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

Percentage Accurate: 99.5% → 99.7%

Time: 16.5s

Alternatives: 16

Speedup: 1.9×

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

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x + (((y - x) * 6.0d0) * ((2.0d0 / 3.0d0) - z))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x + (((y - x) * 6.0d0) * ((2.0d0 / 3.0d0) - z))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. lift--.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. +-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. lift--.f64 N/A

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

   9. sub-neg N/A

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

   10. distribute-lft-in N/A

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

   11. associate-+l+ N/A

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

   12. lift-*.f64 N/A

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

   13. associate-*l* N/A

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

   14. lower-fma.f64 N/A

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

   15. lift-/.f64 N/A

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

   16. metadata-eval N/A

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

   17. metadata-eval N/A

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

4. Applied rewrites 99.8%

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

Alternative 2: 98.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{3} - z\\ t_1 := z \cdot \left(\left(y - x\right) \cdot -6\right)\\ \mathbf{if}\;t\_0 \leq -50:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0.666667:\\ \;\;\;\;\mathsf{fma}\left(-3, x, y \cdot 4\right)\\ \mathbf{elif}\;t\_0 \leq 500000000:\\ \;\;\;\;y \cdot \mathsf{fma}\left(z, -6, 4\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (/ 2.0 3.0) z)) (t_1 (* z (* (- y x) -6.0))))
   (if (<= t_0 -50.0)
     t_1
     (if (<= t_0 0.666667)
       (fma -3.0 x (* y 4.0))
       (if (<= t_0 500000000.0) (* y (fma z -6.0 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 * ((y - x) * -6.0);
	double tmp;
	if (t_0 <= -50.0) {
		tmp = t_1;
	} else if (t_0 <= 0.666667) {
		tmp = fma(-3.0, x, (y * 4.0));
	} else if (t_0 <= 500000000.0) {
		tmp = y * fma(z, -6.0, 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(z * Float64(Float64(y - x) * -6.0))
	tmp = 0.0
	if (t_0 <= -50.0)
		tmp = t_1;
	elseif (t_0 <= 0.666667)
		tmp = fma(-3.0, x, Float64(y * 4.0));
	elseif (t_0 <= 500000000.0)
		tmp = Float64(y * fma(z, -6.0, 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[(z * N[(N[(y - x), $MachinePrecision] * -6.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -50.0], t$95$1, If[LessEqual[t$95$0, 0.666667], N[(-3.0 * x + N[(y * 4.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 500000000.0], N[(y * N[(z * -6.0 + 4.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < -50 or 5e8 < (-.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

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. sub-neg N/A

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

      10. distribute-lft-in N/A

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

      11. associate-+l+ N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l* N/A

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

      14. lower-fma.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

   4. Applied rewrites 99.8%

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

   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. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 99.0

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

   7. Applied rewrites 99.0%

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

   if -50 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < 0.66666700000000001

   1. Initial program 99.3%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 99.2

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

      10. lift-/.f64 N/A

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

      11. metadata-eval 99.2

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in z around 0

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

   7. Applied rewrites 98.1%

      \[\leadsto \mathsf{fma}\left(\color{blue}{0.6666666666666666}, \left(y - x\right) \cdot 6, x\right) \]
   8. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. neg-mul-1 N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 98.2

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

   9. Applied rewrites 98.2%

      \[\leadsto \mathsf{fma}\left(0.6666666666666666, \color{blue}{\mathsf{fma}\left(-6, x, 6 \cdot y\right)}, x\right) \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-fma.f64 N/A

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

       3. +-commutative N/A

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

       4. lift-fma.f64 N/A

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

       5. distribute-lft-in N/A

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

       6. associate-+r+ N/A

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

       7. associate-*r* N/A

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

       8. metadata-eval N/A

          \[\leadsto \left(x + \color{blue}{-4} \cdot x\right) + \frac{2}{3} \cdot \left(6 \cdot y\right) \]

       9. distribute-rgt1-in N/A

          \[\leadsto \color{blue}{\left(-4 + 1\right) \cdot x} + \frac{2}{3} \cdot \left(6 \cdot y\right) \]

       10. lower-fma.f64 N/A

           \[\leadsto \color{blue}{\mathsf{fma}\left(-4 + 1, x, \frac{2}{3} \cdot \left(6 \cdot y\right)\right)} \]

       11. metadata-eval N/A

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

       12. *-commutative N/A

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

       13. lift-*.f64 N/A

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

       14. *-commutative N/A

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

       15. associate-*l* N/A

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

       16. metadata-eval N/A

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

       17. metadata-eval N/A

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

       18. lower-*.f64 N/A

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

       19. metadata-eval 98.8

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

   11. Applied rewrites 98.8%

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

   if 0.66666700000000001 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < 5e8

   1. Initial program 99.3%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. associate-*r* N/A

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

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 88.3

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

   5. Applied rewrites 88.3%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{3} - z \leq -50:\\ \;\;\;\;z \cdot \left(\left(y - x\right) \cdot -6\right)\\ \mathbf{elif}\;\frac{2}{3} - z \leq 0.666667:\\ \;\;\;\;\mathsf{fma}\left(-3, x, y \cdot 4\right)\\ \mathbf{elif}\;\frac{2}{3} - z \leq 500000000:\\ \;\;\;\;y \cdot \mathsf{fma}\left(z, -6, 4\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(\left(y - x\right) \cdot -6\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{3} - z\\ t_1 := 6 \cdot \left(z \cdot \left(x - y\right)\right)\\ \mathbf{if}\;t\_0 \leq -50:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0.666667:\\ \;\;\;\;\mathsf{fma}\left(-3, x, y \cdot 4\right)\\ \mathbf{elif}\;t\_0 \leq 500000000:\\ \;\;\;\;y \cdot \mathsf{fma}\left(z, -6, 4\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (/ 2.0 3.0) z)) (t_1 (* 6.0 (* z (- x y)))))
   (if (<= t_0 -50.0)
     t_1
     (if (<= t_0 0.666667)
       (fma -3.0 x (* y 4.0))
       (if (<= t_0 500000000.0) (* y (fma z -6.0 4.0)) t_1)))))

