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

Percentage Accurate: 98.3% → 99.8%

Time: 8.2s

Alternatives: 7

Speedup: 0.7×

• 34.5% 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: 99.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{+35}:\\ \;\;\;\;x \cdot \left(y + 3 \cdot \left(z \cdot \frac{z}{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot z + \left(z \cdot z + \left(z \cdot z + x \cdot y\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= x -1.2e+35)
   (* x (+ y (* 3.0 (* z (/ z x)))))
   (+ (* z z) (+ (* z z) (+ (* z z) (* x y))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.2e+35) {
		tmp = x * (y + (3.0 * (z * (z / x))));
	} else {
		tmp = (z * z) + ((z * z) + ((z * z) + (x * y)));
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
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 (x <= (-1.2d+35)) then
        tmp = x * (y + (3.0d0 * (z * (z / x))))
    else
        tmp = (z * z) + ((z * z) + ((z * z) + (x * y)))
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.2e+35) {
		tmp = x * (y + (3.0 * (z * (z / x))));
	} else {
		tmp = (z * z) + ((z * z) + ((z * z) + (x * y)));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if x <= -1.2e+35:
		tmp = x * (y + (3.0 * (z * (z / x))))
	else:
		tmp = (z * z) + ((z * z) + ((z * z) + (x * y)))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (x <= -1.2e+35)
		tmp = Float64(x * Float64(y + Float64(3.0 * Float64(z * Float64(z / x)))));
	else
		tmp = Float64(Float64(z * z) + Float64(Float64(z * z) + Float64(Float64(z * z) + Float64(x * y))));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.2e+35)
		tmp = x * (y + (3.0 * (z * (z / x))));
	else
		tmp = (z * z) + ((z * z) + ((z * z) + (x * y)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[x, -1.2e+35], N[(x * N[(y + N[(3.0 * N[(z * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(z * z), $MachinePrecision] + N[(N[(z * z), $MachinePrecision] + N[(N[(z * z), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.20000000000000007e35

   1. Initial program 95.3%

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

   3. Taylor expanded in x around inf 98.4%

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

   5. Simplified 98.4%

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

      1. unpow2 98.4%

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

      2. associate-/l* 99.9%

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

   7. Applied egg-rr 99.9%

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

   if -1.20000000000000007e35 < x

   1. Initial program 96.6%

      \[\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{+35}:\\ \;\;\;\;x \cdot \left(y + 3 \cdot \left(z \cdot \frac{z}{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot z + \left(z \cdot z + \left(z \cdot z + x \cdot y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.3% accurate, 0.1× speedup

Math

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

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z) :precision binary64 (fma z z (fma x y (* 2.0 (* z z)))))

C

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

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	return fma(z, z, fma(x, y, Float64(2.0 * Float64(z * z))))
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := N[(z * z + N[(x * y + N[(2.0 * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.3%

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

   1. +-commutative 96.3%

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

   2. fma-define 96.4%

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

   3. associate-+l+ 96.4%

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

   4. fma-define 98.4%

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

   5. count-2 98.4%

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

3. Simplified 98.4%

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

Alternative 3: 99.3% accurate, 0.1× speedup

Math

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

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z) :precision binary64 (fma x y (* z (* z 3.0))))

C

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

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	return fma(x, y, Float64(z * Float64(z * 3.0)))
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := N[(x * y + N[(z * N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.3%

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

   1. associate-+l+ 96.3%

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

   2. associate-+l+ 96.3%

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

   3. fma-define 98.3%

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

   4. associate-+r+ 98.3%

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

   5. distribute-lft-out 98.3%

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

   6. distribute-lft-out 98.3%

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

   7. remove-double-neg 98.3%

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

   8. unsub-neg 98.3%

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

   9. count-2 98.3%

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

   10. neg-mul-1 98.3%

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

   11. distribute-rgt-out-- 98.3%

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

   12. metadata-eval 98.3%

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

3. Simplified 98.3%

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

Alternative 4: 94.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+86} \lor \neg \left(z \cdot z \leq 10^{+302}\right):\\ \;\;\;\;x \cdot \left(y + 3 \cdot \left(z \cdot \frac{z}{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (or (<= (* z z) 5e+86) (not (<= (* z z) 1e+302)))
   (* x (+ y (* 3.0 (* z (/ z x)))))
   (* z (* z 3.0))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (((z * z) <= 5e+86) || !((z * z) <= 1e+302)) {
		tmp = x * (y + (3.0 * (z * (z / x))));
	} else {
		tmp = z * (z * 3.0);
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (((z * z) <= 5d+86) .or. (.not. ((z * z) <= 1d+302))) then
        tmp = x * (y + (3.0d0 * (z * (z / x))))
    else
        tmp = z * (z * 3.0d0)
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (((z * z) <= 5e+86) || !((z * z) <= 1e+302)) {
		tmp = x * (y + (3.0 * (z * (z / x))));
	} else {
		tmp = z * (z * 3.0);
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if ((z * z) <= 5e+86) or not ((z * z) <= 1e+302):
		tmp = x * (y + (3.0 * (z * (z / x))))
	else:
		tmp = z * (z * 3.0)
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if ((Float64(z * z) <= 5e+86) || !(Float64(z * z) <= 1e+302))
		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

