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

Percentage Accurate: 98.3% → 99.5%

Time: 6.9s

Alternatives: 7

Speedup: 1.7×

• 34.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 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-def 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-def 99.2%

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

   5. count-2 99.2%

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

3. Simplified 99.2%

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

4. Final simplification 99.2%

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

Alternative 2: 99.4% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 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-def 99.1%

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

   4. count-2 99.1%

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

   5. distribute-rgt1-in 99.1%

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

   6. metadata-eval 99.1%

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

   7. metadata-eval 99.1%

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

   8. metadata-eval 99.1%

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

   9. *-commutative 99.1%

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

   10. associate-*l* 99.1%

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

   11. metadata-eval 99.1%

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

   12. metadata-eval 99.1%

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

3. Simplified 99.1%

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

4. Final simplification 99.1%

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

Alternative 3: 98.3% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

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 * 2.0d0)))
end function

Java

public static double code(double x, double y, double z) {
	return (x * y) + ((z * z) + (z * (z * 2.0)));
}

Python

def code(x, y, z):
	return (x * y) + ((z * z) + (z * (z * 2.0)))

Julia

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

MATLAB

function tmp = code(x, y, z)
	tmp = (x * y) + ((z * z) + (z * (z * 2.0)));
end

Wolfram

code[x_, y_, z_] := N[(N[(x * y), $MachinePrecision] + N[(N[(z * z), $MachinePrecision] + N[(z * N[(z * 2.0), $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-def 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-def 99.2%

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

   5. count-2 99.2%

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

3. Simplified 99.2%

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

   1. fma-udef 99.1%

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

   2. +-commutative 99.1%

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

   3. count-2 99.1%

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

   4. fma-def 98.3%

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

   5. associate-+l+ 98.3%

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

   6. *-un-lft-identity 98.3%

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

   7. *-un-lft-identity 98.3%

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

   8. associate-+l+ 98.3%

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

   9. associate-+l+ 98.3%

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

   10. count-2 98.3%

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

   11. *-commutative 98.3%

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

   12. associate-*l* 98.3%

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

5. Applied egg-rr 98.3%

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

6. Final simplification 98.3%

   \[\leadsto x \cdot y + \left(z \cdot z + z \cdot \left(z \cdot 2\right)\right) \]

Alternative 4: 85.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-63}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot z\right) \cdot 3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 5e-63) (* x y) (* (* z z) 3.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 5e-63) {
		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-63) 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-63) {
		tmp = x * y;
	} else {
		tmp = (z * z) * 3.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z * z) <= 5e-63:
		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-63)
		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-63)
		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-63], N[(x * y), $MachinePrecision], N[(N[(z * z), $MachinePrecision] * 3.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 5.0000000000000002e-63

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 99.9%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in z around 0 90.5%

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

   if 5.0000000000000002e-63 < (*.f64 z z)

   1. Initial program 96.8%

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

   2. Taylor expanded in x around 0 84.8%

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

   4. Simplified 84.8%

      \[\leadsto \color{blue}{z \cdot \left(z + z\right)} + z \cdot z \]
   5. Step-by-step derivation

      1. distribute-lft-out 84.8%

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

      2. count-2 84.8%

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

      3. *-un-lft-identity 84.8%

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

      4. distribute-rgt-out 84.8%

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

      5. metadata-eval 84.8%

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

      6. associate-*r* 84.8%

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

   6. Applied egg-rr 84.8%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-63}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot z\right) \cdot 3\\ \end{array} \]

Alternative 5: 98.4% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(N[(z * N[(z * 3.0), $MachinePrecision]), $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. 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-def 99.1%

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

   4. count-2 99.1%

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

   5. distribute-rgt1-in 99.1%

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

   6. metadata-eval 99.1%

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

   7. metadata-eval 99.1%

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

   8. metadata-eval 99.1%

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

   9. *-commutative 99.1%

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

   10. associate-*l* 99.1%

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

   11. metadata-eval 99.1%

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

   12. metadata-eval 99.1%

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

3. Simplified 99.1%

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

   1. fma-udef 98.3%

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

   2. +-commutative 98.3%

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

5. Applied egg-rr 98.3%

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

6. Final simplification 98.3%

   \[\leadsto z \cdot \left(z \cdot 3\right) + x \cdot y \]

Alternative 6: 69.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 2.85 \cdot 10^{-31}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 2.85e-31) (* x y) (* z (* z 3.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 2.85e-31) {
		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 <= 2.85d-31) 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 <= 2.85e-31) {
		tmp = x * y;
	} else {
		tmp = z * (z * 3.0);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 2.85e-31:
		tmp = x * y
	else:
		tmp = z * (z * 3.0)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 2.85e-31)
		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 <= 2.85e-31)
		tmp = x * y;
	else
		tmp = z * (z * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 2.85e-31], N[(x * y), $MachinePrecision], N[(z * N[(z * 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 2.84999999999999997e-31

   1. Initial program 98.8%

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

   2. Taylor expanded in x around 0 98.8%

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

   4. Simplified 99.9%

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

   5. Taylor expanded in z around 0 66.2%

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

   if 2.84999999999999997e-31 < z

   1. Initial program 96.8%

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

   2. Taylor expanded in x around 0 84.7%

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

   4. Simplified 84.6%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.85 \cdot 10^{-31}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \end{array} \]

Alternative 7: 52.7% 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 98.3%

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

2. Taylor expanded in x around 0 98.3%

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

4. Simplified 99.1%

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

5. Taylor expanded in z around 0 54.5%

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

6. Final simplification 54.5%

   \[\leadsto x \cdot y \]

Developer target: 98.4% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023279 
(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)))