C

double code(double x, double y, double z) {
	double t_0 = (2.0 / 3.0) - z;
	double t_1 = 6.0 * (z * (x - y));
	double tmp;
	if (t_0 <= -50.0) {
		tmp = t_1;
	} else if (t_0 <= 0.666667) {
		tmp = fma(-3.0, x, (y * 4.0));
	} else if (t_0 <= 500000000.0) {
		tmp = y * fma(z, -6.0, 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(6.0 * Float64(z * Float64(x - y)))
	tmp = 0.0
	if (t_0 <= -50.0)
		tmp = t_1;
	elseif (t_0 <= 0.666667)
		tmp = fma(-3.0, x, Float64(y * 4.0));
	elseif (t_0 <= 500000000.0)
		tmp = Float64(y * fma(z, -6.0, 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[(6.0 * N[(z * N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -50.0], t$95$1, If[LessEqual[t$95$0, 0.666667], N[(-3.0 * x + N[(y * 4.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 500000000.0], N[(y * N[(z * -6.0 + 4.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < -50 or 5e8 < (-.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. metadata-eval N/A

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

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

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

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

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

      4. lower-*.f64 N/A

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

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

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

      6. neg-mul-1 N/A

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

      7. lower-*.f64 N/A

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

      8. sub-neg N/A

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

      9. neg-mul-1 N/A

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

      10. +-commutative N/A

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

      11. distribute-lft-in N/A

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

      12. associate-*r* N/A

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

      13. metadata-eval N/A

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

      14. *-lft-identity N/A

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

      15. mul-1-neg N/A

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

      16. sub-neg N/A

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

      17. lower--.f64 98.9

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

   5. Applied rewrites 98.9%

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

   if -50 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < 0.66666700000000001

   1. Initial program 99.3%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 99.2

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

      10. lift-/.f64 N/A

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

      11. metadata-eval 99.2

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in z around 0

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

   7. Applied rewrites 98.1%

      \[\leadsto \mathsf{fma}\left(\color{blue}{0.6666666666666666}, \left(y - x\right) \cdot 6, x\right) \]
   8. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. neg-mul-1 N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 98.2

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

   9. Applied rewrites 98.2%

      \[\leadsto \mathsf{fma}\left(0.6666666666666666, \color{blue}{\mathsf{fma}\left(-6, x, 6 \cdot y\right)}, x\right) \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-fma.f64 N/A

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

       3. +-commutative N/A

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

       4. lift-fma.f64 N/A

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

       5. distribute-lft-in N/A

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

       6. associate-+r+ N/A

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

       7. associate-*r* N/A

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

       8. metadata-eval N/A

          \[\leadsto \left(x + \color{blue}{-4} \cdot x\right) + \frac{2}{3} \cdot \left(6 \cdot y\right) \]

       9. distribute-rgt1-in N/A

          \[\leadsto \color{blue}{\left(-4 + 1\right) \cdot x} + \frac{2}{3} \cdot \left(6 \cdot y\right) \]

       10. lower-fma.f64 N/A

           \[\leadsto \color{blue}{\mathsf{fma}\left(-4 + 1, x, \frac{2}{3} \cdot \left(6 \cdot y\right)\right)} \]

       11. metadata-eval N/A

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

       12. *-commutative N/A

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

       13. lift-*.f64 N/A

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

       14. *-commutative N/A

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

       15. associate-*l* N/A

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

       16. metadata-eval N/A

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

       17. metadata-eval N/A

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

       18. lower-*.f64 N/A

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

       19. metadata-eval 98.8

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

   11. Applied rewrites 98.8%

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

   if 0.66666700000000001 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < 5e8

   1. Initial program 99.3%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. associate-*r* N/A

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

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 88.3

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

   5. Applied rewrites 88.3%

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

Alternative 4: 74.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \mathsf{fma}\left(z, -6, 4\right)\\ \mathbf{if}\;z \leq -7.2 \cdot 10^{+35}:\\ \;\;\;\;x \cdot \left(z \cdot 6\right)\\ \mathbf{elif}\;z \leq -1.15 \cdot 10^{-7}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(-3, x, y \cdot 4\right)\\ \mathbf{elif}\;z \leq 1.85 \cdot 10^{+134}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 5.1 \cdot 10^{+290}:\\ \;\;\;\;6 \cdot \left(x \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;-6 \cdot \left(y \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (fma z -6.0 4.0))))
   (if (<= z -7.2e+35)
     (* x (* z 6.0))
     (if (<= z -1.15e-7)
       t_0
       (if (<= z 1.6e-10)
         (fma -3.0 x (* y 4.0))
         (if (<= z 1.85e+134)
           t_0
           (if (<= z 5.1e+290) (* 6.0 (* x z)) (* -6.0 (* y z)))))))))

