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

Percentage Accurate: 99.6% → 99.8%

Time: 6.2s

Alternatives: 14

Speedup: 1.9×

• 21.5% 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: 99.6% 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: 99.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(-6, z, 4\right), y - x, x\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fma (fma -6.0 z 4.0) (- y x) x))

C

double code(double x, double y, double z) {
	return fma(fma(-6.0, z, 4.0), (y - x), x);
}

Julia

function code(x, y, z)
	return fma(fma(-6.0, z, 4.0), Float64(y - x), x)
end

Wolfram

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

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

   7. lower-fma.f64 N/A

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

   8. lift--.f64 N/A

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

   9. sub-neg N/A

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

   10. +-commutative N/A

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

   11. distribute-lft-in N/A

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

   12. neg-mul-1 N/A

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

   13. associate-*r* N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval N/A

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

   16. lower-fma.f64 N/A

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

   17. metadata-eval N/A

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

   18. lift-/.f64 N/A

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

   19. metadata-eval N/A

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

   20. metadata-eval 99.8

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

4. Applied rewrites 99.8%

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

Alternative 2: 74.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(6, z, -3\right) \cdot x\\ t_1 := \frac{2}{3} - z\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+250}:\\ \;\;\;\;\left(z \cdot -6\right) \cdot y\\ \mathbf{elif}\;t\_1 \leq 0.6666666666:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 50:\\ \;\;\;\;\mathsf{fma}\left(y - x, 4, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (fma 6.0 z -3.0) x)) (t_1 (- (/ 2.0 3.0) z)))
   (if (<= t_1 -1e+250)
     (* (* z -6.0) y)
     (if (<= t_1 0.6666666666)
       t_0
       (if (<= t_1 50.0) (fma (- y x) 4.0 x) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = fma(6.0, z, -3.0) * x;
	double t_1 = (2.0 / 3.0) - z;
	double tmp;
	if (t_1 <= -1e+250) {
		tmp = (z * -6.0) * y;
	} else if (t_1 <= 0.6666666666) {
		tmp = t_0;
	} else if (t_1 <= 50.0) {
		tmp = fma((y - x), 4.0, x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(fma(6.0, z, -3.0) * x)
	t_1 = Float64(Float64(2.0 / 3.0) - z)
	tmp = 0.0
	if (t_1 <= -1e+250)
		tmp = Float64(Float64(z * -6.0) * y);
	elseif (t_1 <= 0.6666666666)
		tmp = t_0;
	elseif (t_1 <= 50.0)
		tmp = fma(Float64(y - x), 4.0, x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(6.0 * z + -3.0), $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$1 = N[(N[(2.0 / 3.0), $MachinePrecision] - z), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+250], N[(N[(z * -6.0), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$1, 0.6666666666], t$95$0, If[LessEqual[t$95$1, 50.0], N[(N[(y - x), $MachinePrecision] * 4.0 + x), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < -9.9999999999999992e249

   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. 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 99.8

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

      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 99.8

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

      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 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in y around inf

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

      1. metadata-eval N/A

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

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

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. distribute-lft-in N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. associate-*l* N/A

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

      12. distribute-rgt-in N/A

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

      13. mul-1-neg N/A

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

      14. distribute-lft-neg-out N/A

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

      15. *-commutative N/A

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

      16. remove-double-neg N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. +-commutative N/A

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

      20. *-commutative N/A

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

      21. lower-fma.f64 65.0

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

   7. Applied rewrites 65.0%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 65.0%

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

   if -9.9999999999999992e249 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < 0.666666666599999957 or 50 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z)

   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 y around 0

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

      1. metadata-eval N/A

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

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

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. sub-neg N/A

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

      6. *-lft-identity N/A

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

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

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

      8. neg-mul-1 N/A

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

      9. associate-*r* N/A

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

      10. distribute-rgt-in N/A

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

      11. metadata-eval N/A

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

      12. distribute-rgt-in N/A

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

      13. +-commutative N/A

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

      14. metadata-eval N/A

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

      15. sub-neg N/A

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

      16. neg-mul-1 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

   5. Applied rewrites 58.6%

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

   if 0.666666666599999957 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < 50

