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

Percentage Accurate: 98.5% → 99.6%

Time: 8.8s

Alternatives: 9

Speedup: 0.7×

• 34.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.5 \cdot 10^{-87}:\\ \;\;\;\;z \cdot z + \left(z \cdot z + \left(z \cdot z + x \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x + z \cdot \frac{z \cdot 3}{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 2.5e-87)
   (+ (* z z) (+ (* z z) (+ (* z z) (* x y))))
   (* y (+ x (* z (/ (* z 3.0) y))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.5e-87) {
		tmp = (z * z) + ((z * z) + ((z * z) + (x * y)));
	} else {
		tmp = y * (x + (z * ((z * 3.0) / 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 (y <= 2.5d-87) then
        tmp = (z * z) + ((z * z) + ((z * z) + (x * y)))
    else
        tmp = y * (x + (z * ((z * 3.0d0) / 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 (y <= 2.5e-87) {
		tmp = (z * z) + ((z * z) + ((z * z) + (x * y)));
	} else {
		tmp = y * (x + (z * ((z * 3.0) / y)));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= 2.5e-87:
		tmp = (z * z) + ((z * z) + ((z * z) + (x * y)))
	else:
		tmp = y * (x + (z * ((z * 3.0) / y)))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 2.5e-87)
		tmp = Float64(Float64(z * z) + Float64(Float64(z * z) + Float64(Float64(z * z) + Float64(x * y))));
	else
		tmp = Float64(y * Float64(x + Float64(z * Float64(Float64(z * 3.0) / 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 (y <= 2.5e-87)
		tmp = (z * z) + ((z * z) + ((z * z) + (x * y)));
	else
		tmp = y * (x + (z * ((z * 3.0) / 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[y, 2.5e-87], N[(N[(z * z), $MachinePrecision] + N[(N[(z * z), $MachinePrecision] + N[(N[(z * z), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(x + N[(z * N[(N[(z * 3.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.50000000000000021e-87

   1. Initial program 98.1%

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

   if 2.50000000000000021e-87 < y

   1. Initial program 98.6%

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

   3. Taylor expanded in y around inf 99.9%

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

   5. Simplified 99.9%

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

      1. *-commutative 99.9%

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

      2. clear-num 99.9%

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

      3. un-div-inv 99.9%

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

   7. Applied egg-rr 99.9%

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

      1. metadata-eval 99.9%

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

      2. metadata-eval 99.9%

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

      3. sqrt-pow2 99.8%

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

      4. unpow2 99.8%

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

      5. associate-/r* 99.8%

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

      6. un-div-inv 99.8%

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

      7. clear-num 99.8%

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

      8. associate-*r/ 99.7%

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

      9. *-commutative 99.7%

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

      10. associate-/r/ 99.8%

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

      11. sqrt-pow2 99.9%

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

      12. metadata-eval 99.9%

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

      13. metadata-eval 99.9%

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

   9. Applied egg-rr 99.9%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.5 \cdot 10^{-87}:\\ \;\;\;\;z \cdot z + \left(z \cdot z + \left(z \cdot z + x \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x + z \cdot \frac{z \cdot 3}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.5% 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 98.3%

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

   1. +-commutative 98.3%

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

   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 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. Add Preprocessing

5. Final simplification 99.6%

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

Alternative 3: 99.5% 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 98.3%

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

   1. associate-+l+ 98.3%

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

   2. associate-+l+ 98.3%

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

   3. fma-define 99.5%

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

   4. distribute-lft-out 99.5%

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

   5. distribute-lft-out 99.5%

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

   6. count-2 99.5%

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

   7. distribute-rgt1-in 99.5%

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

   8. metadata-eval 99.5%

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

   9. *-commutative 99.5%

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

   10. metadata-eval 99.5%

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

   11. 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. Add Preprocessing

5. Final simplification 99.5%

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

Alternative 4: 97.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-163}:\\ \;\;\;\;x \cdot \left(y + 3 \cdot \left(z \cdot \frac{z}{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x + 3 \cdot \left(z \cdot \frac{z}{y}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= x -1e-163)
   (* x (+ y (* 3.0 (* z (/ z x)))))
   (* y (+ x (* 3.0 (* z (/ z y)))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (x <= -1e-163) {
		tmp = x * (y + (3.0 * (z * (z / x))));
	} else {
		tmp = y * (x + (3.0 * (z * (z / 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 <= (-1d-163)) then
        tmp = x * (y + (3.0d0 * (z * (z / x))))
    else
        tmp = y * (x + (3.0d0 * (z * (z / 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 <= -1e-163) {
		tmp = x * (y + (3.0 * (z * (z / x))));
	} else {
		tmp = y * (x + (3.0 * (z * (z / y))));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if x <= -1e-163:
		tmp = x * (y + (3.0 * (z * (z / x))))
	else:
		tmp = y * (x + (3.0 * (z * (z / y))))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (x <= -1e-163)
		tmp = Float64(x * Float64(y + Float64(3.0 * Float64(z * Float64(z / x)))));
	else
		tmp = Float64(y * Float64(x + Float64(3.0 * Float64(z * Float64(z / 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 <= -1e-163)
		tmp = x * (y + (3.0 * (z * (z / x))));
	else
		tmp = y * (x + (3.0 * (z * (z / 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, -1e-163], N[(x * N[(y + N[(3.0 * N[(z * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(x + N[(3.0 * N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -9.99999999999999923e-164

