applied rewrites 76.9%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 75.4%

      \[\leadsto 1 \cdot x \]

   if 3.0999999999999999e-79 < 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 55.7

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

   5. Applied rewrites 55.7%

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

   7. Applied rewrites 55.6%

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

Alternative 8: 60.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(6 \cdot y\right) \cdot z\\ \mathbf{if}\;z \leq -0.00042:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{-79}:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (* 6.0 y) z)))
   (if (<= z -0.00042) t_0 (if (<= z 3.1e-79) (* 1.0 x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (6.0 * y) * z;
	double tmp;
	if (z <= -0.00042) {
		tmp = t_0;
	} else if (z <= 3.1e-79) {
		tmp = 1.0 * x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (6.0d0 * y) * z
    if (z <= (-0.00042d0)) then
        tmp = t_0
    else if (z <= 3.1d-79) then
        tmp = 1.0d0 * x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (6.0 * y) * z;
	double tmp;
	if (z <= -0.00042) {
		tmp = t_0;
	} else if (z <= 3.1e-79) {
		tmp = 1.0 * x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (6.0 * y) * z
	tmp = 0
	if z <= -0.00042:
		tmp = t_0
	elif z <= 3.1e-79:
		tmp = 1.0 * x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(6.0 * y) * z)
	tmp = 0.0
	if (z <= -0.00042)
		tmp = t_0;
	elseif (z <= 3.1e-79)
		tmp = Float64(1.0 * x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (6.0 * y) * z;
	tmp = 0.0;
	if (z <= -0.00042)
		tmp = t_0;
	elseif (z <= 3.1e-79)
		tmp = 1.0 * x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(6.0 * y), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -0.00042], t$95$0, If[LessEqual[z, 3.1e-79], N[(1.0 * x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -4.2000000000000002e-4 or 3.0999999999999999e-79 < 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 59.8

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

   5. Applied rewrites 59.8%

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

   7. Applied rewrites 59.8%

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

   if -4.2000000000000002e-4 < z < 3.0999999999999999e-79

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      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 76.9

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

   5. Applied rewrites 76.9%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 75.4%

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

Alternative 9: 60.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(z \cdot -6\right) \cdot x\\ \mathbf{if}\;z \leq -0.165:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 0.165:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (* z -6.0) x)))
   (if (<= z -0.165) t_0 (if (<= z 0.165) (* 1.0 x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (z * -6.0) * x;
	double tmp;
	if (z <= -0.165) {
		tmp = t_0;
	} else if (z <= 0.165) {
		tmp = 1.0 * x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (z * (-6.0d0)) * x
    if (z <= (-0.165d0)) then
        tmp = t_0
    else if (z <= 0.165d0) then
        tmp = 1.0d0 * x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (z * -6.0) * x;
	double tmp;
	if (z <= -0.165) {
		tmp = t_0;
	} else if (z <= 0.165) {
		tmp = 1.0 * x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (z * -6.0) * x
	tmp = 0
	if z <= -0.165:
		tmp = t_0
	elif z <= 0.165:
		tmp = 1.0 * x
	else:
		tmp = t_0
	return tmp

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if z < -0.165000000000000008 or 0.165000000000000008 < z

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 45.0

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

   5. Applied rewrites 45.0%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 43.9%

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

   if -0.165000000000000008 < z < 0.165000000000000008

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      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 73.2

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

   5. Applied rewrites 73.2%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 70.2%

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

4. Final simplification 56.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.165:\\ \;\;\;\;\left(z \cdot -6\right) \cdot x\\ \mathbf{elif}\;z \leq 0.165:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot -6\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 60.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(z \cdot x\right) \cdot -6\\ \mathbf{if}\;z \leq -0.165:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 0.165:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (* z x) -6.0)))
   (if (<= z -0.165) t_0 (if (<= z 0.165) (* 1.0 x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (z * x) * -6.0;
	double tmp;
	if (z <= -0.165) {
		tmp = t_0;
	} else if (z <= 0.165) {
		tmp = 1.0 * x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (z * x) * (-6.0d0)
    if (z <= (-0.165d0)) then
        tmp = t_0
    else if (z <= 0.165d0) then
        tmp = 1.0d0 * x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (z * x) * -6.0;
	double tmp;
	if (z <= -0.165) {
		tmp = t_0;
	} else if (z <= 0.165) {
		tmp = 1.0 * x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (z * x) * -6.0
	tmp = 0
	if z <= -0.165:
		tmp = t_0
	elif z <= 0.165:
		tmp = 1.0 * x
	else:
		tmp = t_0
	return tmp

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if z < -0.165000000000000008 or 0.165000000000000008 < z

   1. Initial program 99.8%

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

   3. Taylor expanded in 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 61.4

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

   5. Applied rewrites 61.4%

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

   7. Applied rewrites 61.5%

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

   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. *-commutative N/A

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

      2. associate-*r* N/A

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

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

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

      4. unsub-neg N/A

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

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

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. metadata-eval N/A

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

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

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

      12. unsub-neg N/A

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

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

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

      14. lower-*.f64 N/A

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

      15. lower--.f64 98.7

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

   10. Applied rewrites 98.7%

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

   11. Taylor expanded in x around inf

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

   13. Applied rewrites 43.9%

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

   if -0.165000000000000008 < z < 0.165000000000000008

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      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 73.2

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

   5. Applied rewrites 73.2%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 70.2%

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

Alternative 11: 99.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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

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

   2. *-commutative N/A

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

   3. lift--.f64 N/A

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

   4. sub-neg N/A

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

   5. distribute-rgt-in N/A

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

   6. lower-fma.f64 N/A

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

   7. *-commutative N/A

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

   8. neg-mul-1 N/A

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

   9. associate-*r* N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. lower-*.f64 N/A

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

   13. metadata-eval 99.8

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

4. Applied rewrites 99.8%

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

   1. lift-fma.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 99.8

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

6. Applied rewrites 99.8%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 99.8

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

   5. lift-fma.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 99.8

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 99.8

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

8. Applied rewrites 99.8%

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

9. Final simplification 99.8%

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

Alternative 12: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 13: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

   \[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
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 99.8

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 14: 35.7% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   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 58.5

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

5. Applied rewrites 58.5%

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

6. Taylor expanded in z around 0

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

8. Applied rewrites 35.2%

   \[\leadsto 1 \cdot x \]
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 2024294 
(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)))