   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 96.0

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

   5. Applied rewrites 96.0%

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

Alternative 3: 74.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{3} - z\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+250}:\\ \;\;\;\;\left(z \cdot -6\right) \cdot y\\ \mathbf{elif}\;t\_0 \leq -1000000:\\ \;\;\;\;\left(6 \cdot x\right) \cdot z\\ \mathbf{elif}\;t\_0 \leq 50:\\ \;\;\;\;\mathsf{fma}\left(y - x, 4, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(6 \cdot z\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (/ 2.0 3.0) z)))
   (if (<= t_0 -1e+250)
     (* (* z -6.0) y)
     (if (<= t_0 -1000000.0)
       (* (* 6.0 x) z)
       (if (<= t_0 50.0) (fma (- y x) 4.0 x) (* (* 6.0 z) x))))))

C

double code(double x, double y, double z) {
	double t_0 = (2.0 / 3.0) - z;
	double tmp;
	if (t_0 <= -1e+250) {
		tmp = (z * -6.0) * y;
	} else if (t_0 <= -1000000.0) {
		tmp = (6.0 * x) * z;
	} else if (t_0 <= 50.0) {
		tmp = fma((y - x), 4.0, x);
	} else {
		tmp = (6.0 * z) * x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(2.0 / 3.0) - z)
	tmp = 0.0
	if (t_0 <= -1e+250)
		tmp = Float64(Float64(z * -6.0) * y);
	elseif (t_0 <= -1000000.0)
		tmp = Float64(Float64(6.0 * x) * z);
	elseif (t_0 <= 50.0)
		tmp = fma(Float64(y - x), 4.0, x);
	else
		tmp = Float64(Float64(6.0 * z) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(2.0 / 3.0), $MachinePrecision] - z), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+250], N[(N[(z * -6.0), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$0, -1000000.0], N[(N[(6.0 * x), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[t$95$0, 50.0], N[(N[(y - x), $MachinePrecision] * 4.0 + x), $MachinePrecision], N[(N[(6.0 * z), $MachinePrecision] * x), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < -9.9999999999999992e249

   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. 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 99.8

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

      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 99.8

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

      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 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in y around inf

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

      1. metadata-eval N/A

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

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

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. distribute-lft-in N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. associate-*l* N/A

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

      12. distribute-rgt-in N/A

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

      13. mul-1-neg N/A

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

      14. distribute-lft-neg-out N/A

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

      15. *-commutative N/A

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

      16. remove-double-neg N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. +-commutative N/A

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

      20. *-commutative N/A

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

      21. lower-fma.f64 65.0

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

   7. Applied rewrites 65.0%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 65.0%

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

   if -9.9999999999999992e249 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < -1e6

   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 y around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt1-in N/A

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

      4. +-commutative N/A

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

      5. lower-*.f64 N/A

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

   7. Applied rewrites 57.0%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 56.1%

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

   12. Applied rewrites 56.1%

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

   if -1e6 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < 50

   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 94.7

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

   5. Applied rewrites 94.7%

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

   if 50 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z)

   1. Initial program 99.9%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   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 99.9

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

      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 99.9

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

      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 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt1-in N/A