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (((z * z) <= 5e+86) || ~(((z * z) <= 1e+302)))
		tmp = x * (y + (3.0 * (z * (z / x))));
	else
		tmp = z * (z * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[Or[LessEqual[N[(z * z), $MachinePrecision], 5e+86], N[Not[LessEqual[N[(z * z), $MachinePrecision], 1e+302]], $MachinePrecision]], 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) < 4.9999999999999998e86 or 1.0000000000000001e302 < (*.f64 z z)

   1. Initial program 95.5%

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

   3. Taylor expanded in x around inf 96.6%

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

   5. Simplified 96.6%

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

      1. unpow2 96.6%

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

      2. associate-/l* 98.6%

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

   7. Applied egg-rr 98.6%

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

   if 4.9999999999999998e86 < (*.f64 z z) < 1.0000000000000001e302

   1. Initial program 99.4%

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

      1. associate-+l+ 99.4%

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

      2. associate-+l+ 99.4%

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

      3. fma-define 99.4%

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

      4. associate-+r+ 99.4%

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

      5. distribute-lft-out 99.4%

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

      6. distribute-lft-out 99.6%

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

      7. remove-double-neg 99.6%

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

      8. unsub-neg 99.6%

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

      9. count-2 99.6%

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

      10. neg-mul-1 99.6%

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

      11. distribute-rgt-out-- 99.6%

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

      12. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

      1. add-sqr-sqrt 99.3%

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

      2. pow2 99.3%

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

      3. associate-*r* 99.1%

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

      4. sqrt-prod 99.1%

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

      5. sqrt-prod 45.4%

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

      6. add-sqr-sqrt 99.1%

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

   6. Applied egg-rr 99.1%

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

   7. Taylor expanded in x around 0 87.1%

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

      1. unpow2 87.1%

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

      2. unpow2 87.1%

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

      3. swap-sqr 87.5%

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

      4. unpow2 87.5%

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

   9. Simplified 87.5%

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

       1. *-commutative 87.5%

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

       2. unpow-prod-down 87.1%

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

       3. pow2 87.1%

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

       4. rem-square-sqrt 87.9%

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

       5. unpow2 87.8%

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

       6. associate-*r* 88.0%

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

   11. Applied egg-rr 88.0%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+86} \lor \neg \left(z \cdot z \leq 10^{+302}\right):\\ \;\;\;\;x \cdot \left(y + 3 \cdot \left(z \cdot \frac{z}{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+58}:\\ \;\;\;\;z \cdot z + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 2e+58) (+ (* z z) (* x y)) (* z (* z 3.0))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e+58) {
		tmp = (z * z) + (x * y);
	} else {
		tmp = z * (z * 3.0);
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
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+58) then
        tmp = (z * z) + (x * y)
    else
        tmp = z * (z * 3.0d0)
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e+58) {
		tmp = (z * z) + (x * y);
	} else {
		tmp = z * (z * 3.0);
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if (z * z) <= 2e+58:
		tmp = (z * z) + (x * y)
	else:
		tmp = z * (z * 3.0)
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 2e+58)
		tmp = Float64(Float64(z * z) + Float64(x * y));
	else
		tmp = Float64(z * Float64(z * 3.0));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 2e+58)
		tmp = (z * z) + (x * y);
	else
		tmp = z * (z * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 2e+58], 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) < 1.99999999999999989e58

   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 87.6%

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

   4. Taylor expanded in x around inf 87.2%

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

   if 1.99999999999999989e58 < (*.f64 z z)