C

double code(double x, double y, double z) {
	double t_0 = y * fma(z, -6.0, 4.0);
	double tmp;
	if (z <= -7.2e+35) {
		tmp = x * (z * 6.0);
	} else if (z <= -1.15e-7) {
		tmp = t_0;
	} else if (z <= 1.6e-10) {
		tmp = fma(-3.0, x, (y * 4.0));
	} else if (z <= 1.85e+134) {
		tmp = t_0;
	} else if (z <= 5.1e+290) {
		tmp = 6.0 * (x * z);
	} else {
		tmp = -6.0 * (y * z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(y * fma(z, -6.0, 4.0))
	tmp = 0.0
	if (z <= -7.2e+35)
		tmp = Float64(x * Float64(z * 6.0));
	elseif (z <= -1.15e-7)
		tmp = t_0;
	elseif (z <= 1.6e-10)
		tmp = fma(-3.0, x, Float64(y * 4.0));
	elseif (z <= 1.85e+134)
		tmp = t_0;
	elseif (z <= 5.1e+290)
		tmp = Float64(6.0 * Float64(x * z));
	else
		tmp = Float64(-6.0 * Float64(y * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[(z * -6.0 + 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -7.2e+35], N[(x * N[(z * 6.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -1.15e-7], t$95$0, If[LessEqual[z, 1.6e-10], N[(-3.0 * x + N[(y * 4.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.85e+134], t$95$0, If[LessEqual[z, 5.1e+290], N[(6.0 * N[(x * z), $MachinePrecision]), $MachinePrecision], N[(-6.0 * N[(y * z), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if z < -7.2000000000000001e35

   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

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. distribute-lft-in N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. associate-*r* N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 61.4

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

   5. Applied rewrites 61.4%

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

   6. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 61.3

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

   8. Applied rewrites 61.3%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lower-*.f64 61.4

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 61.4

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

   10. Applied rewrites 61.4%

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

   if -7.2000000000000001e35 < z < -1.14999999999999997e-7 or 1.5999999999999999e-10 < z < 1.85000000000000007e134

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. associate-*r* N/A

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

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 67.4

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

   5. Applied rewrites 67.4%

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

   if -1.14999999999999997e-7 < z < 1.5999999999999999e-10

   1. Initial program 99.3%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 99.2

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

      10. lift-/.f64 N/A

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

      11. metadata-eval 99.2

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in z around 0

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

   7. Applied rewrites 99.0%

      \[\leadsto \mathsf{fma}\left(\color{blue}{0.6666666666666666}, \left(y - x\right) \cdot 6, x\right) \]
   8. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. neg-mul-1 N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 99.1

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

   9. Applied rewrites 99.1%

      \[\leadsto \mathsf{fma}\left(0.6666666666666666, \color{blue}{\mathsf{fma}\left(-6, x, 6 \cdot y\right)}, x\right) \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-fma.f64 N/A

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

       3. +-commutative N/A

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

       4. lift-fma.f64 N/A

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

       5. distribute-lft-in N/A

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

       6. associate-+r+ N/A

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

       7. associate-*r* N/A

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

       8. metadata-eval N/A

          \[\leadsto \left(x + \color{blue}{-4} \cdot x\right) + \frac{2}{3} \cdot \left(6 \cdot y\right) \]

       9. distribute-rgt1-in N/A

          \[\leadsto \color{blue}{\left(-4 + 1\right) \cdot x} + \frac{2}{3} \cdot \left(6 \cdot y\right) \]

       10. lower-fma.f64 N/A

           \[\leadsto \color{blue}{\mathsf{fma}\left(-4 + 1, x, \frac{2}{3} \cdot \left(6 \cdot y\right)\right)} \]

       11. metadata-eval N/A

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

       12. *-commutative N/A

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

       13. lift-*.f64 N/A

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

       14. *-commutative N/A

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

       15. associate-*l* N/A

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

       16. metadata-eval N/A

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

       17. metadata-eval N/A

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

       18. lower-*.f64 N/A

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

       19. metadata-eval 99.7

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

   11. Applied rewrites 99.7%

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

   if 1.85000000000000007e134 < z < 5.10000000000000023e290

   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. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. distribute-lft-in N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. associate-*r* N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 65.8

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

   5. Applied rewrites 65.8%

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

   6. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 65.9

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

   8. Applied rewrites 65.9%

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 65.9

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

   10. Applied rewrites 65.9%

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

   if 5.10000000000000023e290 < 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 x around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. associate-*r* N/A

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

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 99.8

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

   8. Applied rewrites 99.8%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.2 \cdot 10^{+35}:\\ \;\;\;\;x \cdot \left(z \cdot 6\right)\\ \mathbf{elif}\;z \leq -1.15 \cdot 10^{-7}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(z, -6, 4\right)\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(-3, x, y \cdot 4\right)\\ \mathbf{elif}\;z \leq 1.85 \cdot 10^{+134}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(z, -6, 4\right)\\ \mathbf{elif}\;z \leq 5.1 \cdot 10^{+290}:\\ \;\;\;\;6 \cdot \left(x \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;-6 \cdot \left(y \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \mathsf{fma}\left(z, -6, 4\right)\\ \mathbf{if}\;z \leq -7.2 \cdot 10^{+35}:\\ \;\;\;\;x \cdot \left(z \cdot 6\right)\\ \mathbf{elif}\;z \leq -1.15 \cdot 10^{-7}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(4, y - x, x\right)\\ \mathbf{elif}\;z \leq 1.85 \cdot 10^{+134}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 5.1 \cdot 10^{+290}:\\ \;\;\;\;6 \cdot \left(x \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;-6 \cdot \left(y \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (fma z -6.0 4.0))))
   (if (<= z -7.2e+35)
     (* x (* z 6.0))
     (if (<= z -1.15e-7)
       t_0
       (if (<= z 1.6e-10)
         (fma 4.0 (- y x) x)
         (if (<= z 1.85e+134)
           t_0
           (if (<= z 5.1e+290) (* 6.0 (* x z)) (* -6.0 (* y z)))))))))

