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

Percentage Accurate: 99.5% → 99.5%

Time: 10.5s

Alternatives: 15

Speedup: 1.0×

• 21.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+ x (* (* (- y x) 6.0) (- (/ 2.0 3.0) z))))

C

double code(double x, double y, double z) {
	return x + (((y - x) * 6.0) * ((2.0 / 3.0) - 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) * ((2.0d0 / 3.0d0) - z))
end function

Java

public static double code(double x, double y, double z) {
	return x + (((y - x) * 6.0) * ((2.0 / 3.0) - z));
}

Python

def code(x, y, z):
	return x + (((y - x) * 6.0) * ((2.0 / 3.0) - z))

Julia

function code(x, y, z)
	return Float64(x + Float64(Float64(Float64(y - x) * 6.0) * Float64(Float64(2.0 / 3.0) - z)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = x + (((y - x) * 6.0) * ((2.0 / 3.0) - z));
end

Wolfram

code[x_, y_, z_] := N[(x + N[(N[(N[(y - x), $MachinePrecision] * 6.0), $MachinePrecision] * N[(N[(2.0 / 3.0), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+ x (* (* (- y x) 6.0) (- (/ 2.0 3.0) z))))

C

double code(double x, double y, double z) {
	return x + (((y - x) * 6.0) * ((2.0 / 3.0) - 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) * ((2.0d0 / 3.0d0) - z))
end function

Java

public static double code(double x, double y, double z) {
	return x + (((y - x) * 6.0) * ((2.0 / 3.0) - z));
}

Python

def code(x, y, z):
	return x + (((y - x) * 6.0) * ((2.0 / 3.0) - z))

Julia

function code(x, y, z)
	return Float64(x + Float64(Float64(Float64(y - x) * 6.0) * Float64(Float64(2.0 / 3.0) - z)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = x + (((y - x) * 6.0) * ((2.0 / 3.0) - z));
end

Wolfram

code[x_, y_, z_] := N[(x + N[(N[(N[(y - x), $MachinePrecision] * 6.0), $MachinePrecision] * N[(N[(2.0 / 3.0), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+ x (* (* (- y x) 6.0) (- (/ 2.0 3.0) z))))

C

double code(double x, double y, double z) {
	return x + (((y - x) * 6.0) * ((2.0 / 3.0) - 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) * ((2.0d0 / 3.0d0) - z))
end function

Java

public static double code(double x, double y, double z) {
	return x + (((y - x) * 6.0) * ((2.0 / 3.0) - z));
}

Python

def code(x, y, z):
	return x + (((y - x) * 6.0) * ((2.0 / 3.0) - z))

Julia

function code(x, y, z)
	return Float64(x + Float64(Float64(Float64(y - x) * 6.0) * Float64(Float64(2.0 / 3.0) - z)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = x + (((y - x) * 6.0) * ((2.0 / 3.0) - z));
end

Wolfram

code[x_, y_, z_] := N[(x + N[(N[(N[(y - x), $MachinePrecision] * 6.0), $MachinePrecision] * N[(N[(2.0 / 3.0), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

   \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 2: 74.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-6, z, 4\right) \cdot y\\ \mathbf{if}\;z \leq -2.4 \cdot 10^{+216}:\\ \;\;\;\;\left(-6 \cdot y\right) \cdot z\\ \mathbf{elif}\;z \leq -1.65 \cdot 10^{+120}:\\ \;\;\;\;\left(x \cdot 6\right) \cdot z\\ \mathbf{elif}\;z \leq -1.8 \cdot 10^{-12}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 0.245:\\ \;\;\;\;\mathsf{fma}\left(-3, x, 4 \cdot y\right)\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{+63}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot z\right) \cdot 6\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (fma -6.0 z 4.0) y)))
   (if (<= z -2.4e+216)
     (* (* -6.0 y) z)
     (if (<= z -1.65e+120)
       (* (* x 6.0) z)
       (if (<= z -1.8e-12)
         t_0
         (if (<= z 0.245)
           (fma -3.0 x (* 4.0 y))
           (if (<= z 3.8e+63) t_0 (* (* x z) 6.0))))))))

C

double code(double x, double y, double z) {
	double t_0 = fma(-6.0, z, 4.0) * y;
	double tmp;
	if (z <= -2.4e+216) {
		tmp = (-6.0 * y) * z;
	} else if (z <= -1.65e+120) {
		tmp = (x * 6.0) * z;
	} else if (z <= -1.8e-12) {
		tmp = t_0;
	} else if (z <= 0.245) {
		tmp = fma(-3.0, x, (4.0 * y));
	} else if (z <= 3.8e+63) {
		tmp = t_0;
	} else {
		tmp = (x * z) * 6.0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(fma(-6.0, z, 4.0) * y)
	tmp = 0.0
	if (z <= -2.4e+216)
		tmp = Float64(Float64(-6.0 * y) * z);
	elseif (z <= -1.65e+120)
		tmp = Float64(Float64(x * 6.0) * z);
	elseif (z <= -1.8e-12)
		tmp = t_0;
	elseif (z <= 0.245)
		tmp = fma(-3.0, x, Float64(4.0 * y));
	elseif (z <= 3.8e+63)
		tmp = t_0;
	else
		tmp = Float64(Float64(x * z) * 6.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(-6.0 * z + 4.0), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[z, -2.4e+216], N[(N[(-6.0 * y), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[z, -1.65e+120], N[(N[(x * 6.0), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[z, -1.8e-12], t$95$0, If[LessEqual[z, 0.245], N[(-3.0 * x + N[(4.0 * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3.8e+63], t$95$0, N[(N[(x * z), $MachinePrecision] * 6.0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if z < -2.3999999999999999e216

   1. Initial program 99.7%

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

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) + x} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} + x \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{2}{3} - z\right) \cdot \left(\left(y - x\right) \cdot 6\right)} + x \]

      5. lower-fma.f64 99.7

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{2}{3} - z, \left(y - x\right) \cdot 6, x\right)} \]

      6. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{2}{3}} - z, \left(y - x\right) \cdot 6, x\right) \]

      7. metadata-eval 99.7

         \[\leadsto \mathsf{fma}\left(\color{blue}{0.6666666666666666} - z, \left(y - x\right) \cdot 6, x\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{2}{3} - z, \color{blue}{\left(y - x\right) \cdot 6}, x\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{2}{3} - z, \color{blue}{6 \cdot \left(y - x\right)}, x\right) \]

      10. lower-*.f64 99.7

          \[\leadsto \mathsf{fma}\left(0.6666666666666666 - z, \color{blue}{6 \cdot \left(y - x\right)}, x\right) \]

   4. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.6666666666666666 - z, 6 \cdot \left(y - x\right), x\right)} \]

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 99.7

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

   7. Applied rewrites 99.7%

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

   9. Applied rewrites 99.7%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 68.1%

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

   if -2.3999999999999999e216 < z < -1.64999999999999995e120

   1. Initial program 99.8%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(1 + -6 \cdot \left(\frac{2}{3} - z\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(-6 \cdot \left(\frac{2}{3} - z\right) + 1\right)} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(-6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot x + 1 \cdot x} \]

      3. *-lft-identity N/A

         \[\leadsto \left(-6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot x + \color{blue}{x} \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-6 \cdot \left(\frac{2}{3} - z\right), x, x\right)} \]

      5. *-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(-6 \cdot \left(\frac{2}{3} - \color{blue}{1 \cdot z}\right), x, x\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(-6 \cdot \left(\frac{2}{3} - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot z\right), x, x\right) \]

      7. fp-cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{fma}\left(-6 \cdot \color{blue}{\left(\frac{2}{3} + -1 \cdot z\right)}, x, x\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(-6 \cdot \color{blue}{\left(-1 \cdot z + \frac{2}{3}\right)}, x, x\right) \]

      9. distribute-lft-in N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-6 \cdot \left(-1 \cdot z\right) + -6 \cdot \frac{2}{3}}, x, x\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(-6 \cdot \color{blue}{\left(\mathsf{neg}\left(z\right)\right)} + -6 \cdot \frac{2}{3}, x, x\right) \]

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

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(-6 \cdot z\right)\right)} + -6 \cdot \frac{2}{3}, x, x\right) \]

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

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(-6\right)\right) \cdot z} + -6 \cdot \frac{2}{3}, x, x\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{6} \cdot z + -6 \cdot \frac{2}{3}, x, x\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(6 \cdot z + \color{blue}{-4}, x, x\right) \]

      15. lower-fma.f64 74.1

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(6, z, -4\right)}, x, x\right) \]

   5. Applied rewrites 74.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(6, z, -4\right), x, x\right)} \]

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 74.1%

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

   10. Applied rewrites 74.1%

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

   if -1.64999999999999995e120 < z < -1.8e-12 or 0.245 < z < 3.8000000000000001e63

   1. Initial program 99.5%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{6 \cdot \left(y \cdot \left(\frac{2}{3} - z\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto 6 \cdot \color{blue}{\left(\left(\frac{2}{3} - z\right) \cdot y\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y} \]