   1. Initial program 92.3%

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

      1. associate-+l+ 92.3%

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

      2. associate-+l+ 92.3%

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

      3. fma-define 96.4%

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

      4. associate-+r+ 96.4%

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

      5. distribute-lft-out 96.4%

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

      6. distribute-lft-out 96.5%

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

      7. remove-double-neg 96.5%

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

      8. unsub-neg 96.5%

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

      9. count-2 96.5%

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

      10. neg-mul-1 96.5%

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

      11. distribute-rgt-out-- 96.5%

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

      12. metadata-eval 96.5%

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

   3. Simplified 96.5%

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

      1. add-sqr-sqrt 96.3%

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

      2. pow2 96.3%

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

      3. associate-*r* 96.3%

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

      4. sqrt-prod 96.2%

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

      5. sqrt-prod 44.3%

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

      6. add-sqr-sqrt 96.2%

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

   6. Applied egg-rr 96.2%

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

   7. Taylor expanded in x around 0 88.7%

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

      1. unpow2 88.7%

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

      2. unpow2 88.7%

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

      3. swap-sqr 88.9%

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

      4. unpow2 88.9%

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

   9. Simplified 88.9%

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

       1. *-commutative 88.9%

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

       2. unpow-prod-down 88.7%

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

       3. pow2 88.7%

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

       4. rem-square-sqrt 89.1%

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

       5. unpow2 89.1%

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

       6. associate-*r* 89.2%

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

   11. Applied egg-rr 89.2%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+58}:\\ \;\;\;\;z \cdot z + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{-95}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 2e-95) (* x y) (* z (* z 3.0))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e-95) {
		tmp = x * y;
	} else {
		tmp = z * (z * 3.0);
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
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-95) then
        tmp = x * y
    else
        tmp = z * (z * 3.0d0)
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e-95) {
		tmp = x * y;
	} else {
		tmp = z * (z * 3.0);
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if (z * z) <= 2e-95:
		tmp = x * y
	else:
		tmp = z * (z * 3.0)
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 2e-95)
		tmp = Float64(x * y);
	else
		tmp = Float64(z * Float64(z * 3.0));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 2e-95)
		tmp = x * y;
	else
		tmp = z * (z * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 2e-95], N[(x * y), $MachinePrecision], N[(z * N[(z * 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.99999999999999998e-95

   1. Initial program 100.0%

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

      1. associate-+l+ 100.0%

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

      2. associate-+l+ 100.0%

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

      3. fma-define 100.0%

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

      4. associate-+r+ 100.0%

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

      5. distribute-lft-out 100.0%

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

      6. distribute-lft-out 100.0%

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

      7. remove-double-neg 100.0%

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

      8. unsub-neg 100.0%

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

      9. count-2 100.0%

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

      10. neg-mul-1 100.0%

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

      11. distribute-rgt-out-- 100.0%

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

      12. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

      1. add-sqr-sqrt 99.9%

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

      2. pow2 99.9%

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

      3. associate-*r* 99.9%

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

      4. sqrt-prod 99.9%

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

      5. sqrt-prod 58.1%

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

      6. add-sqr-sqrt 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in x around inf 93.2%

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

   if 1.99999999999999998e-95 < (*.f64 z z)

   1. Initial program 93.5%

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

      1. associate-+l+ 93.5%

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

      2. associate-+l+ 93.5%

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

      3. fma-define 96.9%

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

      4. associate-+r+ 96.9%

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

      5. distribute-lft-out 96.9%

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

      6. distribute-lft-out 97.1%

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

      7. remove-double-neg 97.1%

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

      8. unsub-neg 97.1%

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

      9. count-2 97.1%

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

      10. neg-mul-1 97.1%

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

      11. distribute-rgt-out-- 97.1%

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

      12. metadata-eval 97.1%

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

   3. Simplified 97.1%

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

      1. add-sqr-sqrt 96.9%

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

      2. pow2 96.9%

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

      3. associate-*r* 96.8%

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

      4. sqrt-prod 96.8%

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

      5. sqrt-prod 44.1%

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

      6. add-sqr-sqrt 96.8%

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

   6. Applied egg-rr 96.8%

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

   7. Taylor expanded in x around 0 82.5%

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

      1. unpow2 82.5%

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

      2. unpow2 82.5%

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

      3. swap-sqr 82.7%

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

      4. unpow2 82.7%

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

   9. Simplified 82.7%

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

       1. *-commutative 82.7%

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

       2. unpow-prod-down 82.5%

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

       3. pow2 82.5%

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

       4. rem-square-sqrt 82.9%

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

       5. unpow2 82.9%

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

       6. associate-*r* 83.0%

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

   11. Applied egg-rr 83.0%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{-95}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 53.4% accurate, 5.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ x \cdot y \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z) :precision binary64 (* x y))

C

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

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
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

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

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	return x * y

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	return Float64(x * y)
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp = code(x, y, z)
	tmp = x * y;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := N[(x * y), $MachinePrecision]

Derivation

1. Initial program 96.3%

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

   1. associate-+l+ 96.3%

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

   2. associate-+l+ 96.3%

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

   3. fma-define 98.3%

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

   4. associate-+r+ 98.3%

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

   5. distribute-lft-out 98.3%

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

   6. distribute-lft-out 98.3%

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

   7. remove-double-neg 98.3%

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

   8. unsub-neg 98.3%

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

   9. count-2 98.3%

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

   10. neg-mul-1 98.3%

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

   11. distribute-rgt-out-- 98.3%

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

   12. metadata-eval 98.3%

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

3. Simplified 98.3%

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

   1. add-sqr-sqrt 98.2%

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

   2. pow2 98.2%

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

   3. associate-*r* 98.1%

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

   4. sqrt-prod 98.1%

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

   5. sqrt-prod 50.1%

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

   6. add-sqr-sqrt 98.1%

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

6. Applied egg-rr 98.1%

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

7. Taylor expanded in x around inf 51.5%

   \[\leadsto \color{blue}{x \cdot y} \]
8. Add Preprocessing

Developer target: 98.3% 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 2024101 
(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)))