Linear.Quaternion:$c/ from linear-1.19.1.3, A

Percentage Accurate: 98.4% → 99.5%

Time: 9.0s

Alternatives: 10

Speedup: 1.8×

• 35.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return (((x * y) + (z * z)) + (z * z)) + (z * 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) + (z * z)) + (z * z)) + (z * z)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return (((x * y) + (z * z)) + (z * z)) + (z * 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) + (z * z)) + (z * z)) + (z * z)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (x y z) :precision binary64 (fma (+ z z) z (fma x y (* z z))))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.3%

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

   1. lift-+.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. associate-+l+ N/A

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

   4. +-commutative N/A

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

   5. count-2 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*r* N/A

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

   8. count-2 N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-+.f64 98.4

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

   11. lift-+.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. lower-fma.f64 99.6

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

4. Applied rewrites 99.6%

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

Alternative 2: 86.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+62}:\\ \;\;\;\;\mathsf{fma}\left(z + z, z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z + z, z, z \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 1e+62) (fma (+ z z) z (* x y)) (fma (+ z z) z (* z z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 1e+62) {
		tmp = fma((z + z), z, (x * y));
	} else {
		tmp = fma((z + z), z, (z * z));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 1e+62)
		tmp = fma(Float64(z + z), z, Float64(x * y));
	else
		tmp = fma(Float64(z + z), z, Float64(z * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 1e+62], N[(N[(z + z), $MachinePrecision] * z + N[(x * y), $MachinePrecision]), $MachinePrecision], N[(N[(z + z), $MachinePrecision] * z + N[(z * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.00000000000000004e62

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. count-2 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. count-2 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-+.f64 100.0

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

      11. lift-+.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 81.3

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

   7. Applied rewrites 81.3%

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

   if 1.00000000000000004e62 < (*.f64 z z)

   1. Initial program 96.1%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. count-2 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. count-2 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-+.f64 96.2

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

      11. lift-+.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 99.0

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(z + z, z, \color{blue}{{z}^{2}}\right) \]
   6. Step-by-step derivation

      1. unpow2 N/A

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

      2. lower-*.f64 88.7

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

   7. Applied rewrites 88.7%

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

Alternative 3: 86.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+62}:\\ \;\;\;\;\mathsf{fma}\left(z + z, z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot z\right) \cdot 3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 1e+62) (fma (+ z z) z (* x y)) (* (* z z) 3.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 1e+62) {
		tmp = fma((z + z), z, (x * y));
	} else {
		tmp = (z * z) * 3.0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 1e+62)
		tmp = fma(Float64(z + z), z, Float64(x * y));
	else
		tmp = Float64(Float64(z * z) * 3.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 1e+62], N[(N[(z + z), $MachinePrecision] * z + N[(x * y), $MachinePrecision]), $MachinePrecision], N[(N[(z * z), $MachinePrecision] * 3.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.00000000000000004e62

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. count-2 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. count-2 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-+.f64 100.0

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

      11. lift-+.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 81.3

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

   7. Applied rewrites 81.3%

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

   if 1.00000000000000004e62 < (*.f64 z z)

   1. Initial program 96.1%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{2 \cdot {z}^{2} + {z}^{2}} \]
   4. Step-by-step derivation

      1. distribute-lft1-in N/A

         \[\leadsto \color{blue}{\left(2 + 1\right) \cdot {z}^{2}} \]

      2. metadata-eval N/A

         \[\leadsto \color{blue}{3} \cdot {z}^{2} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{{z}^{2} \cdot 3} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{{z}^{2} \cdot 3} \]

      5. unpow2 N/A

         \[\leadsto \color{blue}{\left(z \cdot z\right)} \cdot 3 \]

      6. lower-*.f64 88.6

         \[\leadsto \color{blue}{\left(z \cdot z\right)} \cdot 3 \]

   5. Applied rewrites 88.6%

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

Alternative 4: 86.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+62}:\\ \;\;\;\;\mathsf{fma}\left(z, z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot z\right) \cdot 3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 1e+62) (fma z z (* x y)) (* (* z z) 3.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 1e+62) {
		tmp = fma(z, z, (x * y));
	} else {
		tmp = (z * z) * 3.0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 1e+62)
		tmp = fma(z, z, Float64(x * y));
	else
		tmp = Float64(Float64(z * z) * 3.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 1e+62], N[(z * z + N[(x * y), $MachinePrecision]), $MachinePrecision], N[(N[(z * z), $MachinePrecision] * 3.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.00000000000000004e62