C

double code(double x, double y, double z) {
	double t_0 = y * fma(z, -6.0, 4.0);
	double tmp;
	if (z <= -7.2e+35) {
		tmp = x * (z * 6.0);
	} else if (z <= -1.15e-7) {
		tmp = t_0;
	} else if (z <= 1.6e-10) {
		tmp = fma(4.0, (y - x), x);
	} else if (z <= 1.85e+134) {
		tmp = t_0;
	} else if (z <= 5.1e+290) {
		tmp = 6.0 * (x * z);
	} else {
		tmp = -6.0 * (y * z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(y * fma(z, -6.0, 4.0))
	tmp = 0.0
	if (z <= -7.2e+35)
		tmp = Float64(x * Float64(z * 6.0));
	elseif (z <= -1.15e-7)
		tmp = t_0;
	elseif (z <= 1.6e-10)
		tmp = fma(4.0, Float64(y - x), x);
	elseif (z <= 1.85e+134)
		tmp = t_0;
	elseif (z <= 5.1e+290)
		tmp = Float64(6.0 * Float64(x * z));
	else
		tmp = Float64(-6.0 * Float64(y * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[(z * -6.0 + 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -7.2e+35], N[(x * N[(z * 6.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -1.15e-7], t$95$0, If[LessEqual[z, 1.6e-10], N[(4.0 * N[(y - x), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 1.85e+134], t$95$0, If[LessEqual[z, 5.1e+290], N[(6.0 * N[(x * z), $MachinePrecision]), $MachinePrecision], N[(-6.0 * N[(y * z), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if z < -7.2000000000000001e35

   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

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. distribute-lft-in N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. associate-*r* N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 61.4

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

   5. Applied rewrites 61.4%

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

   6. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 61.3

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

   8. Applied rewrites 61.3%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lower-*.f64 61.4

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 61.4

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

   10. Applied rewrites 61.4%

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

   if -7.2000000000000001e35 < z < -1.14999999999999997e-7 or 1.5999999999999999e-10 < z < 1.85000000000000007e134

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. associate-*r* N/A

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

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 67.4

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

   5. Applied rewrites 67.4%

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

   if -1.14999999999999997e-7 < z < 1.5999999999999999e-10

   1. Initial program 99.3%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   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. lower-fma.f64 N/A

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

      3. lower--.f64 99.6

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

   5. Applied rewrites 99.6%

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

   if 1.85000000000000007e134 < z < 5.10000000000000023e290

   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. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. distribute-lft-in N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. associate-*r* N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 65.8

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

   5. Applied rewrites 65.8%

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

   6. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 65.9

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

   8. Applied rewrites 65.9%

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 65.9

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

   10. Applied rewrites 65.9%

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

   if 5.10000000000000023e290 < 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 x around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. associate-*r* N/A

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

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 99.8

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

   8. Applied rewrites 99.8%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.2 \cdot 10^{+35}:\\ \;\;\;\;x \cdot \left(z \cdot 6\right)\\ \mathbf{elif}\;z \leq -1.15 \cdot 10^{-7}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(z, -6, 4\right)\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(4, y - x, x\right)\\ \mathbf{elif}\;z \leq 1.85 \cdot 10^{+134}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(z, -6, 4\right)\\ \mathbf{elif}\;z \leq 5.1 \cdot 10^{+290}:\\ \;\;\;\;6 \cdot \left(x \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;-6 \cdot \left(y \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -0.0014:\\ \;\;\;\;x \cdot \mathsf{fma}\left(6, z, -3\right)\\ \mathbf{elif}\;z \leq 0.68:\\ \;\;\;\;\mathsf{fma}\left(4, y - x, x\right)\\ \mathbf{elif}\;z \leq 1.85 \cdot 10^{+134}:\\ \;\;\;\;y \cdot \left(-6 \cdot z\right)\\ \mathbf{elif}\;z \leq 5.1 \cdot 10^{+290}:\\ \;\;\;\;6 \cdot \left(x \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;-6 \cdot \left(y \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -0.0014)
   (* x (fma 6.0 z -3.0))
   (if (<= z 0.68)
     (fma 4.0 (- y x) x)
     (if (<= z 1.85e+134)
       (* y (* -6.0 z))
       (if (<= z 5.1e+290) (* 6.0 (* x z)) (* -6.0 (* y z)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -0.0014) {
		tmp = x * fma(6.0, z, -3.0);
	} else if (z <= 0.68) {
		tmp = fma(4.0, (y - x), x);
	} else if (z <= 1.85e+134) {
		tmp = y * (-6.0 * z);
	} else if (z <= 5.1e+290) {
		tmp = 6.0 * (x * z);
	} else {
		tmp = -6.0 * (y * z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -0.0014)
		tmp = Float64(x * fma(6.0, z, -3.0));
	elseif (z <= 0.68)
		tmp = fma(4.0, Float64(y - x), x);
	elseif (z <= 1.85e+134)
		tmp = Float64(y * Float64(-6.0 * z));
	elseif (z <= 5.1e+290)
		tmp = Float64(6.0 * Float64(x * z));
	else
		tmp = Float64(-6.0 * Float64(y * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -0.0014], N[(x * N[(6.0 * z + -3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 0.68], N[(4.0 * N[(y - x), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 1.85e+134], N[(y * N[(-6.0 * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 5.1e+290], N[(6.0 * N[(x * z), $MachinePrecision]), $MachinePrecision], N[(-6.0 * N[(y * z), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if z < -0.00139999999999999999

