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

Percentage Accurate: 98.5% → 99.1%

Time: 5.9s

Alternatives: 8

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: 98.5% 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: 99.1% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.7%

   \[\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z \]
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 98.8

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lift-*.f64 N/A

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

   14. lower-fma.f64 99.6

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 99.6

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

4. Applied rewrites 99.6%

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

5. Final simplification 99.6%

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

Alternative 2: 85.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{-55}:\\ \;\;\;\;\mathsf{fma}\left(z + z, z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z + z, z, z \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 1e-55) (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-55) {
		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-55)
		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-55], 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) < 9.99999999999999995e-56

   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

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

      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. +-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lower-fma.f64 100.0

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 93.0

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

   7. Applied rewrites 93.0%

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

   if 9.99999999999999995e-56 < (*.f64 z z)

   1. Initial program 97.5%

      \[\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z \]
   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 97.6

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lower-fma.f64 99.2

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 99.2

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(z + z, z, \color{blue}{{z}^{2}}\right) \]
   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 83.1

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

   7. Applied rewrites 83.1%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{-55}:\\ \;\;\;\;\mathsf{fma}\left(z + z, z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z + z, z, z \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{-55}:\\ \;\;\;\;\mathsf{fma}\left(z + z, z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 \cdot z\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 1e-55) (fma (+ z z) z (* x y)) (* (* 3.0 z) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 1e-55) {
		tmp = fma((z + z), z, (x * y));
	} else {
		tmp = (3.0 * z) * z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 1e-55)
		tmp = fma(Float64(z + z), z, Float64(x * y));
	else
		tmp = Float64(Float64(3.0 * z) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 1e-55], N[(N[(z + z), $MachinePrecision] * z + N[(x * y), $MachinePrecision]), $MachinePrecision], N[(N[(3.0 * z), $MachinePrecision] * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 9.99999999999999995e-56

   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

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

      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. +-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lower-fma.f64 100.0

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 93.0

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

   7. Applied rewrites 93.0%

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

   if 9.99999999999999995e-56 < (*.f64 z z)

   1. Initial program 97.5%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 21.2

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

   5. Applied rewrites 21.2%

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

   6. Taylor expanded in z around inf

      \[\leadsto \color{blue}{3 \cdot {z}^{2}} \]
   7. Step-by-step derivation

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 83.0

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

   8. Applied rewrites 83.0%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{-55}:\\ \;\;\;\;\mathsf{fma}\left(z + z, z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 \cdot z\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{-55}:\\ \;\;\;\;\mathsf{fma}\left(y, x, z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 \cdot z\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 1e-55) (fma y x (* z z)) (* (* 3.0 z) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 1e-55) {
		tmp = fma(y, x, (z * z));
	} else {
		tmp = (3.0 * z) * z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 1e-55)
		tmp = fma(y, x, Float64(z * z));
	else
		tmp = Float64(Float64(3.0 * z) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 1e-55], N[(y * x + N[(z * z), $MachinePrecision]), $MachinePrecision], N[(N[(3.0 * z), $MachinePrecision] * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 9.99999999999999995e-56

   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

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

      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. +-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lower-fma.f64 100.0

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-fma.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-+r+ N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. flip-+ N/A

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

      14. +-inverses N/A

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

      15. +-inverses N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. associate-*r/ N/A

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

      19. lift-*.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. +-inverses N/A

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

      22. +-inverses N/A

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

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

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

      24. lift-*.f64 N/A

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

      25. lift-*.f64 N/A

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

   6. Applied rewrites 74.1%

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

   7. Taylor expanded in z around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 92.8

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

   9. Applied rewrites 92.8%

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

   if 9.99999999999999995e-56 < (*.f64 z z)

