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

Percentage Accurate: 98.3% → 99.5%

Time: 9.4s

Alternatives: 9

Speedup: 0.7×

• 33.6% 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: 98.3% 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: 99.5% 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 97.5%

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

   1. +-commutative 97.5%

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

   2. fma-define 97.6%

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

   3. associate-+l+ 97.6%

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

   4. fma-define 100.0%

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

   5. count-2 100.0%

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

3. Simplified 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 99.4% 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 97.5%

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

   1. associate-+l+ 97.5%

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

   2. associate-+l+ 97.5%

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

   3. fma-define 99.9%

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

   4. associate-+r+ 99.9%

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

   5. distribute-lft-out 99.9%

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

   6. distribute-lft-out 99.9%

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

   7. remove-double-neg 99.9%

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

   8. unsub-neg 99.9%

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

   9. count-2 99.9%

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

   10. neg-mul-1 99.9%

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

   11. distribute-rgt-out-- 99.9%

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

   12. metadata-eval 99.9%

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

3. Simplified 99.9%

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

5. Final simplification 99.9%

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

Alternative 3: 99.4% accurate, 0.1× speedup

Math

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

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(Float64(z * z) * 3.0))
end

Wolfram

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

Derivation

1. Initial program 97.5%

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

   1. associate-+l+ 97.5%

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

   2. associate-+l+ 97.5%

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

   3. fma-define 99.9%

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

   4. *-lft-identity 99.9%

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

   5. metadata-eval 99.9%

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

   6. count-2 99.9%

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

   7. distribute-rgt-out 99.9%

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

   8. metadata-eval 99.9%

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

   9. metadata-eval 99.9%

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

3. Simplified 99.9%

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

5. Final simplification 99.9%

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

Alternative 4: 99.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 4 \cdot 10^{+292}:\\ \;\;\;\;z \cdot z + \left(z \cdot z + \left(z \cdot z + x \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 4e+292)
   (+ (* z z) (+ (* z z) (+ (* z z) (* x y))))
   (* z (* z 3.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 4e+292) {
		tmp = (z * z) + ((z * z) + ((z * z) + (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) <= 4d+292) then
        tmp = (z * z) + ((z * z) + ((z * z) + (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) <= 4e+292) {
		tmp = (z * z) + ((z * z) + ((z * z) + (x * y)));
	} else {
		tmp = z * (z * 3.0);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z * z) <= 4e+292:
		tmp = (z * z) + ((z * z) + ((z * z) + (x * y)))
	else:
		tmp = z * (z * 3.0)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 4e+292)
		tmp = Float64(Float64(z * z) + Float64(Float64(z * z) + Float64(Float64(z * z) + 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 * z) <= 4e+292)
		tmp = (z * z) + ((z * z) + ((z * z) + (x * y)));
	else
		tmp = z * (z * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 4e+292], N[(N[(z * z), $MachinePrecision] + N[(N[(z * z), $MachinePrecision] + N[(N[(z * z), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * N[(z * 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 4.0000000000000001e292

   1. Initial program 99.8%

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

   if 4.0000000000000001e292 < (*.f64 z z)

   1. Initial program 91.5%

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

   3. Taylor expanded in x around 0 100.0%

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

   5. Simplified 100.0%

      \[\leadsto \color{blue}{{z}^{2} \cdot 3} \]
   6. Step-by-step derivation

      1. add-sqr-sqrt 100.0%

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

      2. sqrt-unprod 96.4%

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

      3. swap-sqr 96.4%

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

      4. pow-prod-up 96.4%

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

      5. metadata-eval 96.4%

         \[\leadsto \sqrt{{z}^{\color{blue}{4}} \cdot \left(3 \cdot 3\right)} \]

      6. metadata-eval 96.4%

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

   7. Applied egg-rr 96.4%

      \[\leadsto \color{blue}{\sqrt{{z}^{4} \cdot 9}} \]
   8. Step-by-step derivation

      1. sqrt-prod 96.4%

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

      2. metadata-eval 96.4%

         \[\leadsto \sqrt{{z}^{4}} \cdot \color{blue}{3} \]