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

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

      1. remove-double-neg N/A

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

      2. neg-mul-1 N/A

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

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

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

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

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

      5. distribute-neg-in N/A

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

      6. metadata-eval N/A

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

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

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. metadata-eval N/A

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

      11. sub-neg N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. neg-mul-1 N/A

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

      15. lower-*.f64 N/A

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

      16. sub-neg N/A

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

      17. metadata-eval N/A

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

      18. distribute-neg-in N/A

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

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

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

      20. metadata-eval N/A

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

   5. Applied rewrites 56.2%

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

   if -0.00139999999999999999 < z < 0.680000000000000049

   1. Initial program 99.3%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   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. lower-fma.f64 N/A

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

      3. lower--.f64 96.2

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

   5. Applied rewrites 96.2%

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

   if 0.680000000000000049 < z < 1.85000000000000007e134

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. associate-*r* N/A

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

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 66.6

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

   5. Applied rewrites 66.6%

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

   6. Taylor expanded in z around inf

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

      1. lower-*.f64 63.4

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

   8. Applied rewrites 63.4%

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

   if 1.85000000000000007e134 < z < 5.10000000000000023e290

   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. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. distribute-lft-in N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. associate-*r* N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 65.8

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

   5. Applied rewrites 65.8%

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

   6. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 65.9

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

   8. Applied rewrites 65.9%

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 65.9

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

   10. Applied rewrites 65.9%

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

   if 5.10000000000000023e290 < 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 x around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. associate-*r* N/A

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

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 99.8

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

   8. Applied rewrites 99.8%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.0014:\\ \;\;\;\;x \cdot \mathsf{fma}\left(6, z, -3\right)\\ \mathbf{elif}\;z \leq 0.68:\\ \;\;\;\;\mathsf{fma}\left(4, y - x, x\right)\\ \mathbf{elif}\;z \leq 1.85 \cdot 10^{+134}:\\ \;\;\;\;y \cdot \left(-6 \cdot z\right)\\ \mathbf{elif}\;z \leq 5.1 \cdot 10^{+290}:\\ \;\;\;\;6 \cdot \left(x \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;-6 \cdot \left(y \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 6 \cdot \left(x \cdot z\right)\\ \mathbf{if}\;z \leq -490000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 0.68:\\ \;\;\;\;\mathsf{fma}\left(4, y - x, x\right)\\ \mathbf{elif}\;z \leq 1.85 \cdot 10^{+134}:\\ \;\;\;\;y \cdot \left(-6 \cdot z\right)\\ \mathbf{elif}\;z \leq 5.1 \cdot 10^{+290}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;-6 \cdot \left(y \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* 6.0 (* x z))))
   (if (<= z -490000000.0)
     t_0
     (if (<= z 0.68)
       (fma 4.0 (- y x) x)
       (if (<= z 1.85e+134)
         (* y (* -6.0 z))
         (if (<= z 5.1e+290) t_0 (* -6.0 (* y z))))))))

C

double code(double x, double y, double z) {
	double t_0 = 6.0 * (x * z);
	double tmp;
	if (z <= -490000000.0) {
		tmp = t_0;
	} else if (z <= 0.68) {
		tmp = fma(4.0, (y - x), x);
	} else if (z <= 1.85e+134) {
		tmp = y * (-6.0 * z);
	} else if (z <= 5.1e+290) {
		tmp = t_0;
	} else {
		tmp = -6.0 * (y * z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(6.0 * Float64(x * z))
	tmp = 0.0
	if (z <= -490000000.0)
		tmp = t_0;
	elseif (z <= 0.68)
		tmp = fma(4.0, Float64(y - x), x);
	elseif (z <= 1.85e+134)
		tmp = Float64(y * Float64(-6.0 * z));
	elseif (z <= 5.1e+290)
		tmp = t_0;
	else
		tmp = Float64(-6.0 * Float64(y * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(6.0 * N[(x * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -490000000.0], t$95$0, If[LessEqual[z, 0.68], N[(4.0 * N[(y - x), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 1.85e+134], N[(y * N[(-6.0 * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 5.1e+290], t$95$0, N[(-6.0 * N[(y * z), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -4.9e8 or 1.85000000000000007e134 < z < 5.10000000000000023e290

   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

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. distribute-lft-in N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. associate-*r* N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 61.4

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

   5. Applied rewrites 61.4%

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

   6. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 61.4

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

   8. Applied rewrites 61.4%

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 61.4

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

   10. Applied rewrites 61.4%

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

   if -4.9e8 < z < 0.680000000000000049

   1. Initial program 99.3%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   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. lower-fma.f64 N/A