   1. Initial program 97.5%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 21.2

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

   5. Applied rewrites 21.2%

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

   6. Taylor expanded in z around inf

      \[\leadsto \color{blue}{3 \cdot {z}^{2}} \]
   7. Step-by-step derivation

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 83.0

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

   8. Applied rewrites 83.0%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{-55}:\\ \;\;\;\;\mathsf{fma}\left(y, x, z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 \cdot z\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{-55}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(3 \cdot z\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 1e-55) (* x y) (* (* 3.0 z) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 1e-55) {
		tmp = x * y;
	} else {
		tmp = (3.0 * 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) <= 1d-55) then
        tmp = x * y
    else
        tmp = (3.0d0 * z) * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 1e-55) {
		tmp = x * y;
	} else {
		tmp = (3.0 * z) * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z * z) <= 1e-55:
		tmp = x * y
	else:
		tmp = (3.0 * z) * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 1e-55)
		tmp = Float64(x * y);
	else
		tmp = Float64(Float64(3.0 * z) * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 1e-55)
		tmp = x * y;
	else
		tmp = (3.0 * z) * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 1e-55], N[(x * y), $MachinePrecision], N[(N[(3.0 * z), $MachinePrecision] * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 9.99999999999999995e-56

   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 z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 91.8

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

   5. Applied rewrites 91.8%

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

   if 9.99999999999999995e-56 < (*.f64 z z)

   1. Initial program 97.5%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 21.2

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

   5. Applied rewrites 21.2%

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

   6. Taylor expanded in z around inf

      \[\leadsto \color{blue}{3 \cdot {z}^{2}} \]
   7. Step-by-step derivation

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 83.0

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

   8. Applied rewrites 83.0%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{-55}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(3 \cdot z\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+300}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= (* z z) 1e+300) (* x y) (* z z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 1e+300) {
		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) <= 1d+300) 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) <= 1e+300) {
		tmp = x * y;
	} else {
		tmp = z * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z * z) <= 1e+300:
		tmp = x * y
	else:
		tmp = z * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 1e+300)
		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) <= 1e+300)
		tmp = x * y;
	else
		tmp = z * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 1e+300], N[(x * y), $MachinePrecision], N[(z * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.0000000000000001e300

   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 z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 69.7

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

   5. Applied rewrites 69.7%

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

   if 1.0000000000000001e300 < (*.f64 z z)

   1. Initial program 95.1%

      \[\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z \]
   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 95.1

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lower-fma.f64 98.4

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 98.4

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

   4. Applied rewrites 98.4%

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

      1. lift-fma.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-+r+ N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. flip-+ N/A

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

      14. +-inverses N/A

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

      15. +-inverses N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. associate-*r/ N/A

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

      19. lift-*.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. +-inverses N/A

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

      22. +-inverses N/A

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

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

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

      24. lift-*.f64 N/A

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

      25. lift-*.f64 N/A

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

   6. Applied rewrites 95.4%

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

   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 95.4

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

   9. Applied rewrites 95.4%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+300}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.0% accurate, 1.8× speedup

Math

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

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(Float64(3.0 * z), z, Float64(x * y))
end

Wolfram

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

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in z around 0

   \[\leadsto \color{blue}{3 \cdot {z}^{2} + x \cdot y} \]
4. Step-by-step derivation

   1. unpow2 N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 99.5

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

5. Applied rewrites 99.5%

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

6. Final simplification 99.5%

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

Alternative 8: 33.7% 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 98.7%

   \[\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z \]
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 98.8

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lift-*.f64 N/A

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

   14. lower-fma.f64 99.6

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 99.6

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

4. Applied rewrites 99.6%

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

   1. lift-fma.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-+r+ N/A

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

   5. +-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. lower-fma.f64 N/A

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

   8. +-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. lower-fma.f64 N/A

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

   11. *-commutative N/A

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

   12. lift-+.f64 N/A

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

   13. flip-+ N/A

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

   14. +-inverses N/A

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

   15. +-inverses N/A

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

   16. lift-*.f64 N/A

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

   17. lift-*.f64 N/A

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

   18. associate-*r/ N/A

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

   19. lift-*.f64 N/A

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

   20. lift-*.f64 N/A

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

   21. +-inverses N/A

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

   22. +-inverses N/A

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

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

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

   24. lift-*.f64 N/A

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

   25. lift-*.f64 N/A

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

6. Applied rewrites 69.5%

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

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 32.8

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

9. Applied rewrites 32.8%

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

Developer Target 1: 98.5% 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 2024240 
(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)))