   1. Initial program 98.8%

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

   3. Taylor expanded in x around inf 97.8%

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

   5. Simplified 97.8%

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

      1. unpow2 97.8%

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

      2. associate-/l* 97.7%

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

   7. Applied egg-rr 97.7%

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

   if -9.99999999999999923e-164 < x

   1. Initial program 98.0%

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

   3. Taylor expanded in y around inf 93.7%

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

   5. Simplified 93.7%

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

      1. unpow2 93.7%

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

      2. associate-/l* 94.3%

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

   7. Applied egg-rr 94.3%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-163}:\\ \;\;\;\;x \cdot \left(y + 3 \cdot \left(z \cdot \frac{z}{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x + 3 \cdot \left(z \cdot \frac{z}{y}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-141}:\\ \;\;\;\;x \cdot \left(y + 3 \cdot \frac{z}{\frac{x}{z}}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x + \left(z \cdot 3\right) \cdot \frac{z}{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= x -2e-141)
   (* x (+ y (* 3.0 (/ z (/ x z)))))
   (* y (+ x (* (* z 3.0) (/ z y))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (x <= -2e-141) {
		tmp = x * (y + (3.0 * (z / (x / z))));
	} else {
		tmp = y * (x + ((z * 3.0) * (z / 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 <= (-2d-141)) then
        tmp = x * (y + (3.0d0 * (z / (x / z))))
    else
        tmp = y * (x + ((z * 3.0d0) * (z / 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 <= -2e-141) {
		tmp = x * (y + (3.0 * (z / (x / z))));
	} else {
		tmp = y * (x + ((z * 3.0) * (z / y)));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if x <= -2e-141:
		tmp = x * (y + (3.0 * (z / (x / z))))
	else:
		tmp = y * (x + ((z * 3.0) * (z / y)))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (x <= -2e-141)
		tmp = Float64(x * Float64(y + Float64(3.0 * Float64(z / Float64(x / z)))));
	else
		tmp = Float64(y * Float64(x + Float64(Float64(z * 3.0) * Float64(z / 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 <= -2e-141)
		tmp = x * (y + (3.0 * (z / (x / z))));
	else
		tmp = y * (x + ((z * 3.0) * (z / 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, -2e-141], N[(x * N[(y + N[(3.0 * N[(z / N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(x + N[(N[(z * 3.0), $MachinePrecision] * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.0000000000000001e-141

