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

Percentage Accurate: 98.5% → 99.4%

Time: 8.6s

Alternatives: 9

Speedup: 0.1×

• 33.5% 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: 98.5% 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: 99.4% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 97.9%

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

   1. +-commutative 97.9%

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

   2. fma-define 98.0%

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

   3. associate-+l+ 98.0%

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

   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: 99.3% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 97.9%

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

   1. associate-+l+ 97.9%

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

   2. associate-+l+ 97.9%

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

   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. distribute-lft-out 99.9%

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

   5. distribute-lft-out 99.9%

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

   6. count-2 99.9%

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

   7. distribute-rgt1-in 99.9%

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

   8. metadata-eval 99.9%

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

   9. *-commutative 99.9%

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

   10. metadata-eval 99.9%

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

   11. 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: 98.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.55 \cdot 10^{-110}:\\ \;\;\;\;z \cdot z + \left(z \cdot z + \left(z \cdot z + x \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x + \frac{z \cdot \left(z \cdot 3\right)}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.55e-110)
   (+ (* z z) (+ (* z z) (+ (* z z) (* x y))))
   (* y (+ x (/ (* z (* z 3.0)) y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.55e-110) {
		tmp = (z * z) + ((z * z) + ((z * z) + (x * y)));
	} else {
		tmp = y * (x + ((z * (z * 3.0)) / y));
	}
	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 (y <= 1.55d-110) 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

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.55e-110) {
		tmp = (z * z) + ((z * z) + ((z * z) + (x * y)));
	} else {
		tmp = y * (x + ((z * (z * 3.0)) / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1.55e-110:
		tmp = (z * z) + ((z * z) + ((z * z) + (x * y)))
	else:
		tmp = y * (x + ((z * (z * 3.0)) / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.55e-110)
		tmp = Float64(Float64(z * z) + Float64(Float64(z * z) + Float64(Float64(z * z) + Float64(x * y))));
	else
		tmp = Float64(y * Float64(x + Float64(Float64(z * Float64(z * 3.0)) / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1.55e-110)
		tmp = (z * z) + ((z * z) + ((z * z) + (x * y)));
	else
		tmp = y * (x + ((z * (z * 3.0)) / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.55e-110], 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[(N[(z * N[(z * 3.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.55000000000000004e-110

   1. Initial program 98.6%

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

   if 1.55000000000000004e-110 < y

   1. Initial program 97.0%

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

   3. Taylor expanded in y around inf 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%

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

      1. unpow2 99.9%

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

      2. associate-/l* 99.9%

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

   7. Applied egg-rr 99.9%

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

      1. associate-*r* 99.8%

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

      2. associate-*r/ 99.9%

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

   9. Applied egg-rr 99.9%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.55 \cdot 10^{-110}:\\ \;\;\;\;z \cdot z + \left(z \cdot z + \left(z \cdot z + x \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x + \frac{z \cdot \left(z \cdot 3\right)}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.2 \cdot 10^{-80}:\\ \;\;\;\;x \cdot \left(y + 3 \cdot \left(z \cdot \frac{z}{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x + 3 \cdot \left(z \cdot \frac{z}{y}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -8.2e-80)
   (* x (+ y (* 3.0 (* z (/ z x)))))
   (* y (+ x (* 3.0 (* z (/ z y)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -8.2e-80) {
		tmp = x * (y + (3.0 * (z * (z / x))));
	} else {
		tmp = y * (x + (3.0 * (z * (z / y))));
	}
	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 (x <= (-8.2d-80)) 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

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -8.2e-80) {
		tmp = x * (y + (3.0 * (z * (z / x))));
	} else {
		tmp = y * (x + (3.0 * (z * (z / y))));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -8.2e-80:
		tmp = x * (y + (3.0 * (z * (z / x))))
	else:
		tmp = y * (x + (3.0 * (z * (z / y))))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -8.2e-80)
		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

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -8.2e-80)
		tmp = x * (y + (3.0 * (z * (z / x))));
	else
		tmp = y * (x + (3.0 * (z * (z / y))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -8.2e-80], 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 < -8.1999999999999999e-80

