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

Percentage Accurate: 99.5% → 99.8%

Time: 11.4s

Alternatives: 14

Speedup: 1.9×

• 20.7% 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: 99.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift--.f64 N/A

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

   5. sub-neg N/A

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

   6. distribute-lft-in N/A

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

   7. associate-+l+ N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*l* N/A

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

   10. lower-fma.f64 N/A

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

   11. lift-/.f64 N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. *-commutative N/A

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. associate-*r* N/A

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

   18. lower-fma.f64 N/A

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

4. Applied rewrites 99.8%

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

Alternative 2: 97.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{3} - z\\ \mathbf{if}\;t\_0 \leq -5 \lor \neg \left(t\_0 \leq 1\right):\\ \;\;\;\;\left(\left(y - x\right) \cdot -6\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-3, x, 4 \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (/ 2.0 3.0) z)))
   (if (or (<= t_0 -5.0) (not (<= t_0 1.0)))
     (* (* (- y x) -6.0) z)
     (fma -3.0 x (* 4.0 y)))))

C

double code(double x, double y, double z) {
	double t_0 = (2.0 / 3.0) - z;
	double tmp;
	if ((t_0 <= -5.0) || !(t_0 <= 1.0)) {
		tmp = ((y - x) * -6.0) * z;
	} else {
		tmp = fma(-3.0, x, (4.0 * y));
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(2.0 / 3.0) - z)
	tmp = 0.0
	if ((t_0 <= -5.0) || !(t_0 <= 1.0))
		tmp = Float64(Float64(Float64(y - x) * -6.0) * z);
	else
		tmp = fma(-3.0, x, Float64(4.0 * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(2.0 / 3.0), $MachinePrecision] - z), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -5.0], N[Not[LessEqual[t$95$0, 1.0]], $MachinePrecision]], N[(N[(N[(y - x), $MachinePrecision] * -6.0), $MachinePrecision] * z), $MachinePrecision], N[(-3.0 * x + N[(4.0 * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < -5 or 1 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z)

   1. Initial program 99.7%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 97.1

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

   5. Applied rewrites 97.1%

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

   7. Applied rewrites 97.1%

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

   if -5 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < 1

   1. Initial program 99.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. distribute-lft-in N/A

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

      7. associate-+l+ N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-*r* N/A

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

      18. lower-fma.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around 0

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. mul-1-neg N/A

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

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

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

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

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

      8. metadata-eval N/A

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

      9. associate-+r+ N/A

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

      10. distribute-rgt1-in N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 98.4

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

   7. Applied rewrites 98.4%

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

4. Final simplification 97.8%

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

Alternative 3: 97.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{3} - z\\ \mathbf{if}\;t\_0 \leq -5:\\ \;\;\;\;\left(\left(y - x\right) \cdot z\right) \cdot -6\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\mathsf{fma}\left(-3, x, 4 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y - x\right) \cdot -6\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (/ 2.0 3.0) z)))
   (if (<= t_0 -5.0)
     (* (* (- y x) z) -6.0)
     (if (<= t_0 1.0) (fma -3.0 x (* 4.0 y)) (* (* (- y x) -6.0) z)))))

C

double code(double x, double y, double z) {
	double t_0 = (2.0 / 3.0) - z;
	double tmp;
	if (t_0 <= -5.0) {
		tmp = ((y - x) * z) * -6.0;
	} else if (t_0 <= 1.0) {
		tmp = fma(-3.0, x, (4.0 * y));
	} else {
		tmp = ((y - x) * -6.0) * z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(2.0 / 3.0) - z)
	tmp = 0.0
	if (t_0 <= -5.0)
		tmp = Float64(Float64(Float64(y - x) * z) * -6.0);
	elseif (t_0 <= 1.0)
		tmp = fma(-3.0, x, Float64(4.0 * y));
	else
		tmp = Float64(Float64(Float64(y - x) * -6.0) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(2.0 / 3.0), $MachinePrecision] - z), $MachinePrecision]}, If[LessEqual[t$95$0, -5.0], N[(N[(N[(y - x), $MachinePrecision] * z), $MachinePrecision] * -6.0), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[(-3.0 * x + N[(4.0 * y), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y - x), $MachinePrecision] * -6.0), $MachinePrecision] * z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   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 inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 97.1

