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

Percentage Accurate: 98.4% → 99.5%

Time: 6.6s

Alternatives: 5

Speedup: 1.8×

• 33.9% 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, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.6%

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

   1. lift-+.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. associate-+l+ N/A

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

   5. associate-+l+ N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. count-2 N/A

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

   10. distribute-lft1-in N/A

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

   11. metadata-eval N/A

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

   12. lower-*.f64 99.5

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

4. Applied rewrites 99.5%

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

Alternative 2: 83.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-129}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, z + z, z \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 5e-129) (* y x) (fma z (+ z z) (* z z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 5e-129) {
		tmp = y * x;
	} else {
		tmp = fma(z, (z + z), (z * z));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 5e-129)
		tmp = Float64(y * x);
	else
		tmp = fma(z, Float64(z + z), Float64(z * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 5e-129], N[(y * x), $MachinePrecision], N[(z * N[(z + z), $MachinePrecision] + N[(z * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 5.00000000000000027e-129

   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. lift-+.f64 N/A

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

      4. associate-+l+ N/A

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

      5. associate-+l+ N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. count-2 N/A

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

      10. distribute-lft1-in N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, 3 \cdot \left(z \cdot z\right)\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. flip-+ N/A

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

      4. lower-/.f64 N/A

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

   6. Applied rewrites 48.4%

      \[\leadsto \color{blue}{\frac{{\left(x \cdot y\right)}^{2} - 9 \cdot {z}^{4}}{\mathsf{fma}\left(x, y, -3 \cdot \left(z \cdot z\right)\right)}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{-9 \cdot {z}^{4}}}{\mathsf{fma}\left(x, y, -3 \cdot \left(z \cdot z\right)\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{{z}^{4} \cdot -9}}{\mathsf{fma}\left(x, y, -3 \cdot \left(z \cdot z\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{{z}^{4} \cdot -9}}{\mathsf{fma}\left(x, y, -3 \cdot \left(z \cdot z\right)\right)} \]

      3. lower-pow.f64 5.8

         \[\leadsto \frac{\color{blue}{{z}^{4}} \cdot -9}{\mathsf{fma}\left(x, y, -3 \cdot \left(z \cdot z\right)\right)} \]

   9. Applied rewrites 5.8%

      \[\leadsto \frac{\color{blue}{{z}^{4} \cdot -9}}{\mathsf{fma}\left(x, y, -3 \cdot \left(z \cdot z\right)\right)} \]

   10. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 92.8

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

   12. Applied rewrites 92.8%

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

   if 5.00000000000000027e-129 < (*.f64 z z)

   1. Initial program 97.7%

      \[\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z \]
   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. count-2 N/A

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

      11. lower-*.f64 97.9

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lower-fma.f64 98.5

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 98.5

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

   4. Applied rewrites 98.5%

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

   5. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 83.1

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

   7. Applied rewrites 83.1%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. count-2 N/A

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

      5. distribute-lft-out N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 83.1

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

   9. Applied rewrites 83.1%

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

Alternative 3: 82.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-129}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(3 \cdot z\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 5e-129) (* y x) (* (* 3.0 z) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 5e-129) {
		tmp = y * x;
	} 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) <= 5d-129) then
        tmp = y * x
    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) <= 5e-129) {
		tmp = y * x;
	} else {
		tmp = (3.0 * z) * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z * z) <= 5e-129:
		tmp = y * x
	else:
		tmp = (3.0 * z) * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 5e-129)
		tmp = Float64(y * x);
	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) <= 5e-129)
		tmp = y * x;
	else
		tmp = (3.0 * z) * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 5e-129], N[(y * x), $MachinePrecision], N[(N[(3.0 * z), $MachinePrecision] * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 5.00000000000000027e-129

   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. lift-+.f64 N/A

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

      4. associate-+l+ N/A

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

      5. associate-+l+ N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. count-2 N/A

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

      10. distribute-lft1-in N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, 3 \cdot \left(z \cdot z\right)\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. flip-+ N/A

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

      4. lower-/.f64 N/A

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

   6. Applied rewrites 48.4%

      \[\leadsto \color{blue}{\frac{{\left(x \cdot y\right)}^{2} - 9 \cdot {z}^{4}}{\mathsf{fma}\left(x, y, -3 \cdot \left(z \cdot z\right)\right)}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{-9 \cdot {z}^{4}}}{\mathsf{fma}\left(x, y, -3 \cdot \left(z \cdot z\right)\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{{z}^{4} \cdot -9}}{\mathsf{fma}\left(x, y, -3 \cdot \left(z \cdot z\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{{z}^{4} \cdot -9}}{\mathsf{fma}\left(x, y, -3 \cdot \left(z \cdot z\right)\right)} \]

