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

Percentage Accurate: 98.1% → 99.2%

Time: 7.4s

Alternatives: 9

Speedup: 1.0×

• 33.9% 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: 98.1% 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: 99.2% 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 98.7%

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

   1. +-commutative 98.7%

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

   2. fma-def 98.8%

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

   3. associate-+l+ 98.8%

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

   4. fma-def 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: 99.2% 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 98.7%

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

   1. associate-+l+ 98.7%

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

   2. associate-+l+ 98.7%

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

   3. fma-def 99.9%

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

   4. associate-+r+ 99.9%

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

   5. distribute-lft-out 99.9%

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

   6. distribute-lft-out 99.9%

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

   7. remove-double-neg 99.9%

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

   8. unsub-neg 99.9%

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

   9. count-2 99.9%

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

   10. neg-mul-1 99.9%

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

   11. distribute-rgt-out-- 99.9%

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

   12. metadata-eval 99.9%

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

3. Simplified 99.9%

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

5. Final simplification 99.9%

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

Alternative 3: 99.1% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.7%

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

   1. associate-+l+ 98.7%

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

   2. associate-+l+ 98.7%

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

   3. fma-def 99.9%

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

   4. *-lft-identity 99.9%

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

   5. metadata-eval 99.9%

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

   6. count-2 99.9%

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

   7. distribute-rgt-out 99.9%

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

   8. metadata-eval 99.9%

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

   9. metadata-eval 99.9%

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

3. Simplified 99.9%

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

5. Final simplification 99.9%

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

Alternative 4: 86.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+28}:\\ \;\;\;\;z \cdot z + \left(z \cdot z + x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot z\right) \cdot 3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 5e+28) (+ (* z z) (+ (* z z) (* x y))) (* (* z z) 3.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 5e+28) {
		tmp = (z * z) + ((z * z) + (x * y));
	} else {
		tmp = (z * z) * 3.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z * z) <= 5d+28) then
        tmp = (z * z) + ((z * z) + (x * y))
    else
        tmp = (z * z) * 3.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 5e+28) {
		tmp = (z * z) + ((z * z) + (x * y));
	} else {
		tmp = (z * z) * 3.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z * z) <= 5e+28:
		tmp = (z * z) + ((z * z) + (x * y))
	else:
		tmp = (z * z) * 3.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 5e+28)
		tmp = Float64(Float64(z * z) + Float64(Float64(z * z) + Float64(x * y)));
	else
		tmp = Float64(Float64(z * z) * 3.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 5e+28)
		tmp = (z * z) + ((z * z) + (x * y));
	else
		tmp = (z * z) * 3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 5e+28], N[(N[(z * z), $MachinePrecision] + N[(N[(z * z), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(z * z), $MachinePrecision] * 3.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 4.99999999999999957e28

   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 88.7%

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

   if 4.99999999999999957e28 < (*.f64 z z)

   1. Initial program 97.1%

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

   3. Taylor expanded in x around 0 90.2%

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

      1. distribute-lft1-in 90.2%

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

      2. metadata-eval 90.2%

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

      3. *-commutative 90.2%

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

   5. Simplified 90.2%

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

      1. unpow2 90.2%

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

   7. Applied egg-rr 90.2%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+28}:\\ \;\;\;\;z \cdot z + \left(z \cdot z + x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot z\right) \cdot 3\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Final simplification 98.7%

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

Alternative 6: 86.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+28}:\\ \;\;\;\;z \cdot z + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot z\right) \cdot 3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 5e+28) (+ (* z z) (* x y)) (* (* z z) 3.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 5e+28) {
		tmp = (z * z) + (x * y);
	} else {
		tmp = (z * z) * 3.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z * z) <= 5d+28) then
        tmp = (z * z) + (x * y)
    else
        tmp = (z * z) * 3.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 5e+28) {
		tmp = (z * z) + (x * y);
	} else {
		tmp = (z * z) * 3.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z * z) <= 5e+28:
		tmp = (z * z) + (x * y)
	else:
		tmp = (z * z) * 3.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 5e+28)
		tmp = Float64(Float64(z * z) + Float64(x * y));
	else
		tmp = Float64(Float64(z * z) * 3.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 5e+28)
		tmp = (z * z) + (x * y);
	else
		tmp = (z * z) * 3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 5e+28], N[(N[(z * z), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision], N[(N[(z * z), $MachinePrecision] * 3.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 4.99999999999999957e28

   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 88.7%

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

   4. Taylor expanded in x around inf 88.4%

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

   if 4.99999999999999957e28 < (*.f64 z z)

   1. Initial program 97.1%

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

   3. Taylor expanded in x around 0 90.2%

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

      1. distribute-lft1-in 90.2%

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

      2. metadata-eval 90.2%

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

      3. *-commutative 90.2%

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

   5. Simplified 90.2%

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

      1. unpow2 90.2%

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

   7. Applied egg-rr 90.2%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+28}:\\ \;\;\;\;z \cdot z + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot z\right) \cdot 3\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+28}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot z\right) \cdot 3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 5e+28) (* x y) (* (* z z) 3.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 5e+28) {
		tmp = x * y;
	} else {
		tmp = (z * z) * 3.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z * z) <= 5d+28) then
        tmp = x * y
    else
        tmp = (z * z) * 3.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 5e+28) {
		tmp = x * y;
	} else {
		tmp = (z * z) * 3.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z * z) <= 5e+28:
		tmp = x * y
	else:
		tmp = (z * z) * 3.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 5e+28)
		tmp = Float64(x * y);
	else
		tmp = Float64(Float64(z * z) * 3.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 5e+28)
		tmp = x * y;
	else
		tmp = (z * z) * 3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 5e+28], N[(x * y), $MachinePrecision], N[(N[(z * z), $MachinePrecision] * 3.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 4.99999999999999957e28