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. count-2 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. count-2 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-+.f64 100.0

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

      11. lift-+.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lift-+.f64 N/A

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

      5. distribute-lft-in N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. distribute-lft-in N/A

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

      11. flip-+ N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. +-inverses N/A

          \[\leadsto \mathsf{fma}\left(x, y, z \cdot z\right) + z \cdot \frac{\color{blue}{0}}{z - z} \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(x, y, z \cdot z\right) + z \cdot \frac{\color{blue}{0 - 0}}{z - z} \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(x, y, z \cdot z\right) + z \cdot \frac{\color{blue}{0 \cdot 0} - 0}{z - z} \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(x, y, z \cdot z\right) + z \cdot \frac{0 \cdot 0 - \color{blue}{0 \cdot 0}}{z - z} \]

      18. +-inverses N/A

          \[\leadsto \mathsf{fma}\left(x, y, z \cdot z\right) + z \cdot \frac{0 \cdot 0 - 0 \cdot 0}{\color{blue}{0}} \]

      19. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(x, y, z \cdot z\right) + z \cdot \frac{0 \cdot 0 - 0 \cdot 0}{\color{blue}{0 + 0}} \]

      20. flip-- N/A

          \[\leadsto \mathsf{fma}\left(x, y, z \cdot z\right) + z \cdot \color{blue}{\left(0 - 0\right)} \]

      21. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(x, y, z \cdot z\right) + z \cdot \color{blue}{0} \]

      22. +-inverses N/A

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

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

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

      24. lift-*.f64 N/A

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

      25. lift-*.f64 N/A

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

   6. Applied rewrites 80.8%

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

   if 1.00000000000000004e62 < (*.f64 z z)

   1. Initial program 96.1%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{2 \cdot {z}^{2} + {z}^{2}} \]
   4. Step-by-step derivation

      1. distribute-lft1-in N/A

         \[\leadsto \color{blue}{\left(2 + 1\right) \cdot {z}^{2}} \]

      2. metadata-eval N/A

         \[\leadsto \color{blue}{3} \cdot {z}^{2} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{{z}^{2} \cdot 3} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{{z}^{2} \cdot 3} \]

      5. unpow2 N/A

         \[\leadsto \color{blue}{\left(z \cdot z\right)} \cdot 3 \]

      6. lower-*.f64 88.6

         \[\leadsto \color{blue}{\left(z \cdot z\right)} \cdot 3 \]

   5. Applied rewrites 88.6%

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

Alternative 5: 83.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-89}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot z\right) \cdot 3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 5e-89) (* x y) (* (* z z) 3.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 5e-89) {
		tmp = x * y;
	} else {
		tmp = (z * z) * 3.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z * z) <= 5d-89) then
        tmp = x * y
    else
        tmp = (z * z) * 3.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 5e-89) {
		tmp = x * y;
	} else {
		tmp = (z * z) * 3.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z * z) <= 5e-89:
		tmp = x * y
	else:
		tmp = (z * z) * 3.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 5e-89)
		tmp = Float64(x * y);
	else
		tmp = Float64(Float64(z * z) * 3.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 5e-89)
		tmp = x * y;
	else
		tmp = (z * z) * 3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 5e-89], N[(x * y), $MachinePrecision], N[(N[(z * z), $MachinePrecision] * 3.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 4.99999999999999967e-89

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 87.9

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

   5. Applied rewrites 87.9%

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

   if 4.99999999999999967e-89 < (*.f64 z z)

   1. Initial program 97.1%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{2 \cdot {z}^{2} + {z}^{2}} \]
   4. Step-by-step derivation

      1. distribute-lft1-in N/A

         \[\leadsto \color{blue}{\left(2 + 1\right) \cdot {z}^{2}} \]

      2. metadata-eval N/A

         \[\leadsto \color{blue}{3} \cdot {z}^{2} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{{z}^{2} \cdot 3} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{{z}^{2} \cdot 3} \]

      5. unpow2 N/A

         \[\leadsto \color{blue}{\left(z \cdot z\right)} \cdot 3 \]

      6. lower-*.f64 78.5

         \[\leadsto \color{blue}{\left(z \cdot z\right)} \cdot 3 \]

