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

Percentage Accurate: 98.2% → 99.2%

Time: 4.2s

Alternatives: 5

Speedup: 1.7×

• 34.1% 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: 98.2% 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

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

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 99.0%

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

   1. associate-+l+ 99.0%

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

   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. metadata-eval 99.5%

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

   7. metadata-eval 99.5%

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

   8. metadata-eval 99.5%

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

   9. *-commutative 99.5%

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

   10. associate-*l* 99.6%

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

   11. metadata-eval 99.6%

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

   12. metadata-eval 99.6%

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

3. Simplified 99.6%

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

4. Final simplification 99.6%

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

Alternative 2: 84.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 8.6 \cdot 10^{-89} \lor \neg \left(z \cdot z \leq 1.75 \cdot 10^{-74}\right) \land z \cdot z \leq 8 \cdot 10^{-12}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(z \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= (* z z) 8.6e-89)
         (and (not (<= (* z z) 1.75e-74)) (<= (* z z) 8e-12)))
   (* x y)
   (* 3.0 (* z z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (((z * z) <= 8.6e-89) || (!((z * z) <= 1.75e-74) && ((z * z) <= 8e-12))) {
		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) <= 8.6d-89) .or. (.not. ((z * z) <= 