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

   5. Applied rewrites 97.1%

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

   if -5 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < 1

   1. Initial program 99.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. distribute-lft-in N/A

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

      7. associate-+l+ N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-*r* N/A

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

      18. lower-fma.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around 0

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. mul-1-neg N/A

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

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

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

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

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

      8. metadata-eval N/A

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

      9. associate-+r+ N/A

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

      10. distribute-rgt1-in N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 98.4

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

   7. Applied rewrites 98.4%

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

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

   1. Initial program 99.8%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 97.0

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

   5. Applied rewrites 97.0%

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

   7. Applied rewrites 97.1%

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

Alternative 4: 75.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-6, z, 4\right) \cdot y\\ \mathbf{if}\;z \leq -1.72 \cdot 10^{-15}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(-3, x, 4 \cdot y\right)\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{+78}:\\ \;\;\;\;\mathsf{fma}\left(6, z, -3\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (fma -6.0 z 4.0) y)))
   (if (<= z -1.72e-15)
     t_0
     (if (<= z 1.7e-10)
       (fma -3.0 x (* 4.0 y))
       (if (<= z 9.5e+78) (* (fma 6.0 z -3.0) x) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = fma(-6.0, z, 4.0) * y;
	double tmp;
	if (z <= -1.72e-15) {
		tmp = t_0;
	} else if (z <= 1.7e-10) {
		tmp = fma(-3.0, x, (4.0 * y));
	} else if (z <= 9.5e+78) {
		tmp = fma(6.0, z, -3.0) * x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(fma(-6.0, z, 4.0) * y)
	tmp = 0.0
	if (z <= -1.72e-15)
		tmp = t_0;
	elseif (z <= 1.7e-10)
		tmp = fma(-3.0, x, Float64(4.0 * y));
	elseif (z <= 9.5e+78)
		tmp = Float64(fma(6.0, z, -3.0) * x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(-6.0 * z + 4.0), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[z, -1.72e-15], t$95$0, If[LessEqual[z, 1.7e-10], N[(-3.0 * x + N[(4.0 * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 9.5e+78], N[(N[(6.0 * z + -3.0), $MachinePrecision] * x), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.7199999999999999e-15 or 9.5000000000000006e78 < 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. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. sub-neg N/A

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

      5. mul-1-neg N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 64.5

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

   5. Applied rewrites 64.5%

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

   if -1.7199999999999999e-15 < z < 1.70000000000000007e-10

   1. Initial program 99.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. distribute-lft-in N/A

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

      7. associate-+l+ N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-*r* N/A

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

      18. lower-fma.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around 0

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. mul-1-neg N/A

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

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

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

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

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

      8. metadata-eval N/A

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

      9. associate-+r+ N/A

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

      10. distribute-rgt1-in N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 99.8

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

   7. Applied rewrites 99.8%

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

   if 1.70000000000000007e-10 < z < 9.5000000000000006e78

   1. Initial program 99.4%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. associate-*r* N/A

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

      7. mul-1-neg N/A

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

      8. metadata-eval N/A

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

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

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. distribute-neg-in N/A

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

      13. distribute-rgt-in N/A

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

      14. metadata-eval N/A

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

      15. sub-neg N/A

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

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

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

   5. Applied rewrites 65.8%

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

Alternative 5: 75.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-6, z, 4\right) \cdot y\\ \mathbf{if}\;z \leq -1.72 \cdot 10^{-15}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(y - x, 4, x\right)\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{+78}:\\ \;\;\;\;\mathsf{fma}\left(6, z, -3\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (fma -6.0 z 4.0) y)))
   (if (<= z -1.72e-15)
     t_0
     (if (<= z 1.7e-10)
       (fma (- y x) 4.0 x)
       (if (<= z 9.5e+78) (* (fma 6.0 z -3.0) x) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = fma(-6.0, z, 4.0) * y;
	double tmp;
	if (z <= -1.72e-15) {
		tmp = t_0;
	} else if (z <= 1.7e-10) {
		tmp = fma((y - x), 4.0, x);
	} else if (z <= 9.5e+78) {
		tmp = fma(6.0, z, -3.0) * x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(fma(-6.0, z, 4.0) * y)
	tmp = 0.0
	if (z <= -1.72e-15)
		tmp = t_0;
	elseif (z <= 1.7e-10)
		tmp = fma(Float64(y - x), 4.0, x);
	elseif (z <= 9.5e+78)
		tmp = Float64(fma(6.0, z, -3.0) * x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(-6.0 * z + 4.0), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[z, -1.72e-15], t$95$0, If[LessEqual[z, 1.7e-10], N[(N[(y - x), $MachinePrecision] * 4.0 + x), $MachinePrecision], If[LessEqual[z, 9.5e+78], N[(N[(6.0 * z + -3.0), $MachinePrecision] * x), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.7199999999999999e-15 or 9.5000000000000006e78 < 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. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. sub-neg N/A

