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

Percentage Accurate: 98.0% → 99.4%

Time: 9.3s

Alternatives: 5

Speedup: 1.7×

• 34.6% 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: 98.0% 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, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_] := N[(z * z + N[(x * y + N[(2.0 * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

   1. +-commutative 99.5%

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

   2. fma-def 99.6%

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

   3. associate-+l+ 99.6%

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

   4. fma-def 99.6%

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

   5. count-2 99.6%

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

3. Simplified 99.6%

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

4. Final simplification 99.6%

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

Alternative 2: 99.3% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. associate-+l+ 99.5%

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

   2. associate-+l+ 99.5%

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

   3. fma-def 99.5%

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

   4. count-2 99.5%

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

   5. distribute-lft1-in 99.5%

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

   6. metadata-eval 99.5%

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

   7. *-commutative 99.5%

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

   8. associate-*l* 99.5%

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

   9. metadata-eval 99.5%

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

   10. metadata-eval 99.5%

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

3. Simplified 99.5%

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

4. Final simplification 99.5%

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

Alternative 3: 98.1% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. associate-+l+ 99.5%

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

   2. associate-+l+ 99.5%

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

   3. fma-def 99.5%

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

   4. count-2 99.5%

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

   5. distribute-lft1-in 99.5%

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

   6. metadata-eval 99.5%

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

   7. *-commutative 99.5%

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

   8. associate-*l* 99.5%

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

   9. metadata-eval 99.5%

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

   10. metadata-eval 99.5%

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

3. Simplified 99.5%

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

   1. fma-udef 99.5%

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

   2. flip-+ 36.8%

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

   3. pow2 36.8%

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

   4. associate-*r* 36.8%

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

   5. associate-*r* 36.7%

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

   6. swap-sqr 36.6%

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

   7. pow2 36.6%

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

   8. pow2 36.6%

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

   9. pow-sqr 36.7%

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

   10. metadata-eval 36.7%

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

   11. metadata-eval 36.7%

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

   12. associate-*r* 36.7%

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

   13. pow2 36.7%

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

5. Applied egg-rr 36.7%

   \[\leadsto \color{blue}{\frac{{\left(x \cdot y\right)}^{2} - {z}^{4} \cdot 9}{x \cdot y - {z}^{2} \cdot 3}} \]
6. Step-by-step derivation

   1. unpow2 36.7%

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

   2. add-sqr-sqrt 36.7%

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

   3. add-sqr-sqrt 36.6%

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

   4. sqrt-unprod 36.1%

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

   5. swap-sqr 36.1%

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

   6. pow-sqr 36.1%

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

   7. metadata-eval 36.1%

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

   8. metadata-eval 36.1%

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

   9. flip-+ 88.9%

      \[\leadsto \color{blue}{x \cdot y + \sqrt{{z}^{4} \cdot 9}} \]

   10. +-commutative 88.9%

       \[\leadsto \color{blue}{\sqrt{{z}^{4} \cdot 9} + x \cdot y} \]

   11. metadata-eval 88.9%

       \[\leadsto \sqrt{{z}^{\color{blue}{\left(2 \cdot 2\right)}} \cdot 9} + x \cdot y \]

   12. pow-sqr 88.9%

       \[\leadsto \sqrt{\color{blue}{\left({z}^{2} \cdot {z}^{2}\right)} \cdot 9} + x \cdot y \]

   13. metadata-eval 88.9%

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

   14. swap-sqr 88.9%

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

   15. sqrt-unprod 99.4%

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

7. Applied egg-rr 99.7%

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

   1. fma-udef 99.3%

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

   2. unpow2 99.3%

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

9. Simplified 99.3%

   \[\leadsto \color{blue}{{\left(z \cdot \sqrt{3}\right)}^{2} + x \cdot y} \]
10. Step-by-step derivation

    1. unpow2 54.0%

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

    2. *-commutative 54.0%

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

    3. *-commutative 54.0%

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

    4. swap-sqr 53.9%

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

    5. rem-square-sqrt 54.1%

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

    6. associate-*r* 54.1%

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

11. Applied egg-rr 99.5%

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

12. Final simplification 99.5%

    \[\leadsto z \cdot \left(z \cdot 3\right) + x \cdot y \]

