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

Percentage Accurate: 98.4% → 99.5%

Time: 5.3s

Alternatives: 9

Speedup: 1.8×

• 34.7% 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 96.0%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   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. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-+l+ N/A

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

   7. +-commutative N/A

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

   8. count-2 N/A

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

   9. lift-*.f64 N/A

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

   10. associate-*r* N/A

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

   11. count-2 N/A

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

   12. lower-fma.f64 N/A

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

   13. lower-+.f64 96.1

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

   14. lift-+.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. lower-fma.f64 99.2

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

4. Applied rewrites 99.2%

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

Alternative 2: 99.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+306}:\\ \;\;\;\;\mathsf{fma}\left(3, z \cdot z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 1e+306) (fma 3.0 (* z z) (* x y)) (* z z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 1e+306) {
		tmp = fma(3.0, (z * z), (x * y));
	} else {
		tmp = z * z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 1e+306)
		tmp = fma(3.0, Float64(z * z), Float64(x * y));
	else
		tmp = Float64(z * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 1e+306], N[(3.0 * N[(z * z), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision], N[(z * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.00000000000000002e306

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      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. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. associate-+l+ N/A

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

      8. associate-+l+ N/A

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

      9. +-commutative N/A

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

      10. count-2 N/A

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

      11. distribute-lft1-in N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 99.8

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

   4. Applied rewrites 99.8%

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

   if 1.00000000000000002e306 < (*.f64 z z)

   1. Initial program 87.3%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 87.3

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

   5. Applied rewrites 87.3%

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

   6. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 97.5

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

   8. Applied rewrites 97.5%

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

Alternative 3: 84.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-85}:\\ \;\;\;\;\mathsf{fma}\left(z + z, z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z + z, z, z \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 5e-85) (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) <= 5e-85) {
		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) <= 5e-85)
		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], 5e-85], 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) < 5.0000000000000002e-85

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      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. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-+l+ N/A

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

      7. +-commutative N/A

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

      8. count-2 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. count-2 N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-+.f64 100.0

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

      14. lift-+.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. 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 92.3

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

   7. Applied rewrites 92.3%

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

   if 5.0000000000000002e-85 < (*.f64 z z)

   1. Initial program 93.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      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. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-+l+ N/A

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

      7. +-commutative N/A

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

      8. count-2 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. count-2 N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-+.f64 93.8

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

      14. lift-+.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lower-fma.f64 98.7

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

   4. Applied rewrites 98.7%

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

   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 83.9

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

   7. Applied rewrites 83.9%

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

Alternative 4: 84.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-85}:\\ \;\;\;\;\mathsf{fma}\left(z + z, z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 5e-85) (fma (+ z z) z (* x y)) (* z (* z 3.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 5e-85) {
		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) <= 5e-85)
		tmp = fma(Float64(z + z), z, Float64(x * y));
	else
		tmp = Float64(z * Float64(z * 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 5e-85], N[(N[(z + z), $MachinePrecision] * z + N[(x * y), $MachinePrecision]), $MachinePrecision], N[(z * N[(z * 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 5.0000000000000002e-85

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      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. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-+l+ N/A

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

      7. +-commutative N/A

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

      8. count-2 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. count-2 N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-+.f64 100.0

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

      14. lift-+.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. 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 92.3

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

   7. Applied rewrites 92.3%

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

   if 5.0000000000000002e-85 < (*.f64 z z)

   1. Initial program 93.6%

      \[\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z \]
   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 83.8

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

   5. Applied rewrites 83.8%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 83.8

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

   7. Applied rewrites 83.8%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-85}:\\ \;\;\;\;\mathsf{fma}\left(z + z, z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-85}:\\ \;\;\;\;\mathsf{fma}\left(z, z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 5e-85) (fma z z (* x y)) (* z (* z 3.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 5e-85) {
		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) <= 5e-85)
		tmp = fma(z, z, Float64(x * y));
	else
		tmp = Float64(z * Float64(z * 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 5e-85], N[(z * z + N[(x * y), $MachinePrecision]), $MachinePrecision], N[(z * N[(z * 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 5.0000000000000002e-85

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 92.1

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

   5. Applied rewrites 92.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lower-fma.f64 92.1

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

   7. Applied rewrites 92.1%

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

   if 5.0000000000000002e-85 < (*.f64 z z)

   1. Initial program 93.6%

      \[\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z \]
   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 83.8

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

   5. Applied rewrites 83.8%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 83.8

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

   7. Applied rewrites 83.8%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-85}:\\ \;\;\;\;\mathsf{fma}\left(z, z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-85}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 5e-85) (* x y) (* z (* z 3.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 5e-85) {
		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-85) 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-85) {
		tmp = x * y;
	} else {
		tmp = z * (z * 3.0);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z * z) <= 5e-85:
		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-85)
		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-85)
		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-85], N[(x * y), $MachinePrecision], N[(z * N[(z * 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 5.0000000000000002e-85

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 91.1

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

   5. Applied rewrites 91.1%

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

   if 5.0000000000000002e-85 < (*.f64 z z)

   1. Initial program 93.6%

      \[\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z \]
   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 83.8

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

   5. Applied rewrites 83.8%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 83.8

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

   7. Applied rewrites 83.8%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-85}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 4 \cdot 10^{+248}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= (* z z) 4e+248) (* x y) (* z z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 4e+248) {
		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) <= 4d+248) 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) <= 4e+248) {
		tmp = x * y;
	} else {
		tmp = z * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z * z) <= 4e+248:
		tmp = x * y
	else:
		tmp = z * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 4e+248)
		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) <= 4e+248)
		tmp = x * y;
	else
		tmp = z * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 4e+248], N[(x * y), $MachinePrecision], N[(z * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 4.00000000000000018e248

   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 66.4

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

   5. Applied rewrites 66.4%

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

   if 4.00000000000000018e248 < (*.f64 z z)

   1. Initial program 88.9%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 79.7

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

   5. Applied rewrites 79.7%

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

   6. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 87.5

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

   8. Applied rewrites 87.5%

      \[\leadsto \color{blue}{z \cdot z} \]
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 96.0%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   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. lift-*.f64 N/A

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

   6. associate-+l+ N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. count-2 N/A

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

   11. distribute-lft1-in N/A

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

   12. metadata-eval N/A

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

   13. lower-*.f64 99.1

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

4. Applied rewrites 99.1%

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

5. Final simplification 99.1%

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

Alternative 9: 52.5% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ x \cdot y \end{array} \]

FPCore

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

C

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

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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.0%

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

3. Taylor expanded in x around inf

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

   1. lower-*.f64 46.5

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

5. Applied rewrites 46.5%

   \[\leadsto \color{blue}{x \cdot y} \]
6. 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 2024214 
(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)))