      3. sqrt-pow1 100.0%

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

      4. metadata-eval 100.0%

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

      5. pow2 100.0%

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

      6. associate-*r* 100.0%

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

      7. *-commutative 100.0%

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

   9. Applied egg-rr 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 4 \cdot 10^{+292}:\\ \;\;\;\;z \cdot z + \left(z \cdot z + \left(z \cdot z + x \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+118}:\\ \;\;\;\;x \cdot \left(y + 3 \cdot \left(z \cdot \frac{z}{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 1e+118) (* x (+ y (* 3.0 (* z (/ z x))))) (* z (* z 3.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 1e+118) {
		tmp = x * (y + (3.0 * (z * (z / x))));
	} 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+118) then
        tmp = x * (y + (3.0d0 * (z * (z / x))))
    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+118) {
		tmp = x * (y + (3.0 * (z * (z / x))));
	} else {
		tmp = z * (z * 3.0);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z * z) <= 1e+118:
		tmp = x * (y + (3.0 * (z * (z / x))))
	else:
		tmp = z * (z * 3.0)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 1e+118)
		tmp = Float64(x * Float64(y + Float64(3.0 * Float64(z * Float64(z / x)))));
	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 * z) <= 1e+118)
		tmp = x * (y + (3.0 * (z * (z / x))));
	else
		tmp = z * (z * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 1e+118], N[(x * N[(y + N[(3.0 * N[(z * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * N[(z * 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 9.99999999999999967e117

   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. Taylor expanded in x around inf 99.9%

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

   5. Simplified 99.9%

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

      1. unpow2 99.9%

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

      2. associate-/l* 99.8%

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

   7. Applied egg-rr 99.8%

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

   if 9.99999999999999967e117 < (*.f64 z z)

   1. Initial program 93.9%

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

   3. Taylor expanded in x around 0 97.2%

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

   5. Simplified 97.2%

      \[\leadsto \color{blue}{{z}^{2} \cdot 3} \]
   6. Step-by-step derivation

      1. add-sqr-sqrt 97.0%

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

      2. sqrt-unprod 80.6%

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

      3. swap-sqr 80.6%

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

      4. pow-prod-up 80.7%

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

      5. metadata-eval 80.7%

         \[\leadsto \sqrt{{z}^{\color{blue}{4}} \cdot \left(3 \cdot 3\right)} \]

      6. metadata-eval 80.7%

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

   7. Applied egg-rr 80.7%

      \[\leadsto \color{blue}{\sqrt{{z}^{4} \cdot 9}} \]
   8. Step-by-step derivation

      1. sqrt-prod 80.6%

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

      2. metadata-eval 80.6%

         \[\leadsto \sqrt{{z}^{4}} \cdot \color{blue}{3} \]

      3. sqrt-pow1 97.2%

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

      4. metadata-eval 97.2%

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

      5. pow2 97.2%

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

      6. associate-*r* 97.2%

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

      7. *-commutative 97.2%

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

   9. Applied egg-rr 97.2%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+118}:\\ \;\;\;\;x \cdot \left(y + 3 \cdot \left(z \cdot \frac{z}{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 95.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+118}:\\ \;\;\;\;x \cdot \left(y + z \cdot \left(z \cdot \frac{3}{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 1e+118) (* x (+ y (* z (* z (/ 3.0 x))))) (* z (* z 3.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 1e+118) {
		tmp = x * (y + (z * (z * (3.0 / x))));
	} 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+118) then
        tmp = x * (y + (z * (z * (3.0d0 / x))))
    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+118) {
		tmp = x * (y + (z * (z * (3.0 / x))));
	} else {
		tmp = z * (z * 3.0);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z * z) <= 1e+118:
		tmp = x * (y + (z * (z * (3.0 / x))))
	else:
		tmp = z * (z * 3.0)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 1e+118)
		tmp = Float64(x * Float64(y + Float64(z * Float64(z * Float64(3.0 / x)))));
	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 * z) <= 1e+118)
		tmp = x * (y + (z * (z * (3.0 / x))));
	else
		tmp = z * (z * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 1e+118], N[(x * N[(y + N[(z * N[(z * N[(3.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * N[(z * 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 9.99999999999999967e117

