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

Percentage Accurate: 98.3% → 98.7%

Time: 6.1s

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: 98.3% 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: 98.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(z, \mathsf{fma}\left(2, z, z\right), x \cdot y\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fma z (fma 2.0 z z) (* x y)))

C

double code(double x, double y, double z) {
	return fma(z, fma(2.0, z, z), (x * y));
}

Julia

function code(x, y, z)
	return fma(z, fma(2.0, z, z), Float64(x * y))
end

Wolfram

code[x_, y_, z_] := N[(z * N[(2.0 * z + z), $MachinePrecision] + 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. 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. lift-+.f64 N/A

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

   6. +-commutative N/A

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

   7. associate-+r+ N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. distribute-lft-out N/A

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

   11. lift-*.f64 N/A

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

   12. distribute-lft-out N/A

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

   13. lower-fma.f64 N/A

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

   14. count-2 N/A

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

   15. lower-fma.f64 98.7

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

   16. lift-*.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 98.7

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

4. Applied rewrites 98.7%

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

5. Final simplification 98.7%

   \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(2, z, z\right), x \cdot y\right) \]
6. Add Preprocessing

Alternative 2: 83.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{-128}:\\ \;\;\;\;\mathsf{fma}\left(y, x, z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z + z, z, z \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 1e-128) (fma y x (* z z)) (fma (+ z z) z (* z z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 1e-128) {
		tmp = fma(y, x, (z * z));
	} else {
		tmp = fma((z + z), z, (z * z));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 1e-128)
		tmp = fma(y, x, Float64(z * z));
	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-128], N[(y * x + N[(z * z), $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) < 1.00000000000000005e-128

   1. Initial program 100.0%

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

   4. Applied rewrites 94.3%

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

   if 1.00000000000000005e-128 < (*.f64 z z)

   1. Initial program 96.3%

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

   3. Taylor expanded in x around 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 79.5

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

   5. Applied rewrites 79.5%

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

   7. Applied rewrites 79.6%

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

   9. Applied rewrites 79.6%

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

Alternative 3: 83.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{-128}:\\ \;\;\;\;\mathsf{fma}\left(y, x, z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot z\right) \cdot 3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 1e-128) (fma y x (* z z)) (* (* z z) 3.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 1e-128) {
		tmp = fma(y, x, (z * z));
	} else {
		tmp = (z * z) * 3.0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 1e-128)
		tmp = fma(y, x, Float64(z * z));
	else
		tmp = Float64(Float64(z * z) * 3.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 1e-128], N[(y * x + N[(z * z), $MachinePrecision]), $MachinePrecision], N[(N[(z * z), $MachinePrecision] * 3.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.00000000000000005e-128

   1. Initial program 100.0%

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

   4. Applied rewrites 94.3%

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

   if 1.00000000000000005e-128 < (*.f64 z z)

   1. Initial program 96.3%

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

   3. Taylor expanded in x around 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 79.5

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

   5. Applied rewrites 79.5%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{-128}:\\ \;\;\;\;\mathsf{fma}\left(y, x, z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot z\right) \cdot 3\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{-128}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot z\right) \cdot 3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 1e-128) (* x y) (* (* z z) 3.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 1e-128) {
		tmp = x * y;
	} else {
		tmp = (z * z) * 3.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z * z) <= 1d-128) then
        tmp = x * y
    else
        tmp = (z * z) * 3.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 1e-128) {
		tmp = x * y;
	} else {
		tmp = (z * z) * 3.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z * z) <= 1e-128:
		tmp = x * y
	else:
		tmp = (z * z) * 3.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 1e-128)
		tmp = Float64(x * y);
	else
		tmp = Float64(Float64(z * z) * 3.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 1e-128)
		tmp = x * y;
	else
		tmp = (z * z) * 3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 1e-128], N[(x * y), $MachinePrecision], N[(N[(z * z), $MachinePrecision] * 3.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.00000000000000005e-128

   1. Initial program 100.0%

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

   4. Applied rewrites 94.3%

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

   5. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 12.8

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

   7. Applied rewrites 12.8%

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

   8. Taylor expanded in x around inf

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

      1. lower-*.f64 93.5

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

   10. Applied rewrites 93.5%

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

   if 1.00000000000000005e-128 < (*.f64 z z)

   1. Initial program 96.3%

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

   3. Taylor expanded in x around 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 79.5

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

   5. Applied rewrites 79.5%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{-128}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot z\right) \cdot 3\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 7.6 \cdot 10^{+290}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= (* z z) 7.6e+290) (* x y) (* z z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 7.6e+290) {
		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) <= 7.6d+290) 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) <= 7.6e+290) {
		tmp = x * y;
	} else {
		tmp = z * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z * z) <= 7.6e+290:
		tmp = x * y
	else:
		tmp = z * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 7.6e+290)
		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) <= 7.6e+290)
		tmp = x * y;
	else
		tmp = z * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 7.6e+290], N[(x * y), $MachinePrecision], N[(z * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 7.6000000000000001e290

   1. Initial program 99.9%

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

   4. Applied rewrites 72.5%

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

   5. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 12.9

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

   7. Applied rewrites 12.9%

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

   8. Taylor expanded in x around inf

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

      1. lower-*.f64 67.9

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

   10. Applied rewrites 67.9%

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

   if 7.6000000000000001e290 < (*.f64 z z)

   1. Initial program 91.6%

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

   4. Applied rewrites 93.9%

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

   5. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 93.9

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

   7. Applied rewrites 93.9%

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

Alternative 6: 98.3% accurate, 1.8× speedup

Math

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

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

Wolfram

code[x_, y_, z_] := N[(3.0 * N[(z * z), $MachinePrecision] + 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. 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. lift-+.f64 N/A

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

   6. +-commutative N/A

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

   7. associate-+r+ N/A

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

   8. count-2 N/A

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

   9. distribute-lft1-in N/A

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

   10. metadata-eval N/A

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

   11. lower-fma.f64 98.0

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 98.0

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

4. Applied rewrites 98.0%

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

5. Final simplification 98.0%

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

Alternative 7: 52.1% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x * y
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.9%

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

4. Applied rewrites 77.5%

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

5. Taylor expanded in x around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 31.9

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

7. Applied rewrites 31.9%

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

8. Taylor expanded in x around inf

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

   1. lower-*.f64 54.7

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

10. Applied rewrites 54.7%

    \[\leadsto \color{blue}{x \cdot y} \]
11. 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 2024298 
(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)))