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

      3. lower--.f64 93.5

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

   5. Applied rewrites 93.5%

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

   if 0.680000000000000049 < z < 1.85000000000000007e134

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. associate-*r* N/A

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

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 66.6

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

   5. Applied rewrites 66.6%

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

   6. Taylor expanded in z around inf

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

      1. lower-*.f64 63.4

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

   8. Applied rewrites 63.4%

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

   if 5.10000000000000023e290 < 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 x around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. associate-*r* N/A

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

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 99.8

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

   8. Applied rewrites 99.8%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -490000000:\\ \;\;\;\;6 \cdot \left(x \cdot z\right)\\ \mathbf{elif}\;z \leq 0.68:\\ \;\;\;\;\mathsf{fma}\left(4, y - x, x\right)\\ \mathbf{elif}\;z \leq 1.85 \cdot 10^{+134}:\\ \;\;\;\;y \cdot \left(-6 \cdot z\right)\\ \mathbf{elif}\;z \leq 5.1 \cdot 10^{+290}:\\ \;\;\;\;6 \cdot \left(x \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;-6 \cdot \left(y \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(x \cdot 6\right)\\ \mathbf{if}\;z \leq -490000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 0.68:\\ \;\;\;\;\mathsf{fma}\left(4, y - x, x\right)\\ \mathbf{elif}\;z \leq 1.85 \cdot 10^{+134}:\\ \;\;\;\;y \cdot \left(-6 \cdot z\right)\\ \mathbf{elif}\;z \leq 5.1 \cdot 10^{+290}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;-6 \cdot \left(y \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (* x 6.0))))
   (if (<= z -490000000.0)
     t_0
     (if (<= z 0.68)
       (fma 4.0 (- y x) x)
       (if (<= z 1.85e+134)
         (* y (* -6.0 z))
         (if (<= z 5.1e+290) t_0 (* -6.0 (* y z))))))))

C

double code(double x, double y, double z) {
	double t_0 = z * (x * 6.0);
	double tmp;
	if (z <= -490000000.0) {
		tmp = t_0;
	} else if (z <= 0.68) {
		tmp = fma(4.0, (y - x), x);
	} else if (z <= 1.85e+134) {
		tmp = y * (-6.0 * z);
	} else if (z <= 5.1e+290) {
		tmp = t_0;
	} else {
		tmp = -6.0 * (y * z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(x * 6.0))
	tmp = 0.0
	if (z <= -490000000.0)
		tmp = t_0;
	elseif (z <= 0.68)
		tmp = fma(4.0, Float64(y - x), x);
	elseif (z <= 1.85e+134)
		tmp = Float64(y * Float64(-6.0 * z));
	elseif (z <= 5.1e+290)
		tmp = t_0;
	else
		tmp = Float64(-6.0 * Float64(y * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(x * 6.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -490000000.0], t$95$0, If[LessEqual[z, 0.68], N[(4.0 * N[(y - x), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 1.85e+134], N[(y * N[(-6.0 * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 5.1e+290], t$95$0, N[(-6.0 * N[(y * z), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -4.9e8 or 1.85000000000000007e134 < z < 5.10000000000000023e290

   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

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. distribute-lft-in N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. associate-*r* N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 61.4

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

   5. Applied rewrites 61.4%

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

   6. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 61.4

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

   8. Applied rewrites 61.4%

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

   if -4.9e8 < z < 0.680000000000000049

   1. Initial program 99.3%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   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. lower-fma.f64 N/A

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

      3. lower--.f64 93.5

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

   5. Applied rewrites 93.5%

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

   if 0.680000000000000049 < z < 1.85000000000000007e134

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. associate-*r* N/A

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

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 66.6

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

   5. Applied rewrites 66.6%

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

   6. Taylor expanded in z around inf

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

      1. lower-*.f64 63.4

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

   8. Applied rewrites 63.4%

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

   if 5.10000000000000023e290 < 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 x around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. associate-*r* N/A

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

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 99.8

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

   8. Applied rewrites 99.8%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -490000000:\\ \;\;\;\;z \cdot \left(x \cdot 6\right)\\ \mathbf{elif}\;z \leq 0.68:\\ \;\;\;\;\mathsf{fma}\left(4, y - x, x\right)\\ \mathbf{elif}\;z \leq 1.85 \cdot 10^{+134}:\\ \;\;\;\;y \cdot \left(-6 \cdot z\right)\\ \mathbf{elif}\;z \leq 5.1 \cdot 10^{+290}:\\ \;\;\;\;z \cdot \left(x \cdot 6\right)\\ \mathbf{else}:\\ \;\;\;\;-6 \cdot \left(y \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 75.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(-6 \cdot z\right)\\ \mathbf{if}\;z \leq -28.5:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 0.68:\\ \;\;\;\;\mathsf{fma}\left(4, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (* -6.0 z))))
   (if (<= z -28.5) t_0 (if (<= z 0.68) (fma 4.0 (- y x) x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = y * (-6.0 * z);
	double tmp;
	if (z <= -28.5) {
		tmp = t_0;
	} else if (z <= 0.68) {
		tmp = fma(4.0, (y - x), x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(y * Float64(-6.0 * z))
	tmp = 0.0
	if (z <= -28.5)
		tmp = t_0;
	elseif (z <= 0.68)
		tmp = fma(4.0, Float64(y - x), x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[(-6.0 * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -28.5], t$95$0, If[LessEqual[z, 0.68], N[(4.0 * N[(y - x), $MachinePrecision] + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -28.5 or 0.680000000000000049 < 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 x around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. associate-*r* N/A