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

      4. +-commutative N/A

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

      5. lower-*.f64 N/A

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

   7. Applied rewrites 58.2%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 57.3%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{3} - z \leq -1 \cdot 10^{+250}:\\ \;\;\;\;\left(z \cdot -6\right) \cdot y\\ \mathbf{elif}\;\frac{2}{3} - z \leq -1000000:\\ \;\;\;\;\left(6 \cdot x\right) \cdot z\\ \mathbf{elif}\;\frac{2}{3} - z \leq 50:\\ \;\;\;\;\mathsf{fma}\left(y - x, 4, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(6 \cdot z\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{3} - z\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+250}:\\ \;\;\;\;\left(y \cdot z\right) \cdot -6\\ \mathbf{elif}\;t\_0 \leq -1000000:\\ \;\;\;\;\left(6 \cdot x\right) \cdot z\\ \mathbf{elif}\;t\_0 \leq 50:\\ \;\;\;\;\mathsf{fma}\left(y - x, 4, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(6 \cdot z\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (/ 2.0 3.0) z)))
   (if (<= t_0 -1e+250)
     (* (* y z) -6.0)
     (if (<= t_0 -1000000.0)
       (* (* 6.0 x) z)
       (if (<= t_0 50.0) (fma (- y x) 4.0 x) (* (* 6.0 z) x))))))

C

double code(double x, double y, double z) {
	double t_0 = (2.0 / 3.0) - z;
	double tmp;
	if (t_0 <= -1e+250) {
		tmp = (y * z) * -6.0;
	} else if (t_0 <= -1000000.0) {
		tmp = (6.0 * x) * z;
	} else if (t_0 <= 50.0) {
		tmp = fma((y - x), 4.0, x);
	} else {
		tmp = (6.0 * z) * x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(2.0 / 3.0) - z)
	tmp = 0.0
	if (t_0 <= -1e+250)
		tmp = Float64(Float64(y * z) * -6.0);
	elseif (t_0 <= -1000000.0)
		tmp = Float64(Float64(6.0 * x) * z);
	elseif (t_0 <= 50.0)
		tmp = fma(Float64(y - x), 4.0, x);
	else
		tmp = Float64(Float64(6.0 * z) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(2.0 / 3.0), $MachinePrecision] - z), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+250], N[(N[(y * z), $MachinePrecision] * -6.0), $MachinePrecision], If[LessEqual[t$95$0, -1000000.0], N[(N[(6.0 * x), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[t$95$0, 50.0], N[(N[(y - x), $MachinePrecision] * 4.0 + x), $MachinePrecision], N[(N[(6.0 * z), $MachinePrecision] * x), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < -9.9999999999999992e249

   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 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. lower-*.f64 N/A

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

      4. lower--.f64 99.6

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 64.7%

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

   if -9.9999999999999992e249 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < -1e6

   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 y around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt1-in N/A

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

      4. +-commutative N/A

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

      5. lower-*.f64 N/A

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

   7. Applied rewrites 57.0%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 56.1%

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

   12. Applied rewrites 56.1%

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

   if -1e6 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < 50

   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 94.7

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

   5. Applied rewrites 94.7%

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

   if 50 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z)

   1. Initial program 99.9%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   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 99.9

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

      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 99.9

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

      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 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt1-in N/A