   1. Initial program 96.5%

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

   3. Taylor expanded in x around inf 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%

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

      1. unpow2 99.9%

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

      2. associate-/l* 100.0%

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

   7. Applied egg-rr 100.0%

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

   if -8.1999999999999999e-80 < x

   1. Initial program 98.7%

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

   3. Taylor expanded in y around inf 97.1%

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

   5. Simplified 97.1%

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

      1. unpow2 97.1%

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

      2. associate-/l* 97.1%

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

   7. Applied egg-rr 97.1%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.2 \cdot 10^{-80}:\\ \;\;\;\;x \cdot \left(y + 3 \cdot \left(z \cdot \frac{z}{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x + 3 \cdot \left(z \cdot \frac{z}{y}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-86}:\\ \;\;\;\;x \cdot \left(y + 3 \cdot \left(z \cdot \frac{z}{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x + z \cdot \left(3 \cdot \frac{z}{y}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1e-86)
   (* x (+ y (* 3.0 (* z (/ z x)))))
   (* y (+ x (* z (* 3.0 (/ z y)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1e-86) {
		tmp = x * (y + (3.0 * (z * (z / x))));
	} else {
		tmp = y * (x + (z * (3.0 * (z / y))));
	}
	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 (x <= (-1d-86)) then
        tmp = x * (y + (3.0d0 * (z * (z / x))))
    else
        tmp = y * (x + (z * (3.0d0 * (z / y))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1e-86) {
		tmp = x * (y + (3.0 * (z * (z / x))));
	} else {
		tmp = y * (x + (z * (3.0 * (z / y))));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1e-86:
		tmp = x * (y + (3.0 * (z * (z / x))))
	else:
		tmp = y * (x + (z * (3.0 * (z / y))))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1e-86)
		tmp = Float64(x * Float64(y + Float64(3.0 * Float64(z * Float64(z / x)))));
	else
		tmp = Float64(y * Float64(x + Float64(z * Float64(3.0 * Float64(z / y)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1e-86)
		tmp = x * (y + (3.0 * (z * (z / x))));
	else
		tmp = y * (x + (z * (3.0 * (z / y))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1e-86], N[(x * N[(y + N[(3.0 * N[(z * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(x + N[(z * N[(3.0 * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.00000000000000008e-86

   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. 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%

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

      1. unpow2 99.9%

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

      2. associate-/l* 100.0%

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

   7. Applied egg-rr 100.0%

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

   if -1.00000000000000008e-86 < x

   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 97.0%

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

   5. Simplified 97.0%

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

      1. unpow2 97.0%

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

      2. associate-/l* 97.0%

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

   7. Applied egg-rr 97.0%

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

      1. associate-*r* 97.0%

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

      2. associate-*r/ 97.1%

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

   9. Applied egg-rr 97.1%

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

       1. associate-/l* 97.0%

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

       2. *-commutative 97.0%

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

       3. associate-*l* 97.0%

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

   11. Applied egg-rr 97.0%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-86}:\\ \;\;\;\;x \cdot \left(y + 3 \cdot \left(z \cdot \frac{z}{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x + z \cdot \left(3 \cdot \frac{z}{y}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 94.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 6.5 \cdot 10^{-176}:\\ \;\;\;\;x \cdot \left(y + 3 \cdot \left(z \cdot \frac{z}{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x + \frac{z \cdot \left(z \cdot 3\right)}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 6.5e-176)
   (* x (+ y (* 3.0 (* z (/ z x)))))
   (* y (+ x (/ (* z (* z 3.0)) y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 6.5e-176) {
		tmp = x * (y + (3.0 * (z * (z / x))));
	} else {
		tmp = y * (x + ((z * (z * 3.0)) / y));
	}
	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 (y <= 6.5d-176) then
        tmp = x * (y + (3.0d0 * (z * (z / x))))
    else
        tmp = y * (x + ((z * (z * 3.0d0)) / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 6.5e-176) {
		tmp = x * (y + (3.0 * (z * (z / x))));
	} else {
		tmp = y * (x + ((z * (z * 3.0)) / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 6.5e-176:
		tmp = x * (y + (3.0 * (z * (z / x))))
	else:
		tmp = y * (x + ((z * (z * 3.0)) / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 6.5e-176)
		tmp = Float64(x * Float64(y + Float64(3.0 * Float64(z * Float64(z / x)))));
	else
		tmp = Float64(y * Float64(x + Float64(Float64(z * Float64(z * 3.0)) / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 6.5e-176)
		tmp = x * (y + (3.0 * (z * (z / x))));
	else
		tmp = y * (x + ((z * (z * 3.0)) / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 6.5e-176], N[(x * N[(y + N[(3.0 * N[(z * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(x + N[(N[(z * N[(z * 3.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.5e-176

   1. Initial program 98.5%

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

   3. Taylor expanded in x around inf 95.2%

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

   5. Simplified 95.2%

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

      1. unpow2 95.2%

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

      2. associate-/l* 95.2%

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

   7. Applied egg-rr 95.2%

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

   if 6.5e-176 < y

   1. Initial program 97.2%

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

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

      1. unpow2 99.9%

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

      2. associate-/l* 99.9%

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

   7. Applied egg-rr 99.9%

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

      1. associate-*r* 99.8%

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

      2. associate-*r/ 99.9%

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

   9. Applied egg-rr 99.9%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.5 \cdot 10^{-176}:\\ \;\;\;\;x \cdot \left(y + 3 \cdot \left(z \cdot \frac{z}{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x + \frac{z \cdot \left(z \cdot 3\right)}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 93.9% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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 97.9%

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

3. Taylor expanded in x around inf 93.3%

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

5. Simplified 93.3%

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

   1. unpow2 93.3%

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

   2. associate-/l* 93.3%

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

7. Applied egg-rr 93.3%

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

8. Final simplification 93.3%

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

Alternative 8: 77.5% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.9%

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

3. Taylor expanded in x around inf 78.4%

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

4. Taylor expanded in x around inf 77.8%

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

5. Final simplification 77.8%

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

Alternative 9: 53.3% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.9%

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

3. Taylor expanded in y around inf 95.5%

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

5. Simplified 95.5%

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

6. Taylor expanded in x around inf 54.4%

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

7. Final simplification 54.4%

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

Developer target: 98.5% 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 2024076 
(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)))