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

Percentage Accurate: 98.5% → 99.5%

Time: 4.1s

Alternatives: 6

Speedup: 1.1×

• 34.6% 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: 99.5% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 97.1%

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

   1. +-commutative 97.1%

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

   2. fma-def 97.2%

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

   3. associate-+l+ 97.2%

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

   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. distribute-lft-out 99.2%

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

3. Simplified 99.2%

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

4. Final simplification 99.2%

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

Alternative 2: 99.5% 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 97.1%

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

   1. associate-+l+ 97.1%

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

   2. associate-+l+ 97.1%

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

   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. *-commutative 99.1%

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

   7. associate-*l* 99.1%

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

   8. 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: 99.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+281}:\\ \;\;\;\;x \cdot y + 3 \cdot \left(z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 2e+281) (+ (* x y) (* 3.0 (* z z))) (* z (* z 3.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e+281) {
		tmp = (x * y) + (3.0 * (z * z));
	} 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) <= 2d+281) then
        tmp = (x * y) + (3.0d0 * (z * z))
    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) <= 2e+281) {
		tmp = (x * y) + (3.0 * (z * z));
	} else {
		tmp = z * (z * 3.0);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z * z) <= 2e+281:
		tmp = (x * y) + (3.0 * (z * z))
	else:
		tmp = z * (z * 3.0)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 2e+281)
		tmp = Float64(Float64(x * y) + Float64(3.0 * Float64(z * z)));
	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 * z) <= 2e+281)
		tmp = (x * y) + (3.0 * (z * z));
	else
		tmp = z * (z * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 2e+281], N[(N[(x * y), $MachinePrecision] + N[(3.0 * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * N[(z * 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 2.0000000000000001e281

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 99.8%

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

      1. unpow2 99.8%

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

      2. +-commutative 99.8%

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

      3. *-commutative 99.8%

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

      4. associate-+r+ 99.8%

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

      5. unpow2 99.8%

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

      6. distribute-rgt1-in 99.8%

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

      7. metadata-eval 99.8%

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

      8. *-commutative 99.8%

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

      9. associate-*r* 99.9%

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

      10. fma-def 99.9%

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

      11. *-commutative 99.9%

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

   4. Simplified 99.9%

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

      1. *-commutative 99.9%

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

      2. fma-def 99.9%

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

      3. +-commutative 99.9%

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

      4. associate-*r* 99.8%

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

      5. *-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

   if 2.0000000000000001e281 < (*.f64 z z)

   1. Initial program 87.9%

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

   2. Taylor expanded in x around 0 96.5%

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

      1. unpow2 96.5%

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

      2. unpow2 96.5%

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

      3. distribute-rgt1-in 96.5%

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

      4. metadata-eval 96.5%

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

      5. *-commutative 96.5%

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

      6. associate-*r* 96.5%

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

   4. Simplified 96.5%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+281}:\\ \;\;\;\;x \cdot y + 3 \cdot \left(z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \end{array} \]

Alternative 4: 99.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(z \cdot 3\right)\\ \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+281}:\\ \;\;\;\;t_0 + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (* z 3.0)))) (if (<= (* z z) 2e+281) (+ t_0 (* x y)) t_0)))

C

double code(double x, double y, double z) {
	double t_0 = z * (z * 3.0);
	double tmp;
	if ((z * z) <= 2e+281) {
		tmp = t_0 + (x * y);
	} else {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = z * (z * 3.0d0)
    if ((z * z) <= 2d+281) then
        tmp = t_0 + (x * y)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = z * (z * 3.0);
	double tmp;
	if ((z * z) <= 2e+281) {
		tmp = t_0 + (x * y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * (z * 3.0)
	tmp = 0
	if (z * z) <= 2e+281:
		tmp = t_0 + (x * y)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(z * 3.0))
	tmp = 0.0
	if (Float64(z * z) <= 2e+281)
		tmp = Float64(t_0 + Float64(x * y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * (z * 3.0);
	tmp = 0.0;
	if ((z * z) <= 2e+281)
		tmp = t_0 + (x * y);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(z * 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(z * z), $MachinePrecision], 2e+281], N[(t$95$0 + N[(x * y), $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 2.0000000000000001e281