Alternative 4: 69.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 510000000000:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 510000000000.0) (* x y) (* z (* z 3.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 510000000000.0) {
		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 <= 510000000000.0d0) 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 <= 510000000000.0) {
		tmp = x * y;
	} else {
		tmp = z * (z * 3.0);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 510000000000.0:
		tmp = x * y
	else:
		tmp = z * (z * 3.0)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 510000000000.0)
		tmp = Float64(x * y);
	else
		tmp = Float64(z * Float64(z * 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 510000000000.0)
		tmp = x * y;
	else
		tmp = z * (z * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 510000000000.0], N[(x * y), $MachinePrecision], N[(z * N[(z * 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 5.1e11

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf 63.8%

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

   if 5.1e11 < z

   1. Initial program 98.2%

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

      1. associate-+l+ 98.2%

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

      2. associate-+l+ 98.2%

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

      3. fma-def 98.2%

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

      4. count-2 98.2%

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

      5. distribute-lft1-in 98.2%

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

      6. metadata-eval 98.2%

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

      7. *-commutative 98.2%

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

      8. associate-*l* 98.2%

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

      9. metadata-eval 98.2%

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

      10. metadata-eval 98.2%

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

   3. Simplified 98.2%

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

      1. fma-udef 98.2%

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

      2. flip-+ 24.0%

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

      3. pow2 24.0%

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

      4. associate-*r* 23.9%

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

      5. associate-*r* 23.9%

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

      6. swap-sqr 23.8%

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

      7. pow2 23.8%

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

      8. pow2 23.8%

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

      9. pow-sqr 24.0%

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

      10. metadata-eval 24.0%

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

      11. metadata-eval 24.0%

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

      12. associate-*r* 23.9%

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

      13. pow2 23.9%

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

   5. Applied egg-rr 23.9%

      \[\leadsto \color{blue}{\frac{{\left(x \cdot y\right)}^{2} - {z}^{4} \cdot 9}{x \cdot y - {z}^{2} \cdot 3}} \]
   6. Step-by-step derivation

      1. unpow2 23.9%

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

      2. add-sqr-sqrt 23.9%

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

      3. add-sqr-sqrt 23.9%

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

      4. sqrt-unprod 22.9%

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

      5. swap-sqr 22.9%

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

      6. pow-sqr 23.0%

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

      7. metadata-eval 23.0%

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

      8. metadata-eval 23.0%

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

      9. flip-+ 77.1%

         \[\leadsto \color{blue}{x \cdot y + \sqrt{{z}^{4} \cdot 9}} \]

      10. +-commutative 77.1%

          \[\leadsto \color{blue}{\sqrt{{z}^{4} \cdot 9} + x \cdot y} \]

      11. metadata-eval 77.1%

          \[\leadsto \sqrt{{z}^{\color{blue}{\left(2 \cdot 2\right)}} \cdot 9} + x \cdot y \]

      12. pow-sqr 77.0%

          \[\leadsto \sqrt{\color{blue}{\left({z}^{2} \cdot {z}^{2}\right)} \cdot 9} + x \cdot y \]

      13. metadata-eval 77.0%

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

      14. swap-sqr 77.0%

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

      15. sqrt-unprod 98.1%

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

   7. Applied egg-rr 99.6%

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

      1. fma-udef 98.0%

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

      2. unpow2 98.0%

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

   9. Simplified 98.0%

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

   10. Taylor expanded in z around inf 85.3%

       \[\leadsto \color{blue}{{z}^{2} \cdot {\left(\sqrt{3}\right)}^{2}} \]
   11. Step-by-step derivation

       1. unpow2 85.3%

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

       2. unpow2 85.3%

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

       3. swap-sqr 85.5%

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

       4. unpow2 85.5%

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

   12. Simplified 85.5%

       \[\leadsto \color{blue}{{\left(z \cdot \sqrt{3}\right)}^{2}} \]
   13. Step-by-step derivation

       1. unpow2 85.5%

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

       2. *-commutative 85.5%

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

       3. *-commutative 85.5%

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

       4. swap-sqr 85.3%

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

       5. rem-square-sqrt 85.7%

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

       6. associate-*r* 85.7%

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

   14. Applied egg-rr 85.7%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 510000000000:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \end{array} \]

Alternative 5: 52.3% 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 99.5%

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

2. Taylor expanded in x around inf 52.9%

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

3. Final simplification 52.9%

   \[\leadsto x \cdot y \]

Developer target: 98.1% accurate, 1.7× 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 2023301 
(FPCore (x y z)
  :name "Linear.Quaternion:$c/ from linear-1.19.1.3, A"
  :precision binary64

  :herbie-target
  (+ (* (* 3.0 z) z) (* y x))

  (+ (+ (+ (* x y) (* z z)) (* z z)) (* z z)))