      4. *-lft-identity N/A

         \[\leadsto \left(6 \cdot \left(\frac{2}{3} - \color{blue}{1 \cdot z}\right)\right) \cdot y \]

      5. metadata-eval N/A

         \[\leadsto \left(6 \cdot \left(\frac{2}{3} - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot z\right)\right) \cdot y \]

      6. fp-cancel-sign-sub-inv N/A

         \[\leadsto \left(6 \cdot \color{blue}{\left(\frac{2}{3} + -1 \cdot z\right)}\right) \cdot y \]

      7. +-commutative N/A

         \[\leadsto \left(6 \cdot \color{blue}{\left(-1 \cdot z + \frac{2}{3}\right)}\right) \cdot y \]

      8. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\left(-1 \cdot z\right) \cdot 6 + \frac{2}{3} \cdot 6\right)} \cdot y \]

      9. metadata-eval N/A

         \[\leadsto \left(\left(-1 \cdot z\right) \cdot 6 + \color{blue}{4}\right) \cdot y \]

      10. mul-1-neg N/A

          \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(z\right)\right)} \cdot 6 + 4\right) \cdot y \]

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

          \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(z \cdot 6\right)\right)} + 4\right) \cdot y \]

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

          \[\leadsto \left(\color{blue}{z \cdot \left(\mathsf{neg}\left(6\right)\right)} + 4\right) \cdot y \]

      13. metadata-eval N/A

          \[\leadsto \left(z \cdot \color{blue}{-6} + 4\right) \cdot y \]

      14. *-commutative N/A

          \[\leadsto \left(\color{blue}{-6 \cdot z} + 4\right) \cdot y \]

      15. lower-fma.f64 70.9

          \[\leadsto \color{blue}{\mathsf{fma}\left(-6, z, 4\right)} \cdot y \]

   5. Applied rewrites 70.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-6, z, 4\right) \cdot y} \]

   if -1.8e-12 < z < 0.245

   1. Initial program 99.5%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{4 \cdot \left(y - x\right) + x} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(y - x\right) \cdot 4} + x \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 4, x\right)} \]

      4. lower--.f64 98.5

         \[\leadsto \mathsf{fma}\left(\color{blue}{y - x}, 4, x\right) \]

   5. Applied rewrites 98.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 4, x\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto -3 \cdot x + \color{blue}{4 \cdot y} \]
   7. Step-by-step derivation

   8. Applied rewrites 98.6%

      \[\leadsto \mathsf{fma}\left(-3, \color{blue}{x}, 4 \cdot y\right) \]

   if 3.8000000000000001e63 < z

   1. Initial program 99.7%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(1 + -6 \cdot \left(\frac{2}{3} - z\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(-6 \cdot \left(\frac{2}{3} - z\right) + 1\right)} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(-6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot x + 1 \cdot x} \]

      3. *-lft-identity N/A

         \[\leadsto \left(-6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot x + \color{blue}{x} \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-6 \cdot \left(\frac{2}{3} - z\right), x, x\right)} \]

      5. *-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(-6 \cdot \left(\frac{2}{3} - \color{blue}{1 \cdot z}\right), x, x\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(-6 \cdot \left(\frac{2}{3} - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot z\right), x, x\right) \]

      7. fp-cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{fma}\left(-6 \cdot \color{blue}{\left(\frac{2}{3} + -1 \cdot z\right)}, x, x\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(-6 \cdot \color{blue}{\left(-1 \cdot z + \frac{2}{3}\right)}, x, x\right) \]

      9. distribute-lft-in N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-6 \cdot \left(-1 \cdot z\right) + -6 \cdot \frac{2}{3}}, x, x\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(-6 \cdot \color{blue}{\left(\mathsf{neg}\left(z\right)\right)} + -6 \cdot \frac{2}{3}, x, x\right) \]

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

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(-6 \cdot z\right)\right)} + -6 \cdot \frac{2}{3}, x, x\right) \]

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

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(-6\right)\right) \cdot z} + -6 \cdot \frac{2}{3}, x, x\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{6} \cdot z + -6 \cdot \frac{2}{3}, x, x\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(6 \cdot z + \color{blue}{-4}, x, x\right) \]

      15. lower-fma.f64 67.4

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(6, z, -4\right)}, x, x\right) \]

   5. Applied rewrites 67.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(6, z, -4\right), x, x\right)} \]

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 67.4%

      \[\leadsto \left(x \cdot z\right) \cdot \color{blue}{6} \]
3. Recombined 5 regimes into one program.

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.4 \cdot 10^{+216}:\\ \;\;\;\;\left(-6 \cdot y\right) \cdot z\\ \mathbf{elif}\;z \leq -1.65 \cdot 10^{+120}:\\ \;\;\;\;\left(x \cdot 6\right) \cdot z\\ \mathbf{elif}\;z \leq -1.8 \cdot 10^{-12}:\\ \;\;\;\;\mathsf{fma}\left(-6, z, 4\right) \cdot y\\ \mathbf{elif}\;z \leq 0.245:\\ \;\;\;\;\mathsf{fma}\left(-3, x, 4 \cdot y\right)\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{+63}:\\ \;\;\;\;\mathsf{fma}\left(-6, z, 4\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot z\right) \cdot 6\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-6 \cdot y\right) \cdot z\\ \mathbf{if}\;z \leq -2.4 \cdot 10^{+216}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq -1.65 \cdot 10^{+120}:\\ \;\;\;\;\left(x \cdot 6\right) \cdot z\\ \mathbf{elif}\;z \leq -38000:\\ \;\;\;\;\left(y \cdot z\right) \cdot -6\\ \mathbf{elif}\;z \leq 0.6:\\ \;\;\;\;\mathsf{fma}\left(-3, x, 4 \cdot y\right)\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{+63}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot z\right) \cdot 6\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (* -6.0 y) z)))
   (if (<= z -2.4e+216)
     t_0
     (if (<= z -1.65e+120)
       (* (* x 6.0) z)
       (if (<= z -38000.0)
         (* (* y z) -6.0)
         (if (<= z 0.6)
           (fma -3.0 x (* 4.0 y))
           (if (<= z 3.8e+63) t_0 (* (* x z) 6.0))))))))

C

double code(double x, double y, double z) {
	double t_0 = (-6.0 * y) * z;
	double tmp;
	if (z <= -2.4e+216) {
		tmp = t_0;
	} else if (z <= -1.65e+120) {
		tmp = (x * 6.0) * z;
	} else if (z <= -38000.0) {
		tmp = (y * z) * -6.0;
	} else if (z <= 0.6) {
		tmp = fma(-3.0, x, (4.0 * y));
	} else if (z <= 3.8e+63) {
		tmp = t_0;
	} else {
		tmp = (x * z) * 6.0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(-6.0 * y) * z)
	tmp = 0.0
	if (z <= -2.4e+216)
		tmp = t_0;
	elseif (z <= -1.65e+120)
		tmp = Float64(Float64(x * 6.0) * z);
	elseif (z <= -38000.0)
		tmp = Float64(Float64(y * z) * -6.0);
	elseif (z <= 0.6)
		tmp = fma(-3.0, x, Float64(4.0 * y));
	elseif (z <= 3.8e+63)
		tmp = t_0;
	else
		tmp = Float64(Float64(x * z) * 6.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(-6.0 * y), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -2.4e+216], t$95$0, If[LessEqual[z, -1.65e+120], N[(N[(x * 6.0), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[z, -38000.0], N[(N[(y * z), $MachinePrecision] * -6.0), $MachinePrecision], If[LessEqual[z, 0.6], N[(-3.0 * x + N[(4.0 * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3.8e+63], t$95$0, N[(N[(x * z), $MachinePrecision] * 6.0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if z < -2.3999999999999999e216 or 0.599999999999999978 < z < 3.8000000000000001e63

   1. Initial program 99.7%

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

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) + x} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} + x \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{2}{3} - z\right) \cdot \left(\left(y - x\right) \cdot 6\right)} + x \]

      5. lower-fma.f64 99.7

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{2}{3} - z, \left(y - x\right) \cdot 6, x\right)} \]

      6. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{2}{3}} - z, \left(y - x\right) \cdot 6, x\right) \]

      7. metadata-eval 99.7

         \[\leadsto \mathsf{fma}\left(\color{blue}{0.6666666666666666} - z, \left(y - x\right) \cdot 6, x\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{2}{3} - z, \color{blue}{\left(y - x\right) \cdot 6}, x\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{2}{3} - z, \color{blue}{6 \cdot \left(y - x\right)}, x\right) \]