   1. Initial program 99.9%

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

      1. associate-+l+ 99.9%

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

      2. associate-+l+ 99.9%

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

      3. fma-def 99.9%

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

      4. associate-+r+ 99.9%

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

      5. distribute-lft-out 99.9%

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

      6. distribute-lft-out 99.9%

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

      7. remove-double-neg 99.9%

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

      8. unsub-neg 99.9%

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

      9. count-2 99.9%

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

      10. neg-mul-1 99.9%

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

      11. distribute-rgt-out-- 99.9%

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

      12. metadata-eval 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, z \cdot \left(z \cdot 3\right)\right)} \]
   4. Add Preprocessing
   5. 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 48.9%

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

      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 86.6%

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

   if 4.99999999999999957e28 < (*.f64 z z)

   1. Initial program 97.1%

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

   3. Taylor expanded in x around 0 90.2%

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

      1. distribute-lft1-in 90.2%

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

      2. metadata-eval 90.2%

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

      3. *-commutative 90.2%

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

   5. Simplified 90.2%

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

      1. unpow2 90.2%

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

   7. Applied egg-rr 90.2%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+28}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot z\right) \cdot 3\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 69.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 4.2 \cdot 10^{+16}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 4.2e+16) (* x y) (* z (* z 3.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 4.2e+16) {
		tmp = x * y;
	} else {
		tmp = z * (z * 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= 4.2d+16) then
        tmp = x * y
    else
        tmp = z * (z * 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= 4.2e+16) {
		tmp = x * y;
	} else {
		tmp = z * (z * 3.0);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 4.2e+16:
		tmp = x * y
	else:
		tmp = z * (z * 3.0)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 4.2e+16)
		tmp = Float64(x * y);
	else
		tmp = Float64(z * Float64(z * 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 4.2e+16)
		tmp = x * y;
	else
		tmp = z * (z * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 4.2e+16], N[(x * y), $MachinePrecision], N[(z * N[(z * 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 4.2e16

   1. Initial program 98.4%

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

      1. associate-+l+ 98.4%

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

      2. associate-+l+ 98.4%

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

      3. fma-def 99.9%

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

      4. associate-+r+ 99.9%

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

      5. distribute-lft-out 99.9%

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

      6. distribute-lft-out 99.9%

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

      7. remove-double-neg 99.9%

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

      8. unsub-neg 99.9%

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

      9. count-2 99.9%

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

      10. neg-mul-1 99.9%

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

      11. distribute-rgt-out-- 99.9%

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

      12. metadata-eval 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, z \cdot \left(z \cdot 3\right)\right)} \]
   4. Add Preprocessing
   5. 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 99.8%

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

      5. sqrt-prod 35.8%

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

      6. add-sqr-sqrt 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in x around inf 67.4%

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

   if 4.2e16 < z

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 91.2%

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

      1. distribute-lft1-in 91.2%

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

      2. metadata-eval 91.2%

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

      3. *-commutative 91.2%

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

   5. Simplified 91.2%

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

      1. metadata-eval 91.2%

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

      2. sqrt-pow1 69.2%

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

      3. metadata-eval 69.2%

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

      4. sqrt-prod 69.3%

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

      5. pow1/2 69.3%

         \[\leadsto \color{blue}{{\left({z}^{4} \cdot 9\right)}^{0.5}} \]

   7. Applied egg-rr 69.3%

      \[\leadsto \color{blue}{{\left({z}^{4} \cdot 9\right)}^{0.5}} \]
   8. Step-by-step derivation

      1. unpow-prod-down 69.2%

         \[\leadsto \color{blue}{{\left({z}^{4}\right)}^{0.5} \cdot {9}^{0.5}} \]

      2. pow-pow 91.2%

         \[\leadsto \color{blue}{{z}^{\left(4 \cdot 0.5\right)}} \cdot {9}^{0.5} \]

      3. metadata-eval 91.2%

         \[\leadsto {z}^{\color{blue}{2}} \cdot {9}^{0.5} \]

      4. pow2 91.2%

         \[\leadsto \color{blue}{\left(z \cdot z\right)} \cdot {9}^{0.5} \]

      5. unpow1/2 91.2%

         \[\leadsto \left(z \cdot z\right) \cdot \color{blue}{\sqrt{9}} \]

      6. metadata-eval 91.2%

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

      7. associate-*l* 91.2%

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

      8. *-commutative 91.2%

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

   9. Applied egg-rr 91.2%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 4.2 \cdot 10^{+16}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \end{array} \]
5. 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 98.7%

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

   1. associate-+l+ 98.7%

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

   2. associate-+l+ 98.7%

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

   3. fma-def 99.9%

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

   4. associate-+r+ 99.9%

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

   5. distribute-lft-out 99.9%

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

   6. distribute-lft-out 99.9%

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

   7. remove-double-neg 99.9%

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

   8. unsub-neg 99.9%

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

   9. count-2 99.9%

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

   10. neg-mul-1 99.9%

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

   11. distribute-rgt-out-- 99.9%

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

   12. metadata-eval 99.9%

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

3. Simplified 99.9%

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

   1. add-sqr-sqrt 99.8%

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

   2. pow2 99.8%

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

   3. associate-*r* 99.8%

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

   4. sqrt-prod 99.8%

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

   5. sqrt-prod 49.4%

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

   6. add-sqr-sqrt 99.8%

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

6. Applied egg-rr 99.8%

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

7. Taylor expanded in x around inf 55.6%

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

8. Final simplification 55.6%

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

Developer target: 98.1% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = ((3.0d0 * z) * z) + (y * x)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024026 
(FPCore (x y z)
  :name "Linear.Quaternion:$c/ from linear-1.19.1.3, A"
  :precision binary64

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

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