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

      4. +-commutative N/A

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

      5. lower-*.f64 N/A

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

   7. Applied rewrites 58.2%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 57.3%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{3} - z \leq -1 \cdot 10^{+250}:\\ \;\;\;\;\left(y \cdot z\right) \cdot -6\\ \mathbf{elif}\;\frac{2}{3} - z \leq -1000000:\\ \;\;\;\;\left(6 \cdot x\right) \cdot z\\ \mathbf{elif}\;\frac{2}{3} - z \leq 50:\\ \;\;\;\;\mathsf{fma}\left(y - x, 4, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(6 \cdot z\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(6 \cdot x\right) \cdot z\\ t_1 := \frac{2}{3} - z\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+250}:\\ \;\;\;\;\left(y \cdot z\right) \cdot -6\\ \mathbf{elif}\;t\_1 \leq -1000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 50:\\ \;\;\;\;\mathsf{fma}\left(y - x, 4, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (* 6.0 x) z)) (t_1 (- (/ 2.0 3.0) z)))
   (if (<= t_1 -1e+250)
     (* (* y z) -6.0)
     (if (<= t_1 -1000000.0) t_0 (if (<= t_1 50.0) (fma (- y x) 4.0 x) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = (6.0 * x) * z;
	double t_1 = (2.0 / 3.0) - z;
	double tmp;
	if (t_1 <= -1e+250) {
		tmp = (y * z) * -6.0;
	} else if (t_1 <= -1000000.0) {
		tmp = t_0;
	} else if (t_1 <= 50.0) {
		tmp = fma((y - x), 4.0, x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(6.0 * x) * z)
	t_1 = Float64(Float64(2.0 / 3.0) - z)
	tmp = 0.0
	if (t_1 <= -1e+250)
		tmp = Float64(Float64(y * z) * -6.0);
	elseif (t_1 <= -1000000.0)
		tmp = t_0;
	elseif (t_1 <= 50.0)
		tmp = fma(Float64(y - x), 4.0, x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(6.0 * x), $MachinePrecision] * z), $MachinePrecision]}, Block[{t$95$1 = N[(N[(2.0 / 3.0), $MachinePrecision] - z), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+250], N[(N[(y * z), $MachinePrecision] * -6.0), $MachinePrecision], If[LessEqual[t$95$1, -1000000.0], t$95$0, If[LessEqual[t$95$1, 50.0], N[(N[(y - x), $MachinePrecision] * 4.0 + x), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < -9.9999999999999992e249

   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 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. lower-*.f64 N/A

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

      4. lower--.f64 99.6

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 64.7%

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

   if -9.9999999999999992e249 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < -1e6 or 50 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z)

   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. 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 99.8

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

      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 99.8

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

      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 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt1-in N/A

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

      4. +-commutative N/A

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

      5. lower-*.f64 N/A

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

   7. Applied rewrites 57.7%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 56.8%

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

   12. Applied rewrites 56.8%

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

   if -1e6 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < 50

   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 94.7

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

   5. Applied rewrites 94.7%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{3} - z \leq -1 \cdot 10^{+250}:\\ \;\;\;\;\left(y \cdot z\right) \cdot -6\\ \mathbf{elif}\;\frac{2}{3} - z \leq -1000000:\\ \;\;\;\;\left(6 \cdot x\right) \cdot z\\ \mathbf{elif}\;\frac{2}{3} - z \leq 50:\\ \;\;\;\;\mathsf{fma}\left(y - x, 4, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(6 \cdot x\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot z\right) \cdot 6\\ t_1 := \frac{2}{3} - z\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+250}:\\ \;\;\;\;\left(y \cdot z\right) \cdot -6\\ \mathbf{elif}\;t\_1 \leq -1000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 50:\\ \;\;\;\;\mathsf{fma}\left(y - x, 4, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (* x z) 6.0)) (t_1 (- (/ 2.0 3.0) z)))
   (if (<= t_1 -1e+250)
     (* (* y z) -6.0)
     (if (<= t_1 -1000000.0) t_0 (if (<= t_1 50.0) (fma (- y x) 4.0 x) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = (x * z) * 6.0;
	double t_1 = (2.0 / 3.0) - z;
	double tmp;
	if (t_1 <= -1e+250) {
		tmp = (y * z) * -6.0;
	} else if (t_1 <= -1000000.0) {
		tmp = t_0;
	} else if (t_1 <= 50.0) {
		tmp = fma((y - x), 4.0, x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x * z) * 6.0)
	t_1 = Float64(Float64(2.0 / 3.0) - z)
	tmp = 0.0
	if (t_1 <= -1e+250)
		tmp = Float64(Float64(y * z) * -6.0);
	elseif (t_1 <= -1000000.0)
		tmp = t_0;
	elseif (t_1 <= 50.0)
		tmp = fma(Float64(y - x), 4.0, x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x * z), $MachinePrecision] * 6.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(2.0 / 3.0), $MachinePrecision] - z), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+250], N[(N[(y * z), $MachinePrecision] * -6.0), $MachinePrecision], If[LessEqual[t$95$1, -1000000.0], t$95$0, If[LessEqual[t$95$1, 50.0], N[(N[(y - x), $MachinePrecision] * 4.0 + x), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < -9.9999999999999992e249