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

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 53.1

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

   5. Applied rewrites 53.1%

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

   6. Taylor expanded in z around inf

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

      1. lower-*.f64 51.2

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

   8. Applied rewrites 51.2%

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

   if -28.5 < z < 0.680000000000000049

   1. Initial program 99.3%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   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. lower-fma.f64 N/A

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

      3. lower--.f64 95.5

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

   5. Applied rewrites 95.5%

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

Alternative 10: 75.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -6 \cdot \left(y \cdot z\right)\\ \mathbf{if}\;z \leq -28.5:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 0.68:\\ \;\;\;\;\mathsf{fma}\left(4, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* -6.0 (* y z))))
   (if (<= z -28.5) t_0 (if (<= z 0.68) (fma 4.0 (- y x) x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = -6.0 * (y * z);
	double tmp;
	if (z <= -28.5) {
		tmp = t_0;
	} else if (z <= 0.68) {
		tmp = fma(4.0, (y - x), x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(-6.0 * Float64(y * z))
	tmp = 0.0
	if (z <= -28.5)
		tmp = t_0;
	elseif (z <= 0.68)
		tmp = fma(4.0, Float64(y - x), x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(-6.0 * N[(y * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -28.5], t$95$0, If[LessEqual[z, 0.68], N[(4.0 * N[(y - x), $MachinePrecision] + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -28.5 or 0.680000000000000049 < 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 x around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. associate-*r* N/A

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

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 53.1

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

   5. Applied rewrites 53.1%

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

   6. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 51.1

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

   8. Applied rewrites 51.1%

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

   if -28.5 < z < 0.680000000000000049

   1. Initial program 99.3%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   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. lower-fma.f64 N/A

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

      3. lower--.f64 95.5

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

   5. Applied rewrites 95.5%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -28.5:\\ \;\;\;\;-6 \cdot \left(y \cdot z\right)\\ \mathbf{elif}\;z \leq 0.68:\\ \;\;\;\;\mathsf{fma}\left(4, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;-6 \cdot \left(y \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 39.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y - x\right) \cdot 4\\ \mathbf{if}\;y \leq -2.65 \cdot 10^{-120}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2.65 \cdot 10^{-34}:\\ \;\;\;\;x \cdot -3\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- y x) 4.0)))
   (if (<= y -2.65e-120) t_0 (if (<= y 2.65e-34) (* x -3.0) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (y - x) * 4.0;
	double tmp;
	if (y <= -2.65e-120) {
		tmp = t_0;
	} else if (y <= 2.65e-34) {
		tmp = x * -3.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (y - x) * 4.0d0
    if (y <= (-2.65d-120)) then
        tmp = t_0
    else if (y <= 2.65d-34) then
        tmp = x * (-3.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (y - x) * 4.0;
	double tmp;
	if (y <= -2.65e-120) {
		tmp = t_0;
	} else if (y <= 2.65e-34) {
		tmp = x * -3.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (y - x) * 4.0
	tmp = 0
	if y <= -2.65e-120:
		tmp = t_0
	elif y <= 2.65e-34:
		tmp = x * -3.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y - x) * 4.0)
	tmp = 0.0
	if (y <= -2.65e-120)
		tmp = t_0;
	elseif (y <= 2.65e-34)
		tmp = Float64(x * -3.0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (y - x) * 4.0;
	tmp = 0.0;
	if (y <= -2.65e-120)
		tmp = t_0;
	elseif (y <= 2.65e-34)
		tmp = x * -3.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y - x), $MachinePrecision] * 4.0), $MachinePrecision]}, If[LessEqual[y, -2.65e-120], t$95$0, If[LessEqual[y, 2.65e-34], N[(x * -3.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.64999999999999999e-120 or 2.6499999999999998e-34 < y

   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

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. sub-neg N/A

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

      10. distribute-lft-in N/A

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

      11. associate-+l+ N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l* N/A

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

      14. lower-fma.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 73.6

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

   7. Applied rewrites 73.6%

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

   8. Taylor expanded in z around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 36.0

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

   10. Applied rewrites 36.0%

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

   if -2.64999999999999999e-120 < y < 2.6499999999999998e-34

   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. lower-fma.f64 N/A

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

      3. lower--.f64 46.0

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

   5. Applied rewrites 46.0%

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

   6. Taylor expanded in y around 0

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

      1. distribute-rgt1-in N/A

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 40.6

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

   8. Applied rewrites 40.6%

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

4. Final simplification 37.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.65 \cdot 10^{-120}:\\ \;\;\;\;\left(y - x\right) \cdot 4\\ \mathbf{elif}\;y \leq 2.65 \cdot 10^{-34}:\\ \;\;\;\;x \cdot -3\\ \mathbf{else}:\\ \;\;\;\;\left(y - x\right) \cdot 4\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 37.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{+114}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;y \leq 2.45 \cdot 10^{-32}:\\ \;\;\;\;x \cdot -3\\ \mathbf{else}:\\ \;\;\;\;y \cdot 4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -7.2e+114) (* y 4.0) (if (<= y 2.45e-32) (* x -3.0) (* y 4.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -7.2e+114) {
		tmp = y * 4.0;
	} else if (y <= 2.45e-32) {
		tmp = x * -3.0;
	} 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 <= (-7.2d+114)) then
        tmp = y * 4.0d0
    else if (y <= 2.45d-32) then
        tmp = x * (-3.0d0)
    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 <= -7.2e+114) {
		tmp = y * 4.0;
	} else if (y <= 2.45e-32) {
		tmp = x * -3.0;
	} else {
		tmp = y * 4.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -7.2e+114:
		tmp = y * 4.0
	elif y <= 2.45e-32:
		tmp = x * -3.0
	else:
		tmp = y * 4.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -7.2e+114)
		tmp = Float64(y * 4.0);
	elseif (y <= 2.45e-32)
		tmp = Float64(x * -3.0);
	else
		tmp = Float64(y * 4.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -7.2e+114)
		tmp = y * 4.0;
	elseif (y <= 2.45e-32)
		tmp = x * -3.0;
	else
		tmp = y * 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -7.2e+114], N[(y * 4.0), $MachinePrecision], If[LessEqual[y, 2.45e-32], N[(x * -3.0), $MachinePrecision], N[(y * 4.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -7.2000000000000001e114 or 2.4499999999999999e-32 < 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. lower-fma.f64 N/A