   1. Initial program 98.8%

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

   3. Taylor expanded in x around inf 97.7%

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

   5. Simplified 97.7%

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

      1. unpow2 97.7%

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

      2. associate-/l* 97.7%

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

   7. Applied egg-rr 97.7%

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

      1. clear-num 97.7%

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

      2. un-div-inv 97.7%

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

   9. Applied egg-rr 97.7%

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

   if -2.0000000000000001e-141 < x

   1. Initial program 98.0%

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

   3. Taylor expanded in y around inf 93.7%

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

   5. Simplified 93.7%

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

      1. *-commutative 93.7%

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

      2. clear-num 93.7%

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

      3. un-div-inv 93.7%

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

   7. Applied egg-rr 93.7%

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

      1. metadata-eval 93.7%

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

      2. metadata-eval 93.7%

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

      3. sqrt-pow2 93.6%

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

      4. clear-num 93.5%

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

      5. associate-/r/ 93.5%

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

      6. clear-num 93.5%

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

      7. unpow2 93.5%

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

      8. associate-*l/ 94.1%

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

      9. associate-*l* 94.1%

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

      10. sqrt-pow2 94.3%

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

      11. metadata-eval 94.3%

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

      12. metadata-eval 94.3%

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

   9. Applied egg-rr 94.3%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-141}:\\ \;\;\;\;x \cdot \left(y + 3 \cdot \frac{z}{\frac{x}{z}}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x + \left(z \cdot 3\right) \cdot \frac{z}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{-143}:\\ \;\;\;\;x \cdot \left(y + 3 \cdot \frac{z}{\frac{x}{z}}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x + z \cdot \frac{z \cdot 3}{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= x -2.1e-143)
   (* x (+ y (* 3.0 (/ z (/ x z)))))
   (* y (+ x (* z (/ (* z 3.0) y))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.1e-143) {
		tmp = x * (y + (3.0 * (z / (x / z))));
	} else {
		tmp = y * (x + (z * ((z * 3.0) / 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 <= (-2.1d-143)) then
        tmp = x * (y + (3.0d0 * (z / (x / z))))
    else
        tmp = y * (x + (z * ((z * 3.0d0) / 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 <= -2.1e-143) {
		tmp = x * (y + (3.0 * (z / (x / z))));
	} else {
		tmp = y * (x + (z * ((z * 3.0) / y)));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if x <= -2.1e-143:
		tmp = x * (y + (3.0 * (z / (x / z))))
	else:
		tmp = y * (x + (z * ((z * 3.0) / y)))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (x <= -2.1e-143)
		tmp = Float64(x * Float64(y + Float64(3.0 * Float64(z / Float64(x / z)))));
	else
		tmp = Float64(y * Float64(x + Float64(z * Float64(Float64(z * 3.0) / 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 <= -2.1e-143)
		tmp = x * (y + (3.0 * (z / (x / z))));
	else
		tmp = y * (x + (z * ((z * 3.0) / 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, -2.1e-143], N[(x * N[(y + N[(3.0 * N[(z / N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(x + N[(z * N[(N[(z * 3.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.1000000000000001e-143

   1. Initial program 98.8%

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

   3. Taylor expanded in x around inf 97.7%

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

   5. Simplified 97.7%

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

      1. unpow2 97.7%

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

      2. associate-/l* 97.7%

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

   7. Applied egg-rr 97.7%

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

      1. clear-num 97.7%

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

      2. un-div-inv 97.7%

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

   9. Applied egg-rr 97.7%

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

   if -2.1000000000000001e-143 < x

   1. Initial program 98.0%

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

   3. Taylor expanded in y around inf 93.7%

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

   5. Simplified 93.7%

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

      1. *-commutative 93.7%

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

      2. clear-num 93.7%

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

      3. un-div-inv 93.7%

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

   7. Applied egg-rr 93.7%

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

      1. metadata-eval 93.7%

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

      2. metadata-eval 93.7%

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

      3. sqrt-pow2 93.6%

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

      4. unpow2 93.6%

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

      5. associate-/r* 94.2%

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

      6. un-div-inv 94.2%

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

      7. clear-num 94.1%

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

      8. associate-*r/ 94.1%

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

      9. *-commutative 94.1%

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

      10. associate-/r/ 94.2%

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

      11. sqrt-pow2 94.3%

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

      12. metadata-eval 94.3%

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

      13. metadata-eval 94.3%

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

   9. Applied egg-rr 94.3%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{-143}:\\ \;\;\;\;x \cdot \left(y + 3 \cdot \frac{z}{\frac{x}{z}}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x + z \cdot \frac{z \cdot 3}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 94.4% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

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 + (3.0d0 * (z * (z / x))))
end function

Java

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

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	return x * (y + (3.0 * (z * (z / x))))

Julia

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

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp = code(x, y, z)
	tmp = x * (y + (3.0 * (z * (z / x))));
end

Wolfram

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

Derivation

1. Initial program 98.3%

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

3. Taylor expanded in x around inf 91.9%

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

5. Simplified 91.9%

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

   1. unpow2 91.9%

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

   2. associate-/l* 92.3%

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

7. Applied egg-rr 92.3%

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

8. Final simplification 92.3%

   \[\leadsto x \cdot \left(y + 3 \cdot \left(z \cdot \frac{z}{x}\right)\right) \]
9. Add Preprocessing

Alternative 8: 77.6% accurate, 2.1× speedup

Math

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

FPCore

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

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	return (z * z) + (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 = (z * z) + (x * y)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.3%

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

3. Taylor expanded in x around inf 71.9%

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

4. Taylor expanded in x around inf 71.2%

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

5. Final simplification 71.2%

   \[\leadsto z \cdot z + x \cdot y \]
6. Add Preprocessing

Alternative 9: 52.9% 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 98.3%

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

   1. associate-+l+ 98.3%

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

   2. associate-+l+ 98.3%

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

   3. fma-define 99.5%

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

   4. distribute-lft-out 99.5%

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

   5. distribute-lft-out 99.5%

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

   6. count-2 99.5%

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

   7. distribute-rgt1-in 99.5%

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

   8. metadata-eval 99.5%

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

   9. *-commutative 99.5%

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

   10. metadata-eval 99.5%

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

   11. 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. 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* 99.3%

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

   4. sqrt-prod 99.2%

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

   5. sqrt-prod 46.3%

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

   6. add-sqr-sqrt 99.2%

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

6. Applied egg-rr 99.2%

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

7. Taylor expanded in x around inf 46.3%

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

8. Final simplification 46.3%

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

Developer target: 98.6% 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 2024095 
(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)))