   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 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. lower-*.f64 N/A

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

      4. lower--.f64 99.6

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 64.7%

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

   if -9.9999999999999992e249 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < -1e6 or 50 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z)

   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 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. lower-*.f64 N/A

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

      4. lower--.f64 98.4

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

   5. Applied rewrites 98.4%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 56.8%

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

   if -1e6 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < 50

   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 94.7

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

   5. Applied rewrites 94.7%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{3} - z \leq -1 \cdot 10^{+250}:\\ \;\;\;\;\left(y \cdot z\right) \cdot -6\\ \mathbf{elif}\;\frac{2}{3} - z \leq -1000000:\\ \;\;\;\;\left(x \cdot z\right) \cdot 6\\ \mathbf{elif}\;\frac{2}{3} - z \leq 50:\\ \;\;\;\;\mathsf{fma}\left(y - x, 4, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot z\right) \cdot 6\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{3} - z\\ \mathbf{if}\;t\_0 \leq -1000000:\\ \;\;\;\;\left(z \cdot -6\right) \cdot \left(y - x\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(-3, x, y \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y - x\right) \cdot -6\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (/ 2.0 3.0) z)))
   (if (<= t_0 -1000000.0)
     (* (* z -6.0) (- y x))
     (if (<= t_0 2.0) (fma -3.0 x (* y 4.0)) (* (* (- y x) -6.0) z)))))

C

double code(double x, double y, double z) {
	double t_0 = (2.0 / 3.0) - z;
	double tmp;
	if (t_0 <= -1000000.0) {
		tmp = (z * -6.0) * (y - x);
	} else if (t_0 <= 2.0) {
		tmp = fma(-3.0, x, (y * 4.0));
	} else {
		tmp = ((y - x) * -6.0) * z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(2.0 / 3.0) - z)
	tmp = 0.0
	if (t_0 <= -1000000.0)
		tmp = Float64(Float64(z * -6.0) * Float64(y - x));
	elseif (t_0 <= 2.0)
		tmp = fma(-3.0, x, Float64(y * 4.0));
	else
		tmp = Float64(Float64(Float64(y - x) * -6.0) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(2.0 / 3.0), $MachinePrecision] - z), $MachinePrecision]}, If[LessEqual[t$95$0, -1000000.0], N[(N[(z * -6.0), $MachinePrecision] * N[(y - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(-3.0 * x + N[(y * 4.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y - x), $MachinePrecision] * -6.0), $MachinePrecision] * z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < -1e6

   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. lower-*.f64 N/A

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

      4. lower--.f64 98.2

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

   5. Applied rewrites 98.2%

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

   7. Applied rewrites 98.4%

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

   if -1e6 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < 2

   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. 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 99.4

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

      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 99.4

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

      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 99.4

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 99.3

         \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\frac{0.6666666666666666}{z}} - 1\right) \cdot z, 6 \cdot \left(y - x\right), x\right) \]

   7. Applied rewrites 99.3%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{0.6666666666666666}{z} - 1\right) \cdot z}, 6 \cdot \left(y - x\right), x\right) \]

   8. Taylor expanded in z around 0

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-lft-in N/A

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

      4. associate-+r+ N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

      9. distribute-lft1-in N/A

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

      10. metadata-eval N/A

          \[\leadsto \color{blue}{-3} \cdot x + 4 \cdot y \]

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 96.3

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

   10. Applied rewrites 96.3%

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

   if 2 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z)

   1. Initial program 99.9%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   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 99.9