   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. Taylor expanded in x around inf 99.9%

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

   5. Simplified 99.9%

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

      1. *-commutative 99.9%

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

      2. clear-num 99.8%

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

      3. un-div-inv 99.8%

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

   7. Applied egg-rr 99.8%

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

      1. associate-/r/ 99.9%

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

      2. unpow2 99.9%

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

      3. associate-*r* 99.8%

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

   9. Applied egg-rr 99.8%

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

   if 9.99999999999999967e117 < (*.f64 z z)

   1. Initial program 93.9%

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

   3. Taylor expanded in x around 0 97.2%

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

   5. Simplified 97.2%

      \[\leadsto \color{blue}{{z}^{2} \cdot 3} \]
   6. Step-by-step derivation

      1. add-sqr-sqrt 97.0%

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

      2. sqrt-unprod 80.6%

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

      3. swap-sqr 80.6%

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

      4. pow-prod-up 80.7%

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

      5. metadata-eval 80.7%

         \[\leadsto \sqrt{{z}^{\color{blue}{4}} \cdot \left(3 \cdot 3\right)} \]

      6. metadata-eval 80.7%

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

   7. Applied egg-rr 80.7%

      \[\leadsto \color{blue}{\sqrt{{z}^{4} \cdot 9}} \]
   8. Step-by-step derivation

      1. sqrt-prod 80.6%

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

      2. metadata-eval 80.6%

         \[\leadsto \sqrt{{z}^{4}} \cdot \color{blue}{3} \]

      3. sqrt-pow1 97.2%

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

      4. metadata-eval 97.2%

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

      5. pow2 97.2%

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

      6. associate-*r* 97.2%

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

      7. *-commutative 97.2%

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

   9. Applied egg-rr 97.2%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+118}:\\ \;\;\;\;x \cdot \left(y + z \cdot \left(z \cdot \frac{3}{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+29}:\\ \;\;\;\;z \cdot z + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 2e+29) (+ (* z z) (* x y)) (* z (* z 3.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e+29) {
		tmp = (z * z) + (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) <= 2d+29) then
        tmp = (z * z) + (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) <= 2e+29) {
		tmp = (z * z) + (x * y);
	} else {
		tmp = z * (z * 3.0);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z * z) <= 2e+29:
		tmp = (z * z) + (x * y)
	else:
		tmp = z * (z * 3.0)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 2e+29)
		tmp = Float64(Float64(z * z) + 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 * z) <= 2e+29)
		tmp = (z * z) + (x * y);
	else
		tmp = z * (z * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 2e+29], N[(N[(z * z), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision], N[(z * N[(z * 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.99999999999999983e29

   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. Taylor expanded in x around inf 83.3%

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

   4. Taylor expanded in x around inf 82.9%

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

   if 1.99999999999999983e29 < (*.f64 z z)

   1. Initial program 95.0%

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

   3. Taylor expanded in x around 0 90.5%

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

   5. Simplified 90.5%

      \[\leadsto \color{blue}{{z}^{2} \cdot 3} \]
   6. Step-by-step derivation

      1. add-sqr-sqrt 90.3%

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

      2. sqrt-unprod 76.9%

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

      3. swap-sqr 76.9%

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

      4. pow-prod-up 76.9%

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

      5. metadata-eval 76.9%

         \[\leadsto \sqrt{{z}^{\color{blue}{4}} \cdot \left(3 \cdot 3\right)} \]

      6. metadata-eval 76.9%

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

   7. Applied egg-rr 76.9%

      \[\leadsto \color{blue}{\sqrt{{z}^{4} \cdot 9}} \]
   8. Step-by-step derivation

      1. sqrt-prod 76.9%

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

      2. metadata-eval 76.9%

         \[\leadsto \sqrt{{z}^{4}} \cdot \color{blue}{3} \]