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

      5. mul-1-neg N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 64.5

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

   5. Applied rewrites 64.5%

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

   if -1.7199999999999999e-15 < z < 1.70000000000000007e-10

   1. Initial program 99.5%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 99.8

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

   5. Applied rewrites 99.8%

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

   if 1.70000000000000007e-10 < z < 9.5000000000000006e78

   1. Initial program 99.4%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. associate-*r* N/A

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

      7. mul-1-neg N/A

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

      8. metadata-eval N/A

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

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

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. distribute-neg-in N/A

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

      13. distribute-rgt-in N/A

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

      14. metadata-eval N/A

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

      15. sub-neg N/A

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

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

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

   5. Applied rewrites 65.8%

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

Alternative 6: 75.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-6, z, 4\right) \cdot y\\ \mathbf{if}\;z \leq -1.72 \cdot 10^{-15}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 0.52:\\ \;\;\;\;\mathsf{fma}\left(y - x, 4, x\right)\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{+78}:\\ \;\;\;\;\left(z \cdot x\right) \cdot 6\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (fma -6.0 z 4.0) y)))
   (if (<= z -1.72e-15)
     t_0
     (if (<= z 0.52)
       (fma (- y x) 4.0 x)
       (if (<= z 9.5e+78) (* (* z x) 6.0) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = fma(-6.0, z, 4.0) * y;
	double tmp;
	if (z <= -1.72e-15) {
		tmp = t_0;
	} else if (z <= 0.52) {
		tmp = fma((y - x), 4.0, x);
	} else if (z <= 9.5e+78) {
		tmp = (z * x) * 6.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(fma(-6.0, z, 4.0) * y)
	tmp = 0.0
	if (z <= -1.72e-15)
		tmp = t_0;
	elseif (z <= 0.52)
		tmp = fma(Float64(y - x), 4.0, x);
	elseif (z <= 9.5e+78)
		tmp = Float64(Float64(z * x) * 6.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(-6.0 * z + 4.0), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[z, -1.72e-15], t$95$0, If[LessEqual[z, 0.52], N[(N[(y - x), $MachinePrecision] * 4.0 + x), $MachinePrecision], If[LessEqual[z, 9.5e+78], N[(N[(z * x), $MachinePrecision] * 6.0), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.7199999999999999e-15 or 9.5000000000000006e78 < 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. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. sub-neg N/A

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

      5. mul-1-neg N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 64.5

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

   5. Applied rewrites 64.5%

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

   if -1.7199999999999999e-15 < z < 0.52000000000000002

   1. Initial program 99.5%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 99.1

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

   5. Applied rewrites 99.1%

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

   if 0.52000000000000002 < z < 9.5000000000000006e78

   1. Initial program 99.4%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 92.4

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

   5. Applied rewrites 92.4%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 59.4%

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

Alternative 7: 74.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-6 \cdot z\right) \cdot y\\ \mathbf{if}\;z \leq -23000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 0.52:\\ \;\;\;\;\mathsf{fma}\left(y - x, 4, x\right)\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{+78}:\\ \;\;\;\;\left(z \cdot x\right) \cdot 6\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (* -6.0 z) y)))
   (if (<= z -23000000.0)
     t_0
     (if (<= z 0.52)
       (fma (- y x) 4.0 x)
       (if (<= z 9.5e+78) (* (* z x) 6.0) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = (-6.0 * z) * y;
	double tmp;
	if (z <= -23000000.0) {
		tmp = t_0;
	} else if (z <= 0.52) {
		tmp = fma((y - x), 4.0, x);
	} else if (z <= 9.5e+78) {
		tmp = (z * x) * 6.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(-6.0 * z) * y)
	tmp = 0.0
	if (z <= -23000000.0)
		tmp = t_0;
	elseif (z <= 0.52)
		tmp = fma(Float64(y - x), 4.0, x);
	elseif (z <= 9.5e+78)
		tmp = Float64(Float64(z * x) * 6.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(-6.0 * z), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[z, -23000000.0], t$95$0, If[LessEqual[z, 0.52], N[(N[(y - x), $MachinePrecision] * 4.0 + x), $MachinePrecision], If[LessEqual[z, 9.5e+78], N[(N[(z * x), $MachinePrecision] * 6.0), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.3e7 or 9.5000000000000006e78 < z