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

      3. lower--.f64 47.9

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

   5. Applied rewrites 47.9%

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

   6. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 38.4

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

   8. Applied rewrites 38.4%

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

   if -7.2000000000000001e114 < y < 2.4499999999999999e-32

   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. lower-fma.f64 N/A

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

      3. lower--.f64 45.3

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

   5. Applied rewrites 45.3%

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

   6. Taylor expanded in y around 0

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

      1. distribute-rgt1-in N/A

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 35.5

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

   8. Applied rewrites 35.5%

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

Alternative 13: 99.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_] := N[(N[(6.0 * z + -4.0), $MachinePrecision] * N[(x - y), $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. Taylor expanded in x around 0

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

5. Applied rewrites 99.7%

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

Alternative 14: 50.4% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_] := N[(4.0 * 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. 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. lower-fma.f64 N/A

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

   3. lower--.f64 46.5

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

5. Applied rewrites 46.5%

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

Alternative 15: 25.6% accurate, 5.2× speedup

Math

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

FPCore

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

C

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

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 * (-3.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(x * -3.0), $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. lower-fma.f64 N/A

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

   3. lower--.f64 46.5

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

5. Applied rewrites 46.5%

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

6. Taylor expanded in y around 0

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

   1. distribute-rgt1-in N/A

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

   2. metadata-eval N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 24.5

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

8. Applied rewrites 24.5%

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

Alternative 16: 7.4% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ x \cdot -4 \end{array} \]

FPCore

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

C

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

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 * (-4.0d0)
end function

Java

public static double code(double x, double y, double z) {
	return x * -4.0;
}

Python

def code(x, y, z):
	return x * -4.0

Julia

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

MATLAB

function tmp = code(x, y, z)
	tmp = x * -4.0;
end

Wolfram

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

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. +-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. lift--.f64 N/A

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

   9. sub-neg N/A

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

   10. distribute-lft-in N/A

       \[\leadsto \color{blue}{\left(\left(\left(y - x\right) \cdot 6\right) \cdot \frac{2}{3} + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right)\right)} + x \]

   11. associate-+l+ N/A

       \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \frac{2}{3} + \left(\left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right) + x\right)} \]

   12. lift-*.f64 N/A

       \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right)} \cdot \frac{2}{3} + \left(\left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right) + x\right) \]

   13. associate-*l* N/A

       \[\leadsto \color{blue}{\left(y - x\right) \cdot \left(6 \cdot \frac{2}{3}\right)} + \left(\left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right) + x\right) \]

   14. lower-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 6 \cdot \frac{2}{3}, \left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right) + x\right)} \]

   15. lift-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(y - x, 6 \cdot \color{blue}{\frac{2}{3}}, \left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right) + x\right) \]

   16. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(y - x, 6 \cdot \color{blue}{\frac{2}{3}}, \left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right) + x\right) \]

   17. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(y - x, \color{blue}{4}, \left(\left(y - x\right) \cdot 6\right) \cdot \left(\mathsf{neg}\left(z\right)\right) + x\right) \]

4. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 4, \mathsf{fma}\left(y - x, -6 \cdot z, x\right)\right)} \]

5. Taylor expanded in y around inf

   \[\leadsto \mathsf{fma}\left(y - x, 4, \color{blue}{-6 \cdot \left(y \cdot z\right)}\right) \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(y - x, 4, \color{blue}{\left(y \cdot z\right) \cdot -6}\right) \]

   2. associate-*r* N/A

      \[\leadsto \mathsf{fma}\left(y - x, 4, \color{blue}{y \cdot \left(z \cdot -6\right)}\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(y - x, 4, y \cdot \color{blue}{\left(-6 \cdot z\right)}\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(y - x, 4, \color{blue}{y \cdot \left(-6 \cdot z\right)}\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(y - x, 4, y \cdot \color{blue}{\left(z \cdot -6\right)}\right) \]

   6. lower-*.f64 56.2

      \[\leadsto \mathsf{fma}\left(y - x, 4, y \cdot \color{blue}{\left(z \cdot -6\right)}\right) \]

7. Applied rewrites 56.2%

   \[\leadsto \mathsf{fma}\left(y - x, 4, \color{blue}{y \cdot \left(z \cdot -6\right)}\right) \]

8. Taylor expanded in y around 0

   \[\leadsto \color{blue}{-4 \cdot x} \]
9. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{x \cdot -4} \]

   2. lower-*.f64 7.2

      \[\leadsto \color{blue}{x \cdot -4} \]

10. Applied rewrites 7.2%

    \[\leadsto \color{blue}{x \cdot -4} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024216 
(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))))