   5. Applied rewrites 78.5%

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

Alternative 6: 83.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-89}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 5e-89) (* x y) (* z (* z 3.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 5e-89) {
		tmp = x * y;
	} else {
		tmp = z * (z * 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z * z) <= 5d-89) then
        tmp = x * y
    else
        tmp = z * (z * 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 5e-89) {
		tmp = x * y;
	} else {
		tmp = z * (z * 3.0);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z * z) <= 5e-89:
		tmp = x * y
	else:
		tmp = z * (z * 3.0)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 5e-89)
		tmp = Float64(x * y);
	else
		tmp = Float64(z * Float64(z * 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 5e-89)
		tmp = x * y;
	else
		tmp = z * (z * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 5e-89], N[(x * y), $MachinePrecision], N[(z * N[(z * 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 4.99999999999999967e-89

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 87.9

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

   5. Applied rewrites 87.9%

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

   if 4.99999999999999967e-89 < (*.f64 z z)

   1. Initial program 97.1%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{2 \cdot {z}^{2} + {z}^{2}} \]
   4. Step-by-step derivation

      1. distribute-lft1-in N/A

         \[\leadsto \color{blue}{\left(2 + 1\right) \cdot {z}^{2}} \]

      2. metadata-eval N/A

         \[\leadsto \color{blue}{3} \cdot {z}^{2} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{{z}^{2} \cdot 3} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{{z}^{2} \cdot 3} \]

      5. unpow2 N/A

         \[\leadsto \color{blue}{\left(z \cdot z\right)} \cdot 3 \]

      6. lower-*.f64 78.5

         \[\leadsto \color{blue}{\left(z \cdot z\right)} \cdot 3 \]

   5. Applied rewrites 78.5%

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

   7. Applied rewrites 78.5%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-89}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 7.5 \cdot 10^{+302}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= (* z z) 7.5e+302) (* x y) (* z z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 7.5e+302) {
		tmp = x * y;
	} else {
		tmp = z * z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z * z) <= 7.5d+302) then
        tmp = x * y
    else
        tmp = z * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 7.5e+302) {
		tmp = x * y;
	} else {
		tmp = z * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z * z) <= 7.5e+302:
		tmp = x * y
	else:
		tmp = z * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 7.5e+302)
		tmp = Float64(x * y);
	else
		tmp = Float64(z * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 7.5e+302)
		tmp = x * y;
	else
		tmp = z * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 7.5e+302], N[(x * y), $MachinePrecision], N[(z * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 7.49999999999999924e302

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 65.3

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

   5. Applied rewrites 65.3%

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

   if 7.49999999999999924e302 < (*.f64 z z)

   1. Initial program 93.4%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. count-2 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. count-2 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-+.f64 93.4

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

      11. lift-+.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 98.4

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

   4. Applied rewrites 98.4%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lift-+.f64 N/A

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

      5. distribute-lft-in N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. distribute-lft-in N/A

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

      11. flip-+ N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. +-inverses N/A

          \[\leadsto \mathsf{fma}\left(x, y, z \cdot z\right) + z \cdot \frac{\color{blue}{0}}{z - z} \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(x, y, z \cdot z\right) + z \cdot \frac{\color{blue}{0 - 0}}{z - z} \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(x, y, z \cdot z\right) + z \cdot \frac{\color{blue}{0 \cdot 0} - 0}{z - z} \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(x, y, z \cdot z\right) + z \cdot \frac{0 \cdot 0 - \color{blue}{0 \cdot 0}}{z - z} \]

      18. +-inverses N/A

          \[\leadsto \mathsf{fma}\left(x, y, z \cdot z\right) + z \cdot \frac{0 \cdot 0 - 0 \cdot 0}{\color{blue}{0}} \]

      19. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(x, y, z \cdot z\right) + z \cdot \frac{0 \cdot 0 - 0 \cdot 0}{\color{blue}{0 + 0}} \]

      20. flip-- N/A

          \[\leadsto \mathsf{fma}\left(x, y, z \cdot z\right) + z \cdot \color{blue}{\left(0 - 0\right)} \]

      21. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(x, y, z \cdot z\right) + z \cdot \color{blue}{0} \]

      22. +-inverses N/A

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

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

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

      24. lift-*.f64 N/A

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

      25. lift-*.f64 N/A

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

   6. Applied rewrites 92.4%

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

   7. Taylor expanded in z around inf

      \[\leadsto \color{blue}{{z}^{2}} \]
   8. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \color{blue}{z \cdot z} \]