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. distribute-lft-in N/A

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

      7. associate-+l+ N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-*r* N/A

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

      18. lower-fma.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 64.0

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

   7. Applied rewrites 64.0%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 63.6%

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

   12. Applied rewrites 63.6%

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

   if -2.3e7 < z < 0.52000000000000002

   1. Initial program 99.5%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 97.1

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

   5. Applied rewrites 97.1%

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

   if 0.52000000000000002 < z < 9.5000000000000006e78

   1. Initial program 99.4%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 92.4

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

   5. Applied rewrites 92.4%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 59.4%

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

Alternative 8: 74.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-6 \cdot y\right) \cdot z\\ \mathbf{if}\;z \leq -23000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 0.52:\\ \;\;\;\;\mathsf{fma}\left(y - x, 4, x\right)\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{+78}:\\ \;\;\;\;\left(z \cdot x\right) \cdot 6\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (* -6.0 y) z)))
   (if (<= z -23000000.0)
     t_0
     (if (<= z 0.52)
       (fma (- y x) 4.0 x)
       (if (<= z 9.5e+78) (* (* z x) 6.0) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = (-6.0 * y) * z;
	double tmp;
	if (z <= -23000000.0) {
		tmp = t_0;
	} else if (z <= 0.52) {
		tmp = fma((y - x), 4.0, x);
	} else if (z <= 9.5e+78) {
		tmp = (z * x) * 6.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(-6.0 * y) * z)
	tmp = 0.0
	if (z <= -23000000.0)
		tmp = t_0;
	elseif (z <= 0.52)
		tmp = fma(Float64(y - x), 4.0, x);
	elseif (z <= 9.5e+78)
		tmp = Float64(Float64(z * x) * 6.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(-6.0 * y), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -23000000.0], t$95$0, If[LessEqual[z, 0.52], N[(N[(y - x), $MachinePrecision] * 4.0 + x), $MachinePrecision], If[LessEqual[z, 9.5e+78], N[(N[(z * x), $MachinePrecision] * 6.0), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.3e7 or 9.5000000000000006e78 < z

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. distribute-lft-in N/A

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

      7. associate-+l+ N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-*r* N/A

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

      18. lower-fma.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 64.0

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

   7. Applied rewrites 64.0%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 63.6%

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

   if -2.3e7 < z < 0.52000000000000002

   1. Initial program 99.5%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 97.1

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

   5. Applied rewrites 97.1%

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

   if 0.52000000000000002 < z < 9.5000000000000006e78

   1. Initial program 99.4%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 92.4

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

   5. Applied rewrites 92.4%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 59.4%

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

Alternative 9: 74.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(z \cdot y\right) \cdot -6\\ \mathbf{if}\;z \leq -23000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 0.52:\\ \;\;\;\;\mathsf{fma}\left(y - x, 4, x\right)\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{+78}:\\ \;\;\;\;\left(z \cdot x\right) \cdot 6\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (* z y) -6.0)))
   (if (<= z -23000000.0)
     t_0
     (if (<= z 0.52)
       (fma (- y x) 4.0 x)
       (if (<= z 9.5e+78) (* (* z x) 6.0) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = (z * y) * -6.0;
	double tmp;
	if (z <= -23000000.0) {
		tmp = t_0;
	} else if (z <= 0.52) {
		tmp = fma((y - x), 4.0, x);
	} else if (z <= 9.5e+78) {
		tmp = (z * x) * 6.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(z * y) * -6.0)
	tmp = 0.0
	if (z <= -23000000.0)
		tmp = t_0;
	elseif (z <= 0.52)
		tmp = fma(Float64(y - x), 4.0, x);
	elseif (z <= 9.5e+78)
		tmp = Float64(Float64(z * x) * 6.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(z * y), $MachinePrecision] * -6.0), $MachinePrecision]}, If[LessEqual[z, -23000000.0], t$95$0, If[LessEqual[z, 0.52], N[(N[(y - x), $MachinePrecision] * 4.0 + x), $MachinePrecision], If[LessEqual[z, 9.5e+78], N[(N[(z * x), $MachinePrecision] * 6.0), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.3e7 or 9.5000000000000006e78 < z