      10. lower-*.f64 99.7

          \[\leadsto \mathsf{fma}\left(0.6666666666666666 - z, \color{blue}{6 \cdot \left(y - x\right)}, x\right) \]

   4. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.6666666666666666 - z, 6 \cdot \left(y - x\right), x\right)} \]

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 96.3

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

   7. Applied rewrites 96.3%

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

   9. Applied rewrites 96.4%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 70.1%

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

   if -2.3999999999999999e216 < z < -1.64999999999999995e120

   1. Initial program 99.8%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(1 + -6 \cdot \left(\frac{2}{3} - z\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(-6 \cdot \left(\frac{2}{3} - z\right) + 1\right)} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(-6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot x + 1 \cdot x} \]

      3. *-lft-identity N/A

         \[\leadsto \left(-6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot x + \color{blue}{x} \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-6 \cdot \left(\frac{2}{3} - z\right), x, x\right)} \]

      5. *-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(-6 \cdot \left(\frac{2}{3} - \color{blue}{1 \cdot z}\right), x, x\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(-6 \cdot \left(\frac{2}{3} - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot z\right), x, x\right) \]

      7. fp-cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{fma}\left(-6 \cdot \color{blue}{\left(\frac{2}{3} + -1 \cdot z\right)}, x, x\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(-6 \cdot \color{blue}{\left(-1 \cdot z + \frac{2}{3}\right)}, x, x\right) \]

      9. distribute-lft-in N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-6 \cdot \left(-1 \cdot z\right) + -6 \cdot \frac{2}{3}}, x, x\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(-6 \cdot \color{blue}{\left(\mathsf{neg}\left(z\right)\right)} + -6 \cdot \frac{2}{3}, x, x\right) \]

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

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(-6 \cdot z\right)\right)} + -6 \cdot \frac{2}{3}, x, x\right) \]

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

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(-6\right)\right) \cdot z} + -6 \cdot \frac{2}{3}, x, x\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{6} \cdot z + -6 \cdot \frac{2}{3}, x, x\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(6 \cdot z + \color{blue}{-4}, x, x\right) \]

      15. lower-fma.f64 74.1

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(6, z, -4\right)}, x, x\right) \]

   5. Applied rewrites 74.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(6, z, -4\right), x, x\right)} \]

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 74.1%

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

   10. Applied rewrites 74.1%

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

   if -1.64999999999999995e120 < z < -38000

   1. Initial program 99.5%

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

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) + x} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} + x \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{2}{3} - z\right) \cdot \left(\left(y - x\right) \cdot 6\right)} + x \]

      5. lower-fma.f64 99.5

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{2}{3} - z, \left(y - x\right) \cdot 6, x\right)} \]

      6. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{2}{3}} - z, \left(y - x\right) \cdot 6, x\right) \]

      7. metadata-eval 99.5

         \[\leadsto \mathsf{fma}\left(\color{blue}{0.6666666666666666} - z, \left(y - x\right) \cdot 6, x\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{2}{3} - z, \color{blue}{\left(y - x\right) \cdot 6}, x\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{2}{3} - z, \color{blue}{6 \cdot \left(y - x\right)}, x\right) \]

      10. lower-*.f64 99.5

          \[\leadsto \mathsf{fma}\left(0.6666666666666666 - z, \color{blue}{6 \cdot \left(y - x\right)}, x\right) \]

   4. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.6666666666666666 - z, 6 \cdot \left(y - x\right), x\right)} \]

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 98.0

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

   7. Applied rewrites 98.0%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 68.5%

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

   if -38000 < z < 0.599999999999999978

   1. Initial program 99.4%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{4 \cdot \left(y - x\right) + x} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(y - x\right) \cdot 4} + x \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 4, x\right)} \]

      4. lower--.f64 96.5

         \[\leadsto \mathsf{fma}\left(\color{blue}{y - x}, 4, x\right) \]

   5. Applied rewrites 96.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 4, x\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto -3 \cdot x + \color{blue}{4 \cdot y} \]
   7. Step-by-step derivation

   8. Applied rewrites 96.6%

      \[\leadsto \mathsf{fma}\left(-3, \color{blue}{x}, 4 \cdot y\right) \]

   if 3.8000000000000001e63 < z

   1. Initial program 99.7%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(1 + -6 \cdot \left(\frac{2}{3} - z\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(-6 \cdot \left(\frac{2}{3} - z\right) + 1\right)} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(-6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot x + 1 \cdot x} \]

      3. *-lft-identity N/A

         \[\leadsto \left(-6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot x + \color{blue}{x} \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-6 \cdot \left(\frac{2}{3} - z\right), x, x\right)} \]

      5. *-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(-6 \cdot \left(\frac{2}{3} - \color{blue}{1 \cdot z}\right), x, x\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(-6 \cdot \left(\frac{2}{3} - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot z\right), x, x\right) \]

      7. fp-cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{fma}\left(-6 \cdot \color{blue}{\left(\frac{2}{3} + -1 \cdot z\right)}, x, x\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(-6 \cdot \color{blue}{\left(-1 \cdot z + \frac{2}{3}\right)}, x, x\right) \]

      9. distribute-lft-in N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-6 \cdot \left(-1 \cdot z\right) + -6 \cdot \frac{2}{3}}, x, x\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(-6 \cdot \color{blue}{\left(\mathsf{neg}\left(z\right)\right)} + -6 \cdot \frac{2}{3}, x, x\right) \]

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

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(-6 \cdot z\right)\right)} + -6 \cdot \frac{2}{3}, x, x\right) \]

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

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(-6\right)\right) \cdot z} + -6 \cdot \frac{2}{3}, x, x\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{6} \cdot z + -6 \cdot \frac{2}{3}, x, x\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(6 \cdot z + \color{blue}{-4}, x, x\right) \]

      15. lower-fma.f64 67.4

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(6, z, -4\right)}, x, x\right) \]

   5. Applied rewrites 67.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(6, z, -4\right), x, x\right)} \]

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 67.4%

      \[\leadsto \left(x \cdot z\right) \cdot \color{blue}{6} \]
3. Recombined 5 regimes into one program.

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.4 \cdot 10^{+216}:\\ \;\;\;\;\left(-6 \cdot y\right) \cdot z\\ \mathbf{elif}\;z \leq -1.65 \cdot 10^{+120}:\\ \;\;\;\;\left(x \cdot 6\right) \cdot z\\ \mathbf{elif}\;z \leq -38000:\\ \;\;\;\;\left(y \cdot z\right) \cdot -6\\ \mathbf{elif}\;z \leq 0.6:\\ \;\;\;\;\mathsf{fma}\left(-3, x, 4 \cdot y\right)\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{+63}:\\ \;\;\;\;\left(-6 \cdot y\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot z\right) \cdot 6\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-6 \cdot y\right) \cdot z\\ \mathbf{if}\;z \leq -2.4 \cdot 10^{+216}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq -1.65 \cdot 10^{+120}:\\ \;\;\;\;\left(x \cdot 6\right) \cdot z\\ \mathbf{elif}\;z \leq -38000:\\ \;\;\;\;\left(y \cdot z\right) \cdot -6\\ \mathbf{elif}\;z \leq 0.6:\\ \;\;\;\;\mathsf{fma}\left(y - x, 4, x\right)\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{+63}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot z\right) \cdot 6\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (* -6.0 y) z)))
   (if (<= z -2.4e+216)
     t_0
     (if (<= z -1.65e+120)
       (* (* x 6.0) z)
       (if (<= z -38000.0)
         (* (* y z) -6.0)
         (if (<= z 0.6)
           (fma (- y x) 4.0 x)
           (if (<= z 3.8e+63) t_0 (* (* x z) 6.0))))))))

C

double code(double x, double y, double z) {
	double t_0 = (-6.0 * y) * z;
	double tmp;
	if (z <= -2.4e+216) {
		tmp = t_0;
	} else if (z <= -1.65e+120) {
		tmp = (x * 6.0) * z;
	} else if (z <= -38000.0) {
		tmp = (y * z) * -6.0;
	} else if (z <= 0.6) {
		tmp = fma((y - x), 4.0, x);
	} else if (z <= 3.8e+63) {
		tmp = t_0;
	} else {
		tmp = (x * z) * 6.0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(-6.0 * y) * z)
	tmp = 0.0
	if (z <= -2.4e+216)
		tmp = t_0;
	elseif (z <= -1.65e+120)
		tmp = Float64(Float64(x * 6.0) * z);
	elseif (z <= -38000.0)
		tmp = Float64(Float64(y * z) * -6.0);
	elseif (z <= 0.6)
		tmp = fma(Float64(y - x), 4.0, x);
	elseif (z <= 3.8e+63)
		tmp = t_0;
	else
		tmp = Float64(Float64(x * z) * 6.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(-6.0 * y), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -2.4e+216], t$95$0, If[LessEqual[z, -1.65e+120], N[(N[(x * 6.0), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[z, -38000.0], N[(N[(y * z), $MachinePrecision] * -6.0), $MachinePrecision], If[LessEqual[z, 0.6], N[(N[(y - x), $MachinePrecision] * 4.0 + x), $MachinePrecision], If[LessEqual[z, 3.8e+63], t$95$0, N[(N[(x * z), $MachinePrecision] * 6.0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if z < -2.3999999999999999e216 or 0.599999999999999978 < z < 3.8000000000000001e63

