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

Percentage Accurate: 98.5% → 99.4%

Time: 6.6s

Alternatives: 7

Speedup: 1.8×

• 34.2% 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: 99.4% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 97.9%

   \[\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z \]
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. lift-+.f64 N/A

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

   4. associate-+l+ N/A

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

   5. associate-+l+ N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. count-2 N/A

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

   10. distribute-lft1-in N/A

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

   11. metadata-eval N/A

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

   12. lower-*.f64 99.5

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

4. Applied rewrites 99.5%

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

5. Final simplification 99.5%

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

Alternative 2: 86.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 1.7 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(z, z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, z + z, z \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 1.7e-7) (fma z z (* x y)) (fma z (+ z z) (* z z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 1.7e-7) {
		tmp = fma(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) <= 1.7e-7)
		tmp = fma(z, z, Float64(x * y));
	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], 1.7e-7], N[(z * z + N[(x * y), $MachinePrecision]), $MachinePrecision], N[(z * N[(z + z), $MachinePrecision] + N[(z * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.69999999999999987e-7

   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. flip-+ N/A

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

      3. div-inv N/A

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

      4. difference-of-squares N/A

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

      5. lift-+.f64 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

   4. Applied rewrites 78.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(3, z \cdot z, y \cdot x\right) \cdot \left(\mathsf{fma}\left(z, z, \mathsf{fma}\left(y, x, 0\right)\right) \cdot {\left(\mathsf{fma}\left(z, z, \mathsf{fma}\left(y, x, 0\right)\right)\right)}^{-1}\right)} \]

   6. Applied rewrites 86.6%

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

   if 1.69999999999999987e-7 < (*.f64 z z)

   1. Initial program 96.1%

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

   3. Taylor expanded in x around 0

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

      1. distribute-lft1-in N/A

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

      2. metadata-eval N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 90.3

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

   5. Applied rewrites 90.3%

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

   7. Applied rewrites 90.4%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 1.7 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(z, z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, z + z, z \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 1.7 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(z, z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot 3\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 1.7e-7) (fma z z (* x y)) (* (* z 3.0) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 1.7e-7) {
		tmp = fma(z, z, (x * y));
	} else {
		tmp = (z * 3.0) * z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 1.7e-7)
		tmp = fma(z, z, Float64(x * y));
	else
		tmp = Float64(Float64(z * 3.0) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 1.7e-7], N[(z * z + N[(x * y), $MachinePrecision]), $MachinePrecision], N[(N[(z * 3.0), $MachinePrecision] * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.69999999999999987e-7

   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. flip-+ N/A

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

      3. div-inv N/A

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

      4. difference-of-squares N/A

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

      5. lift-+.f64 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

   4. Applied rewrites 78.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(3, z \cdot z, y \cdot x\right) \cdot \left(\mathsf{fma}\left(z, z, \mathsf{fma}\left(y, x, 0\right)\right) \cdot {\left(\mathsf{fma}\left(z, z, \mathsf{fma}\left(y, x, 0\right)\right)\right)}^{-1}\right)} \]

   6. Applied rewrites 86.6%

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

   if 1.69999999999999987e-7 < (*.f64 z z)

   1. Initial program 96.1%

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

   3. Taylor expanded in x around 0

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

      1. distribute-lft1-in N/A

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

      2. metadata-eval N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 90.3

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

   5. Applied rewrites 90.3%

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

   7. Applied rewrites 90.4%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 1.7 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(z, z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot 3\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.0% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 97.9%

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

3. Taylor expanded in x around 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 98.3

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

5. Applied rewrites 98.3%

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

6. Final simplification 98.3%

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

Alternative 5: 53.6% accurate, 2.7× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return (z * 3.0) * 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 * 3.0d0) * z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.9%

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

3. Taylor expanded in x around 0

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

   1. distribute-lft1-in N/A

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

   2. metadata-eval N/A

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

   3. lower-*.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 59.5

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

5. Applied rewrites 59.5%

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

7. Applied rewrites 59.6%

   \[\leadsto \left(z \cdot 3\right) \cdot \color{blue}{z} \]
8. Add Preprocessing

Alternative 6: 53.6% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

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) * 3.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.9%

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

3. Taylor expanded in x around 0

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

   1. distribute-lft1-in N/A

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

   2. metadata-eval N/A

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

   3. lower-*.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 59.5

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

5. Applied rewrites 59.5%

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

6. Final simplification 59.5%

   \[\leadsto \left(z \cdot z\right) \cdot 3 \]
7. Add Preprocessing

Alternative 7: 33.3% 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 97.9%

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

3. Taylor expanded in x around 0

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

   1. distribute-lft1-in N/A

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

   2. metadata-eval N/A

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

   3. lower-*.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 59.5

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

5. Applied rewrites 59.5%

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

7. Applied rewrites 59.6%

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

9. Applied rewrites 38.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(z, z, 0\right)} \]
10. Step-by-step derivation

11. Applied rewrites 38.2%

    \[\leadsto z \cdot \color{blue}{z} \]
12. 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 2024294 
(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)))