   1. Initial program 99.8%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 99.3

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 63.5%

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

   if -2.3e7 < z < 0.52000000000000002

   1. Initial program 99.5%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 97.1

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

   5. Applied rewrites 97.1%

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

   if 0.52000000000000002 < z < 9.5000000000000006e78

   1. Initial program 99.4%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 92.4

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

   5. Applied rewrites 92.4%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 59.4%

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

Alternative 10: 73.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -360 \lor \neg \left(z \leq 0.52\right):\\ \;\;\;\;\left(z \cdot x\right) \cdot 6\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - x, 4, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -360.0) (not (<= z 0.52))) (* (* z x) 6.0) (fma (- y x) 4.0 x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -360.0) || !(z <= 0.52)) {
		tmp = (z * x) * 6.0;
	} else {
		tmp = fma((y - x), 4.0, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -360.0) || !(z <= 0.52))
		tmp = Float64(Float64(z * x) * 6.0);
	else
		tmp = fma(Float64(y - x), 4.0, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -360.0], N[Not[LessEqual[z, 0.52]], $MachinePrecision]], N[(N[(z * x), $MachinePrecision] * 6.0), $MachinePrecision], N[(N[(y - x), $MachinePrecision] * 4.0 + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -360 or 0.52000000000000002 < z

   1. Initial program 99.7%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 97.1

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

   5. Applied rewrites 97.1%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 45.0%

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

   if -360 < z < 0.52000000000000002

   1. Initial program 99.5%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 98.3

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

   5. Applied rewrites 98.3%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -360 \lor \neg \left(z \leq 0.52\right):\\ \;\;\;\;\left(z \cdot x\right) \cdot 6\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - x, 4, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 38.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.1 \cdot 10^{-44} \lor \neg \left(x \leq 2.8 \cdot 10^{+38}\right):\\ \;\;\;\;-3 \cdot x\\ \mathbf{else}:\\ \;\;\;\;4 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -7.1e-44) (not (<= x 2.8e+38))) (* -3.0 x) (* 4.0 y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -7.1e-44) || !(x <= 2.8e+38)) {
		tmp = -3.0 * x;
	} else {
		tmp = 4.0 * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-7.1d-44)) .or. (.not. (x <= 2.8d+38))) then
        tmp = (-3.0d0) * x
    else
        tmp = 4.0d0 * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -7.1e-44) || !(x <= 2.8e+38)) {
		tmp = -3.0 * x;
	} else {
		tmp = 4.0 * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -7.1e-44) or not (x <= 2.8e+38):
		tmp = -3.0 * x
	else:
		tmp = 4.0 * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -7.1e-44) || !(x <= 2.8e+38))
		tmp = Float64(-3.0 * x);
	else
		tmp = Float64(4.0 * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -7.1e-44) || ~((x <= 2.8e+38)))
		tmp = -3.0 * x;
	else
		tmp = 4.0 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -7.1e-44], N[Not[LessEqual[x, 2.8e+38]], $MachinePrecision]], N[(-3.0 * x), $MachinePrecision], N[(4.0 * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -7.09999999999999965e-44 or 2.8e38 < x

   1. Initial program 99.6%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 51.2

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

   5. Applied rewrites 51.2%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 42.8%

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

   if -7.09999999999999965e-44 < x < 2.8e38

   1. Initial program 99.6%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 53.2

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

   5. Applied rewrites 53.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 42.3%

      \[\leadsto 4 \cdot \color{blue}{y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.1 \cdot 10^{-44} \lor \neg \left(x \leq 2.8 \cdot 10^{+38}\right):\\ \;\;\;\;-3 \cdot x\\ \mathbf{else}:\\ \;\;\;\;4 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 99.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lift--.f64 N/A

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

   9. sub-neg N/A

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

   10. +-commutative N/A

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

   11. distribute-lft-in N/A

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

   12. neg-mul-1 N/A

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

   13. associate-*r* N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval N/A

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

   16. lower-fma.f64 N/A

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

   17. metadata-eval N/A

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

   18. lift-/.f64 N/A

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

   19. metadata-eval N/A

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

   20. metadata-eval 99.8

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

4. Applied rewrites 99.8%

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

Alternative 13: 50.4% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in z around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower--.f64 52.3

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

5. Applied rewrites 52.3%

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

Alternative 14: 25.9% accurate, 5.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in z around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower--.f64 52.3

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

5. Applied rewrites 52.3%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 26.1%

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

Reproduce

herbie shell --seed 2024318 
(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))))