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

Percentage Accurate: 98.6% → 99.6%

Time: 3.9s

Alternatives: 6

Speedup: 1.7×

• 33.6% 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: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.1× 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 99.1%

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

   1. +-commutative 99.1%

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

   2. fma-def 99.2%

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

   3. associate-+l+ 99.2%

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

   4. fma-def 99.5%

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

   5. distribute-lft-out 99.5%

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

3. Simplified 99.5%

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

4. Final simplification 99.5%

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

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 99.1%

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

   1. associate-+l+ 99.1%

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

   2. associate-+l+ 99.1%

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

   3. fma-def 99.5%

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

   4. count-2 99.5%

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

   5. distribute-rgt1-in 99.5%

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

   6. *-commutative 99.5%

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

   7. associate-*l* 99.5%

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

   8. metadata-eval 99.5%

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

3. Simplified 99.5%

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

4. Final simplification 99.5%

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

Alternative 3: 83.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{-108}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;z \cdot z \leq 5 \cdot 10^{+76}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \mathbf{elif}\;z \cdot z \leq 10^{+118}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(z \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 2e-108)
   (* x y)
   (if (<= (* z z) 5e+76)
     (* z (* z 3.0))
     (if (<= (* z z) 1e+118) (* x y) (* 3.0 (* z z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e-108) {
		tmp = x * y;
	} else if ((z * z) <= 5e+76) {
		tmp = z * (z * 3.0);
	} else if ((z * z) <= 1e+118) {
		tmp = x * y;
	} else {
		tmp = 3.0 * (z * z);
	}
	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-108) then
        tmp = x * y
    else if ((z * z) <= 5d+76) then
        tmp = z * (z * 3.0d0)
    else if ((z * z) <= 1d+118) then
        tmp = x * y
    else
        tmp = 3.0d0 * (z * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e-108) {
		tmp = x * y;
	} else if ((z * z) <= 5e+76) {
		tmp = z * (z * 3.0);
	} else if ((z * z) <= 1e+118) {
		tmp = x * y;
	} else {
		tmp = 3.0 * (z * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z * z) <= 2e-108:
		tmp = x * y
	elif (z * z) <= 5e+76:
		tmp = z * (z * 3.0)
	elif (z * z) <= 1e+118:
		tmp = x * y
	else:
		tmp = 3.0 * (z * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 2e-108)
		tmp = Float64(x * y);
	elseif (Float64(z * z) <= 5e+76)
		tmp = Float64(z * Float64(z * 3.0));
	elseif (Float64(z * z) <= 1e+118)
		tmp = Float64(x * y);
	else
		tmp = Float64(3.0 * Float64(z * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 2e-108)
		tmp = x * y;
	elseif ((z * z) <= 5e+76)
		tmp = z * (z * 3.0);
	elseif ((z * z) <= 1e+118)
		tmp = x * y;
	else
		tmp = 3.0 * (z * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 2e-108], N[(x * y), $MachinePrecision], If[LessEqual[N[(z * z), $MachinePrecision], 5e+76], N[(z * N[(z * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * z), $MachinePrecision], 1e+118], N[(x * y), $MachinePrecision], N[(3.0 * N[(z * z), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z z) < 2.00000000000000008e-108 or 4.99999999999999991e76 < (*.f64 z z) < 9.99999999999999967e117

   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 0 100.0%

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

      1. unpow2 100.0%

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

      2. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-+r+ 100.0%

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

      5. unpow2 100.0%

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

      6. distribute-rgt1-in 100.0%

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

      7. metadata-eval 100.0%

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

      8. *-commutative 100.0%

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

      9. associate-*r* 100.0%

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

      10. fma-def 100.0%

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

      11. *-commutative 100.0%

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

   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 93.3%

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

   if 2.00000000000000008e-108 < (*.f64 z z) < 4.99999999999999991e76

   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 70.1%

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

      1. unpow2 70.1%

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

      2. unpow2 70.1%

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

      3. distribute-rgt1-in 70.1%

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

      4. metadata-eval 70.1%

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

      5. *-commutative 70.1%

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

      6. associate-*r* 70.1%

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

   4. Simplified 70.1%

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

   if 9.99999999999999967e117 < (*.f64 z z)

   1. Initial program 97.9%

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

   2. Taylor expanded in x around 0 92.5%

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

      1. unpow2 92.5%

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

      2. *-commutative 92.5%

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

      3. associate-*l* 92.5%

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

      4. *-commutative 92.5%

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

      5. count-2 92.5%

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

   4. Simplified 92.5%

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

      1. distribute-lft-out 92.4%

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

      2. count-2 92.4%

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

      3. *-un-lft-identity 92.4%

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

      4. distribute-rgt-out 92.4%

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

      5. metadata-eval 92.4%

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

      6. associate-*r* 92.5%

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

   6. Applied egg-rr 92.5%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{-108}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;z \cdot z \leq 5 \cdot 10^{+76}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \mathbf{elif}\;z \cdot z \leq 10^{+118}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(z \cdot z\right)\\ \end{array} \]