      3. lower-pow.f64 5.8

         \[\leadsto \frac{\color{blue}{{z}^{4}} \cdot -9}{\mathsf{fma}\left(x, y, -3 \cdot \left(z \cdot z\right)\right)} \]

   9. Applied rewrites 5.8%

      \[\leadsto \frac{\color{blue}{{z}^{4} \cdot -9}}{\mathsf{fma}\left(x, y, -3 \cdot \left(z \cdot z\right)\right)} \]

   10. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 92.8

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

   12. Applied rewrites 92.8%

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

   if 5.00000000000000027e-129 < (*.f64 z z)

   1. Initial program 97.7%

      \[\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z \]
   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. *-rgt-identity N/A

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

      2. *-inverses N/A

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

      3. associate-/l* N/A

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

      4. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot {z}^{2} + {z}^{2}}{x} \cdot x} \]

      5. distribute-lft1-in N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

         \[\leadsto \color{blue}{\left(3 \cdot \frac{{z}^{2}}{x}\right)} \cdot x \]

      8. associate-*l* N/A

         \[\leadsto \color{blue}{3 \cdot \left(\frac{{z}^{2}}{x} \cdot x\right)} \]

      9. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{{z}^{2}}{x} \cdot x\right) \cdot 3} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\frac{{z}^{2}}{x} \cdot x\right) \cdot 3} \]

      11. associate-*l/ N/A

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

      12. associate-/l* N/A

          \[\leadsto \color{blue}{\left({z}^{2} \cdot \frac{x}{x}\right)} \cdot 3 \]

      13. *-inverses N/A

          \[\leadsto \left({z}^{2} \cdot \color{blue}{1}\right) \cdot 3 \]

      14. *-rgt-identity N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 83.0

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

   5. Applied rewrites 83.0%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{2 \cdot {z}^{2} + {z}^{2}} \]
   7. 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. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 83.0

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

   8. Applied rewrites 83.0%

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

Alternative 4: 98.8% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.6%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. associate-+r+ N/A

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

   3. distribute-lft1-in N/A

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

   4. metadata-eval N/A

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

   5. unpow2 N/A

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

   6. associate-*r* N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 99.1

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

5. Applied rewrites 99.1%

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

Alternative 5: 51.9% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.6%

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

   1. lift-+.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. associate-+l+ N/A

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

   5. associate-+l+ N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. count-2 N/A

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

   10. distribute-lft1-in N/A

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

   11. metadata-eval N/A

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

   12. lower-*.f64 99.5

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

4. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, 3 \cdot \left(z \cdot z\right)\right)} \]
5. Step-by-step derivation

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. flip-+ N/A

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

   4. lower-/.f64 N/A

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

6. Applied rewrites 42.7%

   \[\leadsto \color{blue}{\frac{{\left(x \cdot y\right)}^{2} - 9 \cdot {z}^{4}}{\mathsf{fma}\left(x, y, -3 \cdot \left(z \cdot z\right)\right)}} \]

7. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{-9 \cdot {z}^{4}}}{\mathsf{fma}\left(x, y, -3 \cdot \left(z \cdot z\right)\right)} \]
8. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\color{blue}{{z}^{4} \cdot -9}}{\mathsf{fma}\left(x, y, -3 \cdot \left(z \cdot z\right)\right)} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{{z}^{4} \cdot -9}}{\mathsf{fma}\left(x, y, -3 \cdot \left(z \cdot z\right)\right)} \]

   3. lower-pow.f64 20.8

      \[\leadsto \frac{\color{blue}{{z}^{4}} \cdot -9}{\mathsf{fma}\left(x, y, -3 \cdot \left(z \cdot z\right)\right)} \]

9. Applied rewrites 20.8%

   \[\leadsto \frac{\color{blue}{{z}^{4} \cdot -9}}{\mathsf{fma}\left(x, y, -3 \cdot \left(z \cdot z\right)\right)} \]

10. Taylor expanded in x around inf

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

    1. *-commutative N/A

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

    2. lower-*.f64 51.0

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

12. Applied rewrites 51.0%

    \[\leadsto \color{blue}{y \cdot x} \]
13. 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 2024318 
(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)))