      2. lower-*.f64 95.7

         \[\leadsto \color{blue}{z \cdot z} \]

   9. Applied rewrites 95.7%

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

Alternative 8: 99.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(y, x, \left(z \cdot z\right) \cdot 3\right) \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.3%

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

   1. lift-+.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. associate-+l+ N/A

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

   5. associate-+l+ N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. count-2 N/A

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

   10. distribute-lft1-in N/A

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

   11. metadata-eval N/A

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

   12. lower-*.f64 99.5

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

4. Applied rewrites 99.5%

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

5. Final simplification 99.5%

   \[\leadsto \mathsf{fma}\left(y, x, \left(z \cdot z\right) \cdot 3\right) \]
6. Add Preprocessing

Alternative 9: 98.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(3, z \cdot z, x \cdot y\right) \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.3%

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

   1. lift-+.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. associate-+l+ N/A

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

   5. associate-+l+ N/A

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

   6. +-commutative N/A

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

   7. count-2 N/A

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

   8. distribute-lft1-in N/A

      \[\leadsto \color{blue}{\left(2 + 1\right) \cdot \left(z \cdot z\right)} + x \cdot y \]

   9. metadata-eval N/A

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

   10. lower-fma.f64 98.3

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

4. Applied rewrites 98.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(3, z \cdot z, x \cdot y\right)} \]
5. Add Preprocessing

Alternative 10: 33.5% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ z \cdot z \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.3%

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

   1. lift-+.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. associate-+l+ N/A

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

   4. +-commutative N/A

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

   5. count-2 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*r* N/A

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

   8. count-2 N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-+.f64 98.4

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

   11. lift-+.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. lower-fma.f64 99.6

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

4. Applied rewrites 99.6%

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

   1. lift-fma.f64 N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. lift-+.f64 N/A

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

   5. distribute-lft-in N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. distribute-lft-in N/A

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

   11. flip-+ N/A

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

   12. lift-*.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. +-inverses N/A

       \[\leadsto \mathsf{fma}\left(x, y, z \cdot z\right) + z \cdot \frac{\color{blue}{0}}{z - z} \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(x, y, z \cdot z\right) + z \cdot \frac{\color{blue}{0 - 0}}{z - z} \]

   16. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(x, y, z \cdot z\right) + z \cdot \frac{\color{blue}{0 \cdot 0} - 0}{z - z} \]

   17. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(x, y, z \cdot z\right) + z \cdot \frac{0 \cdot 0 - \color{blue}{0 \cdot 0}}{z - z} \]

   18. +-inverses N/A

       \[\leadsto \mathsf{fma}\left(x, y, z \cdot z\right) + z \cdot \frac{0 \cdot 0 - 0 \cdot 0}{\color{blue}{0}} \]

   19. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(x, y, z \cdot z\right) + z \cdot \frac{0 \cdot 0 - 0 \cdot 0}{\color{blue}{0 + 0}} \]

   20. flip-- N/A

       \[\leadsto \mathsf{fma}\left(x, y, z \cdot z\right) + z \cdot \color{blue}{\left(0 - 0\right)} \]

   21. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(x, y, z \cdot z\right) + z \cdot \color{blue}{0} \]

   22. +-inverses N/A

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

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

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

   24. lift-*.f64 N/A

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

   25. lift-*.f64 N/A

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

6. Applied rewrites 75.6%

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

7. Taylor expanded in z around inf

   \[\leadsto \color{blue}{{z}^{2}} \]
8. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \color{blue}{z \cdot z} \]

   2. lower-*.f64 31.7

      \[\leadsto \color{blue}{z \cdot z} \]

9. Applied rewrites 31.7%

   \[\leadsto \color{blue}{z \cdot z} \]
10. Add Preprocessing

Developer Target 1: 98.4% accurate, 1.6× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return ((3.0 * z) * z) + (y * 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 * z) * z) + (y * x)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024238 
(FPCore (x y z)
  :name "Linear.Quaternion:$c/ from linear-1.19.1.3, A"
  :precision binary64

  :alt
  (! :herbie-platform default (+ (* (* 3 z) z) (* y x)))

  (+ (+ (+ (* x y) (* z z)) (* z z)) (* z z)))