   1. Initial program 99.7%

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

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) + x} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} + x \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{2}{3} - z\right) \cdot \left(\left(y - x\right) \cdot 6\right)} + x \]

      5. lower-fma.f64 99.7

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{2}{3} - z, \left(y - x\right) \cdot 6, x\right)} \]

      6. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{2}{3}} - z, \left(y - x\right) \cdot 6, x\right) \]

      7. metadata-eval 99.7

         \[\leadsto \mathsf{fma}\left(\color{blue}{0.6666666666666666} - z, \left(y - x\right) \cdot 6, x\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{2}{3} - z, \color{blue}{\left(y - x\right) \cdot 6}, x\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{2}{3} - z, \color{blue}{6 \cdot \left(y - x\right)}, x\right) \]

      10. lower-*.f64 99.7

          \[\leadsto \mathsf{fma}\left(0.6666666666666666 - z, \color{blue}{6 \cdot \left(y - x\right)}, x\right) \]

   4. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.6666666666666666 - z, 6 \cdot \left(y - x\right), x\right)} \]

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 96.3

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

   7. Applied rewrites 96.3%

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

   9. Applied rewrites 96.4%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 70.1%

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

   if -2.3999999999999999e216 < z < -1.64999999999999995e120

   1. Initial program 99.8%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(1 + -6 \cdot \left(\frac{2}{3} - z\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(-6 \cdot \left(\frac{2}{3} - z\right) + 1\right)} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(-6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot x + 1 \cdot x} \]

      3. *-lft-identity N/A

         \[\leadsto \left(-6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot x + \color{blue}{x} \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-6 \cdot \left(\frac{2}{3} - z\right), x, x\right)} \]

      5. *-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(-6 \cdot \left(\frac{2}{3} - \color{blue}{1 \cdot z}\right), x, x\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(-6 \cdot \left(\frac{2}{3} - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot z\right), x, x\right) \]

      7. fp-cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{fma}\left(-6 \cdot \color{blue}{\left(\frac{2}{3} + -1 \cdot z\right)}, x, x\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(-6 \cdot \color{blue}{\left(-1 \cdot z + \frac{2}{3}\right)}, x, x\right) \]

      9. distribute-lft-in N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-6 \cdot \left(-1 \cdot z\right) + -6 \cdot \frac{2}{3}}, x, x\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(-6 \cdot \color{blue}{\left(\mathsf{neg}\left(z\right)\right)} + -6 \cdot \frac{2}{3}, x, x\right) \]

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

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(-6 \cdot z\right)\right)} + -6 \cdot \frac{2}{3}, x, x\right) \]

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

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(-6\right)\right) \cdot z} + -6 \cdot \frac{2}{3}, x, x\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{6} \cdot z + -6 \cdot \frac{2}{3}, x, x\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(6 \cdot z + \color{blue}{-4}, x, x\right) \]

      15. lower-fma.f64 74.1

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(6, z, -4\right)}, x, x\right) \]

   5. Applied rewrites 74.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(6, z, -4\right), x, x\right)} \]

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 74.1%

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

   10. Applied rewrites 74.1%

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

   if -1.64999999999999995e120 < z < -38000

   1. Initial program 99.5%

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

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) + x} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} + x \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{2}{3} - z\right) \cdot \left(\left(y - x\right) \cdot 6\right)} + x \]

      5. lower-fma.f64 99.5

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{2}{3} - z, \left(y - x\right) \cdot 6, x\right)} \]

      6. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{2}{3}} - z, \left(y - x\right) \cdot 6, x\right) \]

      7. metadata-eval 99.5

         \[\leadsto \mathsf{fma}\left(\color{blue}{0.6666666666666666} - z, \left(y - x\right) \cdot 6, x\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{2}{3} - z, \color{blue}{\left(y - x\right) \cdot 6}, x\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{2}{3} - z, \color{blue}{6 \cdot \left(y - x\right)}, x\right) \]

      10. lower-*.f64 99.5

          \[\leadsto \mathsf{fma}\left(0.6666666666666666 - z, \color{blue}{6 \cdot \left(y - x\right)}, x\right) \]

   4. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.6666666666666666 - z, 6 \cdot \left(y - x\right), x\right)} \]

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 98.0

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

   7. Applied rewrites 98.0%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 68.5%

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

   if -38000 < z < 0.599999999999999978

   1. Initial program 99.4%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{4 \cdot \left(y - x\right) + x} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(y - x\right) \cdot 4} + x \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 4, x\right)} \]

      4. lower--.f64 96.5

         \[\leadsto \mathsf{fma}\left(\color{blue}{y - x}, 4, x\right) \]

   5. Applied rewrites 96.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 4, x\right)} \]

   if 3.8000000000000001e63 < z

   1. Initial program 99.7%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(1 + -6 \cdot \left(\frac{2}{3} - z\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(-6 \cdot \left(\frac{2}{3} - z\right) + 1\right)} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(-6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot x + 1 \cdot x} \]

      3. *-lft-identity N/A

         \[\leadsto \left(-6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot x + \color{blue}{x} \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-6 \cdot \left(\frac{2}{3} - z\right), x, x\right)} \]

      5. *-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(-6 \cdot \left(\frac{2}{3} - \color{blue}{1 \cdot z}\right), x, x\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(-6 \cdot \left(\frac{2}{3} - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot z\right), x, x\right) \]

      7. fp-cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{fma}\left(-6 \cdot \color{blue}{\left(\frac{2}{3} + -1 \cdot z\right)}, x, x\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(-6 \cdot \color{blue}{\left(-1 \cdot z + \frac{2}{3}\right)}, x, x\right) \]

      9. distribute-lft-in N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-6 \cdot \left(-1 \cdot z\right) + -6 \cdot \frac{2}{3}}, x, x\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(-6 \cdot \color{blue}{\left(\mathsf{neg}\left(z\right)\right)} + -6 \cdot \frac{2}{3}, x, x\right) \]

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

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(-6 \cdot z\right)\right)} + -6 \cdot \frac{2}{3}, x, x\right) \]

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

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(-6\right)\right) \cdot z} + -6 \cdot \frac{2}{3}, x, x\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{6} \cdot z + -6 \cdot \frac{2}{3}, x, x\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(6 \cdot z + \color{blue}{-4}, x, x\right) \]

      15. lower-fma.f64 67.4

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(6, z, -4\right)}, x, x\right) \]

   5. Applied rewrites 67.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(6, z, -4\right), x, x\right)} \]

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 67.4%

      \[\leadsto \left(x \cdot z\right) \cdot \color{blue}{6} \]
3. Recombined 5 regimes into one program.

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.4 \cdot 10^{+216}:\\ \;\;\;\;\left(-6 \cdot y\right) \cdot z\\ \mathbf{elif}\;z \leq -1.65 \cdot 10^{+120}:\\ \;\;\;\;\left(x \cdot 6\right) \cdot z\\ \mathbf{elif}\;z \leq -38000:\\ \;\;\;\;\left(y \cdot z\right) \cdot -6\\ \mathbf{elif}\;z \leq 0.6:\\ \;\;\;\;\mathsf{fma}\left(y - x, 4, x\right)\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{+63}:\\ \;\;\;\;\left(-6 \cdot y\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot z\right) \cdot 6\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-6 \cdot y\right) \cdot z\\ \mathbf{if}\;z \leq -2.4 \cdot 10^{+216}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq -1.65 \cdot 10^{+120}:\\ \;\;\;\;\left(x \cdot 6\right) \cdot z\\ \mathbf{elif}\;z \leq -38000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 0.6:\\ \;\;\;\;\mathsf{fma}\left(y - x, 4, x\right)\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{+63}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot z\right) \cdot 6\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (* -6.0 y) z)))
   (if (<= z -2.4e+216)
     t_0
     (if (<= z -1.65e+120)
       (* (* x 6.0) z)
       (if (<= z -38000.0)
         t_0
         (if (<= z 0.6)
           (fma (- y x) 4.0 x)
           (if (<= z 3.8e+63) t_0 (* (* x z) 6.0))))))))