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

      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 99.9

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

      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 99.9

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

   4. Applied rewrites 99.9%

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

   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

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 97.0

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

   7. Applied rewrites 97.0%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{3} - z \leq -1000000:\\ \;\;\;\;\left(z \cdot -6\right) \cdot \left(y - x\right)\\ \mathbf{elif}\;\frac{2}{3} - z \leq 2:\\ \;\;\;\;\mathsf{fma}\left(-3, x, y \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y - x\right) \cdot -6\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{3} - z\\ t_1 := \left(z \cdot -6\right) \cdot \left(y - x\right)\\ \mathbf{if}\;t\_0 \leq -1000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(-3, x, y \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (/ 2.0 3.0) z)) (t_1 (* (* z -6.0) (- y x))))
   (if (<= t_0 -1000000.0) t_1 (if (<= t_0 2.0) (fma -3.0 x (* y 4.0)) t_1))))

C

double code(double x, double y, double z) {
	double t_0 = (2.0 / 3.0) - z;
	double t_1 = (z * -6.0) * (y - x);
	double tmp;
	if (t_0 <= -1000000.0) {
		tmp = t_1;
	} else if (t_0 <= 2.0) {
		tmp = fma(-3.0, x, (y * 4.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(2.0 / 3.0) - z)
	t_1 = Float64(Float64(z * -6.0) * Float64(y - x))
	tmp = 0.0
	if (t_0 <= -1000000.0)
		tmp = t_1;
	elseif (t_0 <= 2.0)
		tmp = fma(-3.0, x, Float64(y * 4.0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(2.0 / 3.0), $MachinePrecision] - z), $MachinePrecision]}, Block[{t$95$1 = N[(N[(z * -6.0), $MachinePrecision] * N[(y - x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1000000.0], t$95$1, If[LessEqual[t$95$0, 2.0], N[(-3.0 * x + N[(y * 4.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < -1e6 or 2 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z)

   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 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. lower-*.f64 N/A

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

      4. lower--.f64 97.5

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

   5. Applied rewrites 97.5%

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

   7. Applied rewrites 97.6%

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

   if -1e6 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < 2

   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. 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 99.4

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

      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 99.4

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

      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 99.4

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 99.3

         \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\frac{0.6666666666666666}{z}} - 1\right) \cdot z, 6 \cdot \left(y - x\right), x\right) \]

   7. Applied rewrites 99.3%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{0.6666666666666666}{z} - 1\right) \cdot z}, 6 \cdot \left(y - x\right), x\right) \]

   8. Taylor expanded in z around 0

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-lft-in N/A

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

      4. associate-+r+ N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

      9. distribute-lft1-in N/A

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

      10. metadata-eval N/A

          \[\leadsto \color{blue}{-3} \cdot x + 4 \cdot y \]

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 96.3

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

   10. Applied rewrites 96.3%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{3} - z \leq -1000000:\\ \;\;\;\;\left(z \cdot -6\right) \cdot \left(y - x\right)\\ \mathbf{elif}\;\frac{2}{3} - z \leq 2:\\ \;\;\;\;\mathsf{fma}\left(-3, x, y \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot -6\right) \cdot \left(y - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{3} - z\\ t_1 := \left(\left(y - x\right) \cdot z\right) \cdot -6\\ \mathbf{if}\;t\_0 \leq -1000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(-3, x, y \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (/ 2.0 3.0) z)) (t_1 (* (* (- y x) z) -6.0)))
   (if (<= t_0 -1000000.0) t_1 (if (<= t_0 2.0) (fma -3.0 x (* y 4.0)) t_1))))

C

double code(double x, double y, double z) {
	double t_0 = (2.0 / 3.0) - z;
	double t_1 = ((y - x) * z) * -6.0;
	double tmp;
	if (t_0 <= -1000000.0) {
		tmp = t_1;
	} else if (t_0 <= 2.0) {
		tmp = fma(-3.0, x, (y * 4.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(2.0 / 3.0) - z)
	t_1 = Float64(Float64(Float64(y - x) * z) * -6.0)
	tmp = 0.0
	if (t_0 <= -1000000.0)
		tmp = t_1;
	elseif (t_0 <= 2.0)
		tmp = fma(-3.0, x, Float64(y * 4.0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(2.0 / 3.0), $MachinePrecision] - z), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(y - x), $MachinePrecision] * z), $MachinePrecision] * -6.0), $MachinePrecision]}, If[LessEqual[t$95$0, -1000000.0], t$95$1, If[LessEqual[t$95$0, 2.0], N[(-3.0 * x + N[(y * 4.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < -1e6 or 2 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z)