Alternative 4: 84.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{+59} \lor \neg \left(z \leq -3.8 \cdot 10^{+44} \lor \neg \left(z \leq -5.4 \cdot 10^{-26}\right) \land z \leq 1.12 \cdot 10^{-51}\right):\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.1e+59)
         (not
          (or (<= z -3.8e+44) (and (not (<= z -5.4e-26)) (<= z 1.12e-51)))))
   (* z (* z 3.0))
   (* x y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.1e+59) || !((z <= -3.8e+44) || (!(z <= -5.4e-26) && (z <= 1.12e-51)))) {
		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 <= (-1.1d+59)) .or. (.not. (z <= (-3.8d+44)) .or. (.not. (z <= (-5.4d-26))) .and. (z <= 1.12d-51))) 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 <= -1.1e+59) || !((z <= -3.8e+44) || (!(z <= -5.4e-26) && (z <= 1.12e-51)))) {
		tmp = z * (z * 3.0);
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.1e+59) or not ((z <= -3.8e+44) or (not (z <= -5.4e-26) and (z <= 1.12e-51))):
		tmp = z * (z * 3.0)
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.1e+59) || !((z <= -3.8e+44) || (!(z <= -5.4e-26) && (z <= 1.12e-51))))
		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 <= -1.1e+59) || ~(((z <= -3.8e+44) || (~((z <= -5.4e-26)) && (z <= 1.12e-51)))))
		tmp = z * (z * 3.0);
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.1e+59], N[Not[Or[LessEqual[z, -3.8e+44], And[N[Not[LessEqual[z, -5.4e-26]], $MachinePrecision], LessEqual[z, 1.12e-51]]]], $MachinePrecision]], N[(z * N[(z * 3.0), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.1e59 or -3.8000000000000002e44 < z < -5.39999999999999963e-26 or 1.11999999999999998e-51 < z

   1. Initial program 98.4%

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

   2. Taylor expanded in x around 0 86.5%

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

      1. unpow2 86.5%

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

      2. unpow2 86.5%

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

      3. distribute-rgt1-in 86.5%

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

      4. metadata-eval 86.5%

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

      5. *-commutative 86.5%

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

      6. associate-*r* 86.5%

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

   4. Simplified 86.5%

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

   if -1.1e59 < z < -3.8000000000000002e44 or -5.39999999999999963e-26 < z < 1.11999999999999998e-51

   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 0 100.0%

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

      1. unpow2 100.0%

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

      2. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-+r+ 100.0%

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

      5. unpow2 100.0%

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

      6. distribute-rgt1-in 100.0%

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

      7. metadata-eval 100.0%

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

      8. *-commutative 100.0%

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

      9. associate-*r* 100.0%

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

      10. fma-def 100.0%

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

      11. *-commutative 100.0%

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

   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 93.3%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{+59} \lor \neg \left(z \leq -3.8 \cdot 10^{+44} \lor \neg \left(z \leq -5.4 \cdot 10^{-26}\right) \land z \leq 1.12 \cdot 10^{-51}\right):\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 5: 98.6% 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 99.1%

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

   1. associate-+l+ 99.1%

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

   2. associate-+l+ 99.1%

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

   3. fma-def 99.5%

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

   4. count-2 99.5%

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

   5. distribute-rgt1-in 99.5%

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

   6. *-commutative 99.5%

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

   7. associate-*l* 99.5%

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

   8. metadata-eval 99.5%

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

3. Simplified 99.5%

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

   1. fma-udef 99.1%

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

   2. +-commutative 99.1%

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

5. Applied egg-rr 99.1%

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

6. Final simplification 99.1%

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

Alternative 6: 52.1% 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 99.1%

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

2. Taylor expanded in x around 0 99.1%

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

   1. unpow2 99.1%

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

   2. +-commutative 99.1%

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

   3. *-commutative 99.1%

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

   4. associate-+r+ 99.1%

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

   5. unpow2 99.1%

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

   6. distribute-rgt1-in 99.1%

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

   7. metadata-eval 99.1%

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

   8. *-commutative 99.1%

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

   9. associate-*r* 99.1%

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

   10. fma-def 99.5%

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

   11. *-commutative 99.5%

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

4. Simplified 99.5%

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

5. Taylor expanded in z around 0 54.3%

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

6. Final simplification 54.3%

   \[\leadsto x \cdot y \]

Developer target: 98.6% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023193 
(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)))