C

double code(double x, double y, double z) {
	double t_0 = (-6.0 * y) * z;
	double tmp;
	if (z <= -2.4e+216) {
		tmp = t_0;
	} else if (z <= -1.65e+120) {
		tmp = (x * 6.0) * z;
	} else if (z <= -38000.0) {
		tmp = t_0;
	} else if (z <= 0.6) {
		tmp = fma((y - x), 4.0, x);
	} else if (z <= 3.8e+63) {
		tmp = t_0;
	} else {
		tmp = (x * z) * 6.0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(-6.0 * y) * z)
	tmp = 0.0
	if (z <= -2.4e+216)
		tmp = t_0;
	elseif (z <= -1.65e+120)
		tmp = Float64(Float64(x * 6.0) * z);
	elseif (z <= -38000.0)
		tmp = t_0;
	elseif (z <= 0.6)
		tmp = fma(Float64(y - x), 4.0, x);
	elseif (z <= 3.8e+63)
		tmp = t_0;
	else
		tmp = Float64(Float64(x * z) * 6.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(-6.0 * y), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -2.4e+216], t$95$0, If[LessEqual[z, -1.65e+120], N[(N[(x * 6.0), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[z, -38000.0], t$95$0, If[LessEqual[z, 0.6], N[(N[(y - x), $MachinePrecision] * 4.0 + x), $MachinePrecision], If[LessEqual[z, 3.8e+63], t$95$0, N[(N[(x * z), $MachinePrecision] * 6.0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -2.3999999999999999e216 or -1.64999999999999995e120 < z < -38000 or 0.599999999999999978 < z < 3.8000000000000001e63

   1. Initial program 99.6%

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

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) + x} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} + x \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{2}{3} - z\right) \cdot \left(\left(y - x\right) \cdot 6\right)} + x \]

      5. lower-fma.f64 99.6

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{2}{3} - z, \left(y - x\right) \cdot 6, x\right)} \]

      6. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{2}{3}} - z, \left(y - x\right) \cdot 6, x\right) \]

      7. metadata-eval 99.6

         \[\leadsto \mathsf{fma}\left(\color{blue}{0.6666666666666666} - z, \left(y - x\right) \cdot 6, x\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{2}{3} - z, \color{blue}{\left(y - x\right) \cdot 6}, x\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{2}{3} - z, \color{blue}{6 \cdot \left(y - x\right)}, x\right) \]

      10. lower-*.f64 99.6

          \[\leadsto \mathsf{fma}\left(0.6666666666666666 - z, \color{blue}{6 \cdot \left(y - x\right)}, x\right) \]

   4. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.6666666666666666 - z, 6 \cdot \left(y - x\right), x\right)} \]

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 97.0

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

   7. Applied rewrites 97.0%

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

   9. Applied rewrites 97.0%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 69.4%

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

   if -2.3999999999999999e216 < z < -1.64999999999999995e120

   1. Initial program 99.8%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(1 + -6 \cdot \left(\frac{2}{3} - z\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(-6 \cdot \left(\frac{2}{3} - z\right) + 1\right)} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(-6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot x + 1 \cdot x} \]

      3. *-lft-identity N/A

         \[\leadsto \left(-6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot x + \color{blue}{x} \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-6 \cdot \left(\frac{2}{3} - z\right), x, x\right)} \]

      5. *-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(-6 \cdot \left(\frac{2}{3} - \color{blue}{1 \cdot z}\right), x, x\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(-6 \cdot \left(\frac{2}{3} - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot z\right), x, x\right) \]

      7. fp-cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{fma}\left(-6 \cdot \color{blue}{\left(\frac{2}{3} + -1 \cdot z\right)}, x, x\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(-6 \cdot \color{blue}{\left(-1 \cdot z + \frac{2}{3}\right)}, x, x\right) \]

      9. distribute-lft-in N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-6 \cdot \left(-1 \cdot z\right) + -6 \cdot \frac{2}{3}}, x, x\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(-6 \cdot \color{blue}{\left(\mathsf{neg}\left(z\right)\right)} + -6 \cdot \frac{2}{3}, x, x\right) \]

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

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(-6 \cdot z\right)\right)} + -6 \cdot \frac{2}{3}, x, x\right) \]

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

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(-6\right)\right) \cdot z} + -6 \cdot \frac{2}{3}, x, x\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{6} \cdot z + -6 \cdot \frac{2}{3}, x, x\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(6 \cdot z + \color{blue}{-4}, x, x\right) \]

      15. lower-fma.f64 74.1

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(6, z, -4\right)}, x, x\right) \]

   5. Applied rewrites 74.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(6, z, -4\right), x, x\right)} \]

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 74.1%

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

   10. Applied rewrites 74.1%

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

   if -38000 < z < 0.599999999999999978

   1. Initial program 99.4%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{4 \cdot \left(y - x\right) + x} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(y - x\right) \cdot 4} + x \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 4, x\right)} \]

      4. lower--.f64 96.5

         \[\leadsto \mathsf{fma}\left(\color{blue}{y - x}, 4, x\right) \]

   5. Applied rewrites 96.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 4, x\right)} \]

   if 3.8000000000000001e63 < z

   1. Initial program 99.7%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(1 + -6 \cdot \left(\frac{2}{3} - z\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(-6 \cdot \left(\frac{2}{3} - z\right) + 1\right)} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(-6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot x + 1 \cdot x} \]

      3. *-lft-identity N/A

         \[\leadsto \left(-6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot x + \color{blue}{x} \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-6 \cdot \left(\frac{2}{3} - z\right), x, x\right)} \]

      5. *-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(-6 \cdot \left(\frac{2}{3} - \color{blue}{1 \cdot z}\right), x, x\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(-6 \cdot \left(\frac{2}{3} - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot z\right), x, x\right) \]

      7. fp-cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{fma}\left(-6 \cdot \color{blue}{\left(\frac{2}{3} + -1 \cdot z\right)}, x, x\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(-6 \cdot \color{blue}{\left(-1 \cdot z + \frac{2}{3}\right)}, x, x\right) \]

      9. distribute-lft-in N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-6 \cdot \left(-1 \cdot z\right) + -6 \cdot \frac{2}{3}}, x, x\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(-6 \cdot \color{blue}{\left(\mathsf{neg}\left(z\right)\right)} + -6 \cdot \frac{2}{3}, x, x\right) \]

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

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(-6 \cdot z\right)\right)} + -6 \cdot \frac{2}{3}, x, x\right) \]

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

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(-6\right)\right) \cdot z} + -6 \cdot \frac{2}{3}, x, x\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{6} \cdot z + -6 \cdot \frac{2}{3}, x, x\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(6 \cdot z + \color{blue}{-4}, x, x\right) \]

      15. lower-fma.f64 67.4

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(6, z, -4\right)}, x, x\right) \]

   5. Applied rewrites 67.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(6, z, -4\right), x, x\right)} \]

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 67.4%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.4 \cdot 10^{+216}:\\ \;\;\;\;\left(-6 \cdot y\right) \cdot z\\ \mathbf{elif}\;z \leq -1.65 \cdot 10^{+120}:\\ \;\;\;\;\left(x \cdot 6\right) \cdot z\\ \mathbf{elif}\;z \leq -38000:\\ \;\;\;\;\left(-6 \cdot y\right) \cdot z\\ \mathbf{elif}\;z \leq 0.6:\\ \;\;\;\;\mathsf{fma}\left(y - x, 4, x\right)\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{+63}:\\ \;\;\;\;\left(-6 \cdot y\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot z\right) \cdot 6\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -0.55 \lor \neg \left(z \leq 0.58\right):\\ \;\;\;\;\left(\left(y - x\right) \cdot z\right) \cdot -6\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-3, x, 4 \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -0.55) (not (<= z 0.58)))
   (* (* (- y x) z) -6.0)
   (fma -3.0 x (* 4.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -0.55) || !(z <= 0.58)) {
		tmp = ((y - x) * z) * -6.0;
	} else {
		tmp = fma(-3.0, x, (4.0 * y));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -0.55) || !(z <= 0.58))
		tmp = Float64(Float64(Float64(y - x) * z) * -6.0);
	else
		tmp = fma(-3.0, x, Float64(4.0 * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -0.55], N[Not[LessEqual[z, 0.58]], $MachinePrecision]], N[(N[(N[(y - x), $MachinePrecision] * z), $MachinePrecision] * -6.0), $MachinePrecision], N[(-3.0 * x + N[(4.0 * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -0.55000000000000004 or 0.57999999999999996 < z

   1. Initial program 99.7%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 97.5

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

   5. Applied rewrites 97.5%

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

   if -0.55000000000000004 < z < 0.57999999999999996

   1. Initial program 99.4%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{4 \cdot \left(y - x\right) + x} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(y - x\right) \cdot 4} + x \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 4, x\right)} \]

      4. lower--.f64 97.9

         \[\leadsto \mathsf{fma}\left(\color{blue}{y - x}, 4, x\right) \]

   5. Applied rewrites 97.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 4, x\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto -3 \cdot x + \color{blue}{4 \cdot y} \]
   7. Step-by-step derivation