   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 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. lower-*.f64 N/A

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

      4. lower--.f64 97.5

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

   5. Applied rewrites 97.5%

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

   if -1e6 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < 2

   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. 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 99.4

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

      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 99.4

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

      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 99.4

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 99.3

         \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\frac{0.6666666666666666}{z}} - 1\right) \cdot z, 6 \cdot \left(y - x\right), x\right) \]

   7. Applied rewrites 99.3%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{0.6666666666666666}{z} - 1\right) \cdot z}, 6 \cdot \left(y - x\right), x\right) \]

   8. Taylor expanded in z around 0

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-lft-in N/A

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

      4. associate-+r+ N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

      9. distribute-lft1-in N/A

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

      10. metadata-eval N/A

          \[\leadsto \color{blue}{-3} \cdot x + 4 \cdot y \]

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 96.3

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

   10. Applied rewrites 96.3%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{3} - z \leq -1000000:\\ \;\;\;\;\left(\left(y - x\right) \cdot z\right) \cdot -6\\ \mathbf{elif}\;\frac{2}{3} - z \leq 2:\\ \;\;\;\;\mathsf{fma}\left(-3, x, y \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y - x\right) \cdot z\right) \cdot -6\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 74.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{3} - z\\ t_1 := \left(x \cdot z\right) \cdot 6\\ \mathbf{if}\;t\_0 \leq -1000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 50:\\ \;\;\;\;\mathsf{fma}\left(y - x, 4, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (/ 2.0 3.0) z)) (t_1 (* (* x z) 6.0)))
   (if (<= t_0 -1000000.0) t_1 (if (<= t_0 50.0) (fma (- y x) 4.0 x) t_1))))

C

double code(double x, double y, double z) {
	double t_0 = (2.0 / 3.0) - z;
	double t_1 = (x * z) * 6.0;
	double tmp;
	if (t_0 <= -1000000.0) {
		tmp = t_1;
	} else if (t_0 <= 50.0) {
		tmp = fma((y - x), 4.0, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(2.0 / 3.0) - z)
	t_1 = Float64(Float64(x * z) * 6.0)
	tmp = 0.0
	if (t_0 <= -1000000.0)
		tmp = t_1;
	elseif (t_0 <= 50.0)
		tmp = fma(Float64(y - x), 4.0, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(2.0 / 3.0), $MachinePrecision] - z), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * z), $MachinePrecision] * 6.0), $MachinePrecision]}, If[LessEqual[t$95$0, -1000000.0], t$95$1, If[LessEqual[t$95$0, 50.0], N[(N[(y - x), $MachinePrecision] * 4.0 + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < -1e6 or 50 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z)

   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 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. lower-*.f64 N/A