      3. sqrt-pow1 90.5%

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

      4. metadata-eval 90.5%

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

      5. pow2 90.5%

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

      6. associate-*r* 90.5%

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

      7. *-commutative 90.5%

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

   9. Applied egg-rr 90.5%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+29}:\\ \;\;\;\;z \cdot z + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 68.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 470000000:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 470000000.0) (* x y) (* z (* z 3.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 470000000.0) {
		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 <= 470000000.0d0) 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 <= 470000000.0) {
		tmp = x * y;
	} else {
		tmp = z * (z * 3.0);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 470000000.0:
		tmp = x * y
	else:
		tmp = z * (z * 3.0)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 470000000.0)
		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 <= 470000000.0)
		tmp = x * y;
	else
		tmp = z * (z * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 470000000.0], N[(x * y), $MachinePrecision], N[(z * N[(z * 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 4.7e8

   1. Initial program 98.4%

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

      1. +-commutative 98.4%

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

      2. fma-define 98.4%

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

      3. associate-+l+ 98.4%

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

      4. fma-define 100.0%

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

      5. count-2 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, z, \mathsf{fma}\left(x, y, 2 \cdot \left(z \cdot z\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 99.8%

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

      2. pow2 99.8%

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

      3. *-commutative 99.8%

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

      4. sqrt-prod 99.8%

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

      5. sqrt-prod 33.2%

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

      6. add-sqr-sqrt 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in z around 0 58.7%

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

   if 4.7e8 < z

   1. Initial program 94.6%

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

   3. Taylor expanded in x around 0 92.4%

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

   5. Simplified 92.4%

      \[\leadsto \color{blue}{{z}^{2} \cdot 3} \]
   6. Step-by-step derivation

      1. add-sqr-sqrt 92.3%

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

      2. sqrt-unprod 78.1%

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

      3. swap-sqr 78.1%

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

      4. pow-prod-up 78.2%

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

      5. metadata-eval 78.2%

         \[\leadsto \sqrt{{z}^{\color{blue}{4}} \cdot \left(3 \cdot 3\right)} \]

      6. metadata-eval 78.2%

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

   7. Applied egg-rr 78.2%

      \[\leadsto \color{blue}{\sqrt{{z}^{4} \cdot 9}} \]
   8. Step-by-step derivation

      1. sqrt-prod 78.1%

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

      2. metadata-eval 78.1%

         \[\leadsto \sqrt{{z}^{4}} \cdot \color{blue}{3} \]

      3. sqrt-pow1 92.4%

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

      4. metadata-eval 92.4%

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

      5. pow2 92.4%

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

      6. associate-*r* 92.5%

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

      7. *-commutative 92.5%

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

   9. Applied egg-rr 92.5%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 470000000:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 52.5% 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 97.5%

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

   1. +-commutative 97.5%

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

   2. fma-define 97.6%

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

   3. associate-+l+ 97.6%

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

   4. fma-define 100.0%

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

   5. count-2 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(z, z, \mathsf{fma}\left(x, y, 2 \cdot \left(z \cdot z\right)\right)\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-sqr-sqrt 99.8%

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

   2. pow2 99.8%

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

   3. *-commutative 99.8%

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

   4. sqrt-prod 99.7%

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

   5. sqrt-prod 48.3%

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

   6. add-sqr-sqrt 99.7%

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

6. Applied egg-rr 99.7%

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

7. Taylor expanded in z around 0 48.2%

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

8. Final simplification 48.2%

   \[\leadsto x \cdot y \]
9. Add Preprocessing

Developer target: 98.4% 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 2024079 
(FPCore (x y z)
  :name "Linear.Quaternion:$c/ from linear-1.19.1.3, A"
  :precision binary64

  :alt
  (+ (* (* 3.0 z) z) (* y x))

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