   8. Applied rewrites 98.0%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.55 \lor \neg \left(z \leq 0.58\right):\\ \;\;\;\;\left(\left(y - x\right) \cdot z\right) \cdot -6\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-3, x, 4 \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{-76} \lor \neg \left(x \leq 7.8 \cdot 10^{-28}\right):\\ \;\;\;\;\mathsf{fma}\left(6, z, -3\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-6, z, 4\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.45e-76) (not (<= x 7.8e-28)))
   (* (fma 6.0 z -3.0) x)
   (* (fma -6.0 z 4.0) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.45e-76) || !(x <= 7.8e-28)) {
		tmp = fma(6.0, z, -3.0) * x;
	} else {
		tmp = fma(-6.0, z, 4.0) * y;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.45e-76) || !(x <= 7.8e-28))
		tmp = Float64(fma(6.0, z, -3.0) * x);
	else
		tmp = Float64(fma(-6.0, z, 4.0) * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.45e-76], N[Not[LessEqual[x, 7.8e-28]], $MachinePrecision]], N[(N[(6.0 * z + -3.0), $MachinePrecision] * x), $MachinePrecision], N[(N[(-6.0 * z + 4.0), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.4500000000000001e-76 or 7.79999999999999998e-28 < x

   1. Initial program 99.6%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(1 + -6 \cdot \left(\frac{2}{3} - z\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(-6 \cdot \left(\frac{2}{3} - z\right) + 1\right)} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(-6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot x + 1 \cdot x} \]

      3. *-lft-identity N/A

         \[\leadsto \left(-6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot x + \color{blue}{x} \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-6 \cdot \left(\frac{2}{3} - z\right), x, x\right)} \]

      5. *-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(-6 \cdot \left(\frac{2}{3} - \color{blue}{1 \cdot z}\right), x, x\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(-6 \cdot \left(\frac{2}{3} - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot z\right), x, x\right) \]

      7. fp-cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{fma}\left(-6 \cdot \color{blue}{\left(\frac{2}{3} + -1 \cdot z\right)}, x, x\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(-6 \cdot \color{blue}{\left(-1 \cdot z + \frac{2}{3}\right)}, x, x\right) \]

      9. distribute-lft-in N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-6 \cdot \left(-1 \cdot z\right) + -6 \cdot \frac{2}{3}}, x, x\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(-6 \cdot \color{blue}{\left(\mathsf{neg}\left(z\right)\right)} + -6 \cdot \frac{2}{3}, x, x\right) \]

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

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(-6 \cdot z\right)\right)} + -6 \cdot \frac{2}{3}, x, x\right) \]

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

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(-6\right)\right) \cdot z} + -6 \cdot \frac{2}{3}, x, x\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{6} \cdot z + -6 \cdot \frac{2}{3}, x, x\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(6 \cdot z + \color{blue}{-4}, x, x\right) \]

      15. lower-fma.f64 81.1

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(6, z, -4\right)}, x, x\right) \]

   5. Applied rewrites 81.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(6, z, -4\right), x, x\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 81.1%

      \[\leadsto \mathsf{fma}\left(6, z, -3\right) \cdot \color{blue}{x} \]

   if -1.4500000000000001e-76 < x < 7.79999999999999998e-28

   1. Initial program 99.5%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{6 \cdot \left(y \cdot \left(\frac{2}{3} - z\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto 6 \cdot \color{blue}{\left(\left(\frac{2}{3} - z\right) \cdot y\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y} \]

      4. *-lft-identity N/A

         \[\leadsto \left(6 \cdot \left(\frac{2}{3} - \color{blue}{1 \cdot z}\right)\right) \cdot y \]

      5. metadata-eval N/A

         \[\leadsto \left(6 \cdot \left(\frac{2}{3} - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot z\right)\right) \cdot y \]

      6. fp-cancel-sign-sub-inv N/A

         \[\leadsto \left(6 \cdot \color{blue}{\left(\frac{2}{3} + -1 \cdot z\right)}\right) \cdot y \]

      7. +-commutative N/A

         \[\leadsto \left(6 \cdot \color{blue}{\left(-1 \cdot z + \frac{2}{3}\right)}\right) \cdot y \]

      8. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\left(-1 \cdot z\right) \cdot 6 + \frac{2}{3} \cdot 6\right)} \cdot y \]

      9. metadata-eval N/A

         \[\leadsto \left(\left(-1 \cdot z\right) \cdot 6 + \color{blue}{4}\right) \cdot y \]

      10. mul-1-neg N/A

          \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(z\right)\right)} \cdot 6 + 4\right) \cdot y \]

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

          \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(z \cdot 6\right)\right)} + 4\right) \cdot y \]

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

          \[\leadsto \left(\color{blue}{z \cdot \left(\mathsf{neg}\left(6\right)\right)} + 4\right) \cdot y \]

      13. metadata-eval N/A

          \[\leadsto \left(z \cdot \color{blue}{-6} + 4\right) \cdot y \]

      14. *-commutative N/A

          \[\leadsto \left(\color{blue}{-6 \cdot z} + 4\right) \cdot y \]

      15. lower-fma.f64 78.9

          \[\leadsto \color{blue}{\mathsf{fma}\left(-6, z, 4\right)} \cdot y \]

   5. Applied rewrites 78.9%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{-76} \lor \neg \left(x \leq 7.8 \cdot 10^{-28}\right):\\ \;\;\;\;\mathsf{fma}\left(6, z, -3\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-6, z, 4\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.2 \lor \neg \left(z \leq 0.5\right):\\ \;\;\;\;\left(x \cdot z\right) \cdot 6\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - x, 4, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -3.2) (not (<= z 0.5))) (* (* x z) 6.0) (fma (- y x) 4.0 x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -3.2) || !(z <= 0.5)) {
		tmp = (x * z) * 6.0;
	} else {
		tmp = fma((y - x), 4.0, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -3.2) || !(z <= 0.5))
		tmp = Float64(Float64(x * z) * 6.0);
	else
		tmp = fma(Float64(y - x), 4.0, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -3.2], N[Not[LessEqual[z, 0.5]], $MachinePrecision]], N[(N[(x * z), $MachinePrecision] * 6.0), $MachinePrecision], N[(N[(y - x), $MachinePrecision] * 4.0 + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.2000000000000002 or 0.5 < z

   1. Initial program 99.7%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(1 + -6 \cdot \left(\frac{2}{3} - z\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(-6 \cdot \left(\frac{2}{3} - z\right) + 1\right)} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(-6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot x + 1 \cdot x} \]

      3. *-lft-identity N/A

         \[\leadsto \left(-6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot x + \color{blue}{x} \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-6 \cdot \left(\frac{2}{3} - z\right), x, x\right)} \]

      5. *-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(-6 \cdot \left(\frac{2}{3} - \color{blue}{1 \cdot z}\right), x, x\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(-6 \cdot \left(\frac{2}{3} - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot z\right), x, x\right) \]

      7. fp-cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{fma}\left(-6 \cdot \color{blue}{\left(\frac{2}{3} + -1 \cdot z\right)}, x, x\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(-6 \cdot \color{blue}{\left(-1 \cdot z + \frac{2}{3}\right)}, x, x\right) \]

      9. distribute-lft-in N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-6 \cdot \left(-1 \cdot z\right) + -6 \cdot \frac{2}{3}}, x, x\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(-6 \cdot \color{blue}{\left(\mathsf{neg}\left(z\right)\right)} + -6 \cdot \frac{2}{3}, x, x\right) \]

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

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(-6 \cdot z\right)\right)} + -6 \cdot \frac{2}{3}, x, x\right) \]

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

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(-6\right)\right) \cdot z} + -6 \cdot \frac{2}{3}, x, x\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{6} \cdot z + -6 \cdot \frac{2}{3}, x, x\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(6 \cdot z + \color{blue}{-4}, x, x\right) \]

      15. lower-fma.f64 54.2

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(6, z, -4\right)}, x, x\right) \]

   5. Applied rewrites 54.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(6, z, -4\right), x, x\right)} \]

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 52.6%

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

   if -3.2000000000000002 < z < 0.5

   1. Initial program 99.4%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{4 \cdot \left(y - x\right) + x} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(y - x\right) \cdot 4} + x \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 4, x\right)} \]

      4. lower--.f64 97.9

         \[\leadsto \mathsf{fma}\left(\color{blue}{y - x}, 4, x\right) \]

   5. Applied rewrites 97.9%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.2 \lor \neg \left(z \leq 0.5\right):\\ \;\;\;\;\left(x \cdot z\right) \cdot 6\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - x, 4, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 75.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.2 \lor \neg \left(z \leq 0.5\right):\\ \;\;\;\;\left(x \cdot 6\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - x, 4, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -3.2) (not (<= z 0.5))) (* (* x 6.0) z) (fma (- y x) 4.0 x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -3.2) || !(z <= 0.5)) {
		tmp = (x * 6.0) * z;
	} else {
		tmp = fma((y - x), 4.0, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -3.2) || !(z <= 0.5))
		tmp = Float64(Float64(x * 6.0) * z);
	else
		tmp = fma(Float64(y - x), 4.0, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -3.2], N[Not[LessEqual[z, 0.5]], $MachinePrecision]], N[(N[(x * 6.0), $MachinePrecision] * z), $MachinePrecision], N[(N[(y - x), $MachinePrecision] * 4.0 + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.2000000000000002 or 0.5 < z

   1. Initial program 99.7%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(1 + -6 \cdot \left(\frac{2}{3} - z\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(-6 \cdot \left(\frac{2}{3} - z\right) + 1\right)} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(-6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot x + 1 \cdot x} \]

      3. *-lft-identity N/A

         \[\leadsto \left(-6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot x + \color{blue}{x} \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-6 \cdot \left(\frac{2}{3} - z\right), x, x\right)} \]

      5. *-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(-6 \cdot \left(\frac{2}{3} - \color{blue}{1 \cdot z}\right), x, x\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(-6 \cdot \left(\frac{2}{3} - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot z\right), x, x\right) \]

      7. fp-cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{fma}\left(-6 \cdot \color{blue}{\left(\frac{2}{3} + -1 \cdot z\right)}, x, x\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(-6 \cdot \color{blue}{\left(-1 \cdot z + \frac{2}{3}\right)}, x, x\right) \]

      9. distribute-lft-in N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-6 \cdot \left(-1 \cdot z\right) + -6 \cdot \frac{2}{3}}, x, x\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(-6 \cdot \color{blue}{\left(\mathsf{neg}\left(z\right)\right)} + -6 \cdot \frac{2}{3}, x, x\right) \]

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

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(-6 \cdot z\right)\right)} + -6 \cdot \frac{2}{3}, x, x\right) \]

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

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(-6\right)\right) \cdot z} + -6 \cdot \frac{2}{3}, x, x\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{6} \cdot z + -6 \cdot \frac{2}{3}, x, x\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(6 \cdot z + \color{blue}{-4}, x, x\right) \]

      15. lower-fma.f64 54.2

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(6, z, -4\right)}, x, x\right) \]

   5. Applied rewrites 54.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(6, z, -4\right), x, x\right)} \]

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 52.6%

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

   10. Applied rewrites 52.5%

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

   if -3.2000000000000002 < z < 0.5

   1. Initial program 99.4%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{4 \cdot \left(y - x\right) + x} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(y - x\right) \cdot 4} + x \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 4, x\right)} \]

      4. lower--.f64 97.9

         \[\leadsto \mathsf{fma}\left(\color{blue}{y - x}, 4, x\right) \]

   5. Applied rewrites 97.9%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.2 \lor \neg \left(z \leq 0.5\right):\\ \;\;\;\;\left(x \cdot 6\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - x, 4, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 75.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.2:\\ \;\;\;\;\left(6 \cdot z\right) \cdot x\\ \mathbf{elif}\;z \leq 0.5:\\ \;\;\;\;\mathsf{fma}\left(y - x, 4, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot z\right) \cdot 6\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -3.2)
   (* (* 6.0 z) x)
   (if (<= z 0.5) (fma (- y x) 4.0 x) (* (* x z) 6.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -3.2) {
		tmp = (6.0 * z) * x;
	} else if (z <= 0.5) {
		tmp = fma((y - x), 4.0, x);
	} else {
		tmp = (x * z) * 6.0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -3.2)
		tmp = Float64(Float64(6.0 * z) * x);
	elseif (z <= 0.5)
		tmp = fma(Float64(y - x), 4.0, x);
	else
		tmp = Float64(Float64(x * z) * 6.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -3.2], N[(N[(6.0 * z), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[z, 0.5], N[(N[(y - x), $MachinePrecision] * 4.0 + x), $MachinePrecision], N[(N[(x * z), $MachinePrecision] * 6.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.2000000000000002

   1. Initial program 99.7%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(1 + -6 \cdot \left(\frac{2}{3} - z\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(-6 \cdot \left(\frac{2}{3} - z\right) + 1\right)} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(-6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot x + 1 \cdot x} \]

      3. *-lft-identity N/A

         \[\leadsto \left(-6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot x + \color{blue}{x} \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-6 \cdot \left(\frac{2}{3} - z\right), x, x\right)} \]

      5. *-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(-6 \cdot \left(\frac{2}{3} - \color{blue}{1 \cdot z}\right), x, x\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(-6 \cdot \left(\frac{2}{3} - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot z\right), x, x\right) \]

      7. fp-cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{fma}\left(-6 \cdot \color{blue}{\left(\frac{2}{3} + -1 \cdot z\right)}, x, x\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(-6 \cdot \color{blue}{\left(-1 \cdot z + \frac{2}{3}\right)}, x, x\right) \]

      9. distribute-lft-in N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-6 \cdot \left(-1 \cdot z\right) + -6 \cdot \frac{2}{3}}, x, x\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(-6 \cdot \color{blue}{\left(\mathsf{neg}\left(z\right)\right)} + -6 \cdot \frac{2}{3}, x, x\right) \]

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

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(-6 \cdot z\right)\right)} + -6 \cdot \frac{2}{3}, x, x\right) \]

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

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(-6\right)\right) \cdot z} + -6 \cdot \frac{2}{3}, x, x\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{6} \cdot z + -6 \cdot \frac{2}{3}, x, x\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(6 \cdot z + \color{blue}{-4}, x, x\right) \]

      15. lower-fma.f64 49.6

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(6, z, -4\right)}, x, x\right) \]

   5. Applied rewrites 49.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(6, z, -4\right), x, x\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 49.6%

      \[\leadsto \mathsf{fma}\left(6, z, -3\right) \cdot \color{blue}{x} \]

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 47.9%

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

   if -3.2000000000000002 < z < 0.5

   1. Initial program 99.4%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{4 \cdot \left(y - x\right) + x} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(y - x\right) \cdot 4} + x \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 4, x\right)} \]

      4. lower--.f64 97.9

         \[\leadsto \mathsf{fma}\left(\color{blue}{y - x}, 4, x\right) \]

   5. Applied rewrites 97.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 4, x\right)} \]

   if 0.5 < z

   1. Initial program 99.7%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(1 + -6 \cdot \left(\frac{2}{3} - z\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(-6 \cdot \left(\frac{2}{3} - z\right) + 1\right)} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(-6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot x + 1 \cdot x} \]

      3. *-lft-identity N/A

         \[\leadsto \left(-6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot x + \color{blue}{x} \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-6 \cdot \left(\frac{2}{3} - z\right), x, x\right)} \]

      5. *-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(-6 \cdot \left(\frac{2}{3} - \color{blue}{1 \cdot z}\right), x, x\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(-6 \cdot \left(\frac{2}{3} - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot z\right), x, x\right) \]

      7. fp-cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{fma}\left(-6 \cdot \color{blue}{\left(\frac{2}{3} + -1 \cdot z\right)}, x, x\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(-6 \cdot \color{blue}{\left(-1 \cdot z + \frac{2}{3}\right)}, x, x\right) \]

      9. distribute-lft-in N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-6 \cdot \left(-1 \cdot z\right) + -6 \cdot \frac{2}{3}}, x, x\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(-6 \cdot \color{blue}{\left(\mathsf{neg}\left(z\right)\right)} + -6 \cdot \frac{2}{3}, x, x\right) \]

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

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(-6 \cdot z\right)\right)} + -6 \cdot \frac{2}{3}, x, x\right) \]

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

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(-6\right)\right) \cdot z} + -6 \cdot \frac{2}{3}, x, x\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{6} \cdot z + -6 \cdot \frac{2}{3}, x, x\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(6 \cdot z + \color{blue}{-4}, x, x\right) \]

      15. lower-fma.f64 59.8

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(6, z, -4\right)}, x, x\right) \]

   5. Applied rewrites 59.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(6, z, -4\right), x, x\right)} \]

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 58.5%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.2:\\ \;\;\;\;\left(6 \cdot z\right) \cdot x\\ \mathbf{elif}\;z \leq 0.5:\\ \;\;\;\;\mathsf{fma}\left(y - x, 4, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot z\right) \cdot 6\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 38.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.4 \cdot 10^{+56} \lor \neg \left(y \leq 1.6 \cdot 10^{+23}\right):\\ \;\;\;\;4 \cdot y\\ \mathbf{else}:\\ \;\;\;\;-3 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -6.4e+56) (not (<= y 1.6e+23))) (* 4.0 y) (* -3.0 x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -6.4e+56) || !(y <= 1.6e+23)) {
		tmp = 4.0 * y;
	} else {
		tmp = -3.0 * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-6.4d+56)) .or. (.not. (y <= 1.6d+23))) then
        tmp = 4.0d0 * y
    else
        tmp = (-3.0d0) * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -6.4e+56) || !(y <= 1.6e+23)) {
		tmp = 4.0 * y;
	} else {
		tmp = -3.0 * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -6.4e+56) or not (y <= 1.6e+23):
		tmp = 4.0 * y
	else:
		tmp = -3.0 * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -6.4e+56) || !(y <= 1.6e+23))
		tmp = Float64(4.0 * y);
	else
		tmp = Float64(-3.0 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -6.4e+56) || ~((y <= 1.6e+23)))
		tmp = 4.0 * y;
	else
		tmp = -3.0 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -6.4e+56], N[Not[LessEqual[y, 1.6e+23]], $MachinePrecision]], N[(4.0 * y), $MachinePrecision], N[(-3.0 * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -6.40000000000000007e56 or 1.6e23 < y

   1. Initial program 99.7%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{4 \cdot \left(y - x\right) + x} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(y - x\right) \cdot 4} + x \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 4, x\right)} \]

      4. lower--.f64 41.8

         \[\leadsto \mathsf{fma}\left(\color{blue}{y - x}, 4, x\right) \]

   5. Applied rewrites 41.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 4, x\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto 4 \cdot \color{blue}{y} \]
   7. Step-by-step derivation

   8. Applied rewrites 31.7%

      \[\leadsto 4 \cdot \color{blue}{y} \]

   if -6.40000000000000007e56 < y < 1.6e23

   1. Initial program 99.5%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{4 \cdot \left(y - x\right) + x} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(y - x\right) \cdot 4} + x \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 4, x\right)} \]

      4. lower--.f64 51.7

         \[\leadsto \mathsf{fma}\left(\color{blue}{y - x}, 4, x\right) \]

   5. Applied rewrites 51.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 4, x\right)} \]

   6. Taylor expanded in x around inf

      \[\leadsto -3 \cdot \color{blue}{x} \]
   7. Step-by-step derivation

   8. Applied rewrites 40.8%

      \[\leadsto -3 \cdot \color{blue}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.4 \cdot 10^{+56} \lor \neg \left(y \leq 1.6 \cdot 10^{+23}\right):\\ \;\;\;\;4 \cdot y\\ \mathbf{else}:\\ \;\;\;\;-3 \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 99.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(0.6666666666666666 - z, 6 \cdot \left(y - x\right), x\right) \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (fma (- 0.6666666666666666 z) (* 6.0 (- y x)) x))

C

double code(double x, double y, double z) {
	return fma((0.6666666666666666 - z), (6.0 * (y - x)), x);
}

Julia

function code(x, y, z)
	return fma(Float64(0.6666666666666666 - z), Float64(6.0 * Float64(y - x)), x)
end

Wolfram

code[x_, y_, z_] := N[(N[(0.6666666666666666 - z), $MachinePrecision] * N[(6.0 * N[(y - x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

   1. lift-+.f64 N/A

      \[\leadsto \color{blue}{x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} \]

   2. +-commutative N/A

      \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) + x} \]

   3. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} + x \]

   4. *-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{2}{3} - z\right) \cdot \left(\left(y - x\right) \cdot 6\right)} + x \]

   5. lower-fma.f64 99.5

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{2}{3} - z, \left(y - x\right) \cdot 6, x\right)} \]

   6. lift-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{2}{3}} - z, \left(y - x\right) \cdot 6, x\right) \]

   7. metadata-eval 99.5

      \[\leadsto \mathsf{fma}\left(\color{blue}{0.6666666666666666} - z, \left(y - x\right) \cdot 6, x\right) \]

   8. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{2}{3} - z, \color{blue}{\left(y - x\right) \cdot 6}, x\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\frac{2}{3} - z, \color{blue}{6 \cdot \left(y - x\right)}, x\right) \]

   10. lower-*.f64 99.5

       \[\leadsto \mathsf{fma}\left(0.6666666666666666 - z, \color{blue}{6 \cdot \left(y - x\right)}, x\right) \]

4. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.6666666666666666 - z, 6 \cdot \left(y - x\right), x\right)} \]
5. Add Preprocessing

Alternative 13: 99.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\left(0.6666666666666666 - z\right) \cdot \left(y - x\right), 6, x\right) \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (fma (* (- 0.6666666666666666 z) (- y x)) 6.0 x))

C

double code(double x, double y, double z) {
	return fma(((0.6666666666666666 - z) * (y - x)), 6.0, x);
}

Julia

function code(x, y, z)
	return fma(Float64(Float64(0.6666666666666666 - z) * Float64(y - x)), 6.0, x)
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(0.6666666666666666 - z), $MachinePrecision] * N[(y - x), $MachinePrecision]), $MachinePrecision] * 6.0 + x), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

   1. lift-+.f64 N/A

      \[\leadsto \color{blue}{x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} \]

   2. +-commutative N/A

      \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) + x} \]

   3. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} + x \]

   4. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right)} \cdot \left(\frac{2}{3} - z\right) + x \]

   5. associate-*l* N/A

      \[\leadsto \color{blue}{\left(y - x\right) \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right)} + x \]

   6. *-commutative N/A

      \[\leadsto \left(y - x\right) \cdot \color{blue}{\left(\left(\frac{2}{3} - z\right) \cdot 6\right)} + x \]

   7. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot \left(\frac{2}{3} - z\right)\right) \cdot 6} + x \]

   8. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(y - x\right) \cdot \left(\frac{2}{3} - z\right), 6, x\right)} \]

   9. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{2}{3} - z\right) \cdot \left(y - x\right)}, 6, x\right) \]

   10. lower-*.f64 99.5

       \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{2}{3} - z\right) \cdot \left(y - x\right)}, 6, x\right) \]

   11. lift-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\frac{2}{3}} - z\right) \cdot \left(y - x\right), 6, x\right) \]

   12. metadata-eval 99.5

       \[\leadsto \mathsf{fma}\left(\left(\color{blue}{0.6666666666666666} - z\right) \cdot \left(y - x\right), 6, x\right) \]

4. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left(0.6666666666666666 - z\right) \cdot \left(y - x\right), 6, x\right)} \]
5. Add Preprocessing

Alternative 14: 50.9% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(y - x, 4, x\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fma (- y x) 4.0 x))

C

double code(double x, double y, double z) {
	return fma((y - x), 4.0, x);
}

Julia

function code(x, y, z)
	return fma(Float64(y - x), 4.0, x)
end

Wolfram

code[x_, y_, z_] := N[(N[(y - x), $MachinePrecision] * 4.0 + x), $MachinePrecision]

Derivation

1. Initial program 99.6%

   \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
2. Add Preprocessing

3. Taylor expanded in z around 0

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

   1. +-commutative N/A

      \[\leadsto \color{blue}{4 \cdot \left(y - x\right) + x} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\left(y - x\right) \cdot 4} + x \]

   3. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 4, x\right)} \]

   4. lower--.f64 47.8

      \[\leadsto \mathsf{fma}\left(\color{blue}{y - x}, 4, x\right) \]

5. Applied rewrites 47.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 4, x\right)} \]

6. Final simplification 47.8%

   \[\leadsto \mathsf{fma}\left(y - x, 4, x\right) \]
7. Add Preprocessing

Alternative 15: 27.0% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ -3 \cdot x \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* -3.0 x))

C

double code(double x, double y, double z) {
	return -3.0 * 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 = (-3.0d0) * x
end function

Java

public static double code(double x, double y, double z) {
	return -3.0 * x;
}

Python

def code(x, y, z):
	return -3.0 * x

Julia

function code(x, y, z)
	return Float64(-3.0 * x)
end

MATLAB

function tmp = code(x, y, z)
	tmp = -3.0 * x;
end

Wolfram

code[x_, y_, z_] := N[(-3.0 * x), $MachinePrecision]

Derivation

1. Initial program 99.6%

   \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
2. Add Preprocessing

3. Taylor expanded in z around 0

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

   1. +-commutative N/A

      \[\leadsto \color{blue}{4 \cdot \left(y - x\right) + x} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\left(y - x\right) \cdot 4} + x \]

   3. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 4, x\right)} \]

   4. lower--.f64 47.8

      \[\leadsto \mathsf{fma}\left(\color{blue}{y - x}, 4, x\right) \]

5. Applied rewrites 47.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 4, x\right)} \]

6. Taylor expanded in x around inf

   \[\leadsto -3 \cdot \color{blue}{x} \]
7. Step-by-step derivation

8. Applied rewrites 29.1%

   \[\leadsto -3 \cdot \color{blue}{x} \]

9. Final simplification 29.1%

   \[\leadsto -3 \cdot x \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024338 
(FPCore (x y z)
  :name "Data.Colour.RGBSpace.HSL:hsl from colour-2.3.3, D"
  :precision binary64
  (+ x (* (* (- y x) 6.0) (- (/ 2.0 3.0) z))))