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

Percentage Accurate: 98.3% → 98.9%

Time: 9.8s

Alternatives: 8

Speedup: 1.8×

• 34.8% 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: 98.3% 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: 98.9% 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 99.1%

   \[\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z \]
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 99.2

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

   11. lift-+.f64 N/A

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

   12. +-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. Final simplification 99.2%

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

Alternative 2: 86.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+48}:\\ \;\;\;\;\mathsf{fma}\left(z + z, z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z + z, z, z \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 5e+48) (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+48) {
		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+48)
		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+48], 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) < 4.99999999999999973e48

   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. +-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 85.2

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

   7. Applied rewrites 85.2%

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

   if 4.99999999999999973e48 < (*.f64 z z)

   1. Initial program 96.4%

      \[\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z \]
   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 96.5

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

      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 97.6

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

      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 97.6

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

   4. Applied rewrites 97.6%

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

   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 87.1

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

   7. Applied rewrites 87.1%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+48}:\\ \;\;\;\;\mathsf{fma}\left(z + z, z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z + z, z, z \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Developer Target 1: 98.3% 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 2024230 
(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)))