   1. Initial program 99.8%

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

      1. associate-+l+ 99.8%

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

      2. associate-+l+ 99.8%

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

      3. fma-def 99.8%

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

      4. count-2 99.8%

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

      5. distribute-rgt1-in 99.8%

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

      6. *-commutative 99.8%

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

      7. associate-*l* 99.9%

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

      8. 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. Step-by-step derivation

      1. fma-udef 99.9%

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

      2. +-commutative 99.9%

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

   5. Applied egg-rr 99.9%

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

   if 2.0000000000000001e281 < (*.f64 z z)

   1. Initial program 87.9%

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

   2. Taylor expanded in x around 0 96.5%

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

      1. unpow2 96.5%

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

      2. unpow2 96.5%

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

      3. distribute-rgt1-in 96.5%

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

      4. metadata-eval 96.5%

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

      5. *-commutative 96.5%

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

      6. associate-*r* 96.5%

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

   4. Simplified 96.5%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+281}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right) + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \end{array} \]

Alternative 5: 84.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{-65} \lor \neg \left(z \leq 2.15 \cdot 10^{+15}\right):\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1e-65) (not (<= z 2.15e+15))) (* z (* z 3.0)) (* x y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1e-65) || !(z <= 2.15e+15)) {
		tmp = z * (z * 3.0);
	} else {
		tmp = x * 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 ((z <= (-1d-65)) .or. (.not. (z <= 2.15d+15))) then
        tmp = z * (z * 3.0d0)
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1e-65) || !(z <= 2.15e+15)) {
		tmp = z * (z * 3.0);
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1e-65) or not (z <= 2.15e+15):
		tmp = z * (z * 3.0)
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1e-65) || !(z <= 2.15e+15))
		tmp = Float64(z * Float64(z * 3.0));
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1e-65) || ~((z <= 2.15e+15)))
		tmp = z * (z * 3.0);
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1e-65], N[Not[LessEqual[z, 2.15e+15]], $MachinePrecision]], N[(z * N[(z * 3.0), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -9.99999999999999923e-66 or 2.15e15 < z

   1. Initial program 94.2%

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

   2. Taylor expanded in x around 0 85.4%

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

      1. unpow2 85.4%

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

      2. unpow2 85.4%

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

      3. distribute-rgt1-in 85.4%

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

      4. metadata-eval 85.4%

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

      5. *-commutative 85.4%

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

      6. associate-*r* 85.5%

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

   4. Simplified 85.5%

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

   if -9.99999999999999923e-66 < z < 2.15e15

   1. Initial program 100.0%

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

      1. associate-+l+ 100.0%

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

      2. associate-+l+ 100.0%

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

      3. fma-def 100.0%

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

      4. count-2 100.0%

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

      5. distribute-rgt1-in 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*l* 99.9%

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

      8. 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. Step-by-step derivation

      1. fma-udef 99.9%

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

      2. +-commutative 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in z around 0 89.5%

      \[\leadsto \color{blue}{y \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{-65} \lor \neg \left(z \leq 2.15 \cdot 10^{+15}\right):\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 6: 51.4% 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 97.1%

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

   1. associate-+l+ 97.1%

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

   2. associate-+l+ 97.1%

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

   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. *-commutative 99.1%

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

   7. associate-*l* 99.1%

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

   8. 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 97.2%

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

   2. +-commutative 97.2%

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

5. Applied egg-rr 97.2%

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

6. Taylor expanded in z around 0 53.7%

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

7. Final simplification 53.7%

   \[\leadsto x \cdot y \]

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 2023185 
(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)))