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

      4. lower--.f64 98.5

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

   5. Applied rewrites 98.5%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 54.4%

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

   if -1e6 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < 50

   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 94.7

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

   5. Applied rewrites 94.7%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{3} - z \leq -1000000:\\ \;\;\;\;\left(x \cdot z\right) \cdot 6\\ \mathbf{elif}\;\frac{2}{3} - z \leq 50:\\ \;\;\;\;\mathsf{fma}\left(y - x, 4, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot z\right) \cdot 6\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 75.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(z, -6, 4\right) \cdot y\\ \mathbf{if}\;y \leq -1.05 \cdot 10^{-50}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 7.6 \cdot 10^{+61}:\\ \;\;\;\;\mathsf{fma}\left(6, z, -3\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (fma z -6.0 4.0) y)))
   (if (<= y -1.05e-50) t_0 (if (<= y 7.6e+61) (* (fma 6.0 z -3.0) x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma(z, -6.0, 4.0) * y;
	double tmp;
	if (y <= -1.05e-50) {
		tmp = t_0;
	} else if (y <= 7.6e+61) {
		tmp = fma(6.0, z, -3.0) * x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(fma(z, -6.0, 4.0) * y)
	tmp = 0.0
	if (y <= -1.05e-50)
		tmp = t_0;
	elseif (y <= 7.6e+61)
		tmp = Float64(fma(6.0, z, -3.0) * x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(z * -6.0 + 4.0), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[y, -1.05e-50], t$95$0, If[LessEqual[y, 7.6e+61], N[(N[(6.0 * z + -3.0), $MachinePrecision] * x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.05e-50 or 7.5999999999999999e61 < 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 y around inf

      \[\leadsto \color{blue}{6 \cdot \left(y \cdot \left(\frac{2}{3} - z\right)\right)} \]
   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. sub-neg N/A

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

      5. mul-1-neg N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. metadata-eval N/A

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

      9. associate-*r* N/A

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

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 81.3

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

   5. Applied rewrites 81.3%

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

   if -1.05e-50 < y < 7.5999999999999999e61

   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 y around 0

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

      1. metadata-eval N/A

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

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

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. sub-neg N/A

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

      6. *-lft-identity N/A

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

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

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

      8. neg-mul-1 N/A

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

      9. associate-*r* N/A

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

      10. distribute-rgt-in N/A

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

      11. metadata-eval N/A

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

      12. distribute-rgt-in N/A

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

      13. +-commutative N/A

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

      14. metadata-eval N/A

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

      15. sub-neg N/A

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

      16. neg-mul-1 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

   5. Applied rewrites 76.4%

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

Alternative 12: 36.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.9 \cdot 10^{+155}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{+66}:\\ \;\;\;\;-3 \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot 4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2.9e+155) (* y 4.0) (if (<= y 1.1e+66) (* -3.0 x) (* y 4.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.9e+155) {
		tmp = y * 4.0;
	} else if (y <= 1.1e+66) {
		tmp = -3.0 * x;
	} else {
		tmp = y * 4.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-2.9d+155)) then
        tmp = y * 4.0d0
    else if (y <= 1.1d+66) then
        tmp = (-3.0d0) * x
    else
        tmp = y * 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.9e+155) {
		tmp = y * 4.0;
	} else if (y <= 1.1e+66) {
		tmp = -3.0 * x;
	} else {
		tmp = y * 4.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -2.9e+155:
		tmp = y * 4.0
	elif y <= 1.1e+66:
		tmp = -3.0 * x
	else:
		tmp = y * 4.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2.9e+155)
		tmp = Float64(y * 4.0);
	elseif (y <= 1.1e+66)
		tmp = Float64(-3.0 * x);
	else
		tmp = Float64(y * 4.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -2.9e+155)
		tmp = y * 4.0;
	elseif (y <= 1.1e+66)
		tmp = -3.0 * x;
	else
		tmp = y * 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2.9e+155], N[(y * 4.0), $MachinePrecision], If[LessEqual[y, 1.1e+66], N[(-3.0 * x), $MachinePrecision], N[(y * 4.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.8999999999999999e155 or 1.0999999999999999e66 < 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 51.8

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

   5. Applied rewrites 51.8%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 48.4%

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

   if -2.8999999999999999e155 < y < 1.0999999999999999e66

   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 38.2

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

   5. Applied rewrites 38.2%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 29.6%

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

Alternative 13: 50.3% 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 42.7

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

5. Applied rewrites 42.7%

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

Alternative 14: 25.5% 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 42.7

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

5. Applied rewrites 42.7%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 21.5%

   \[\leadsto -3 \cdot \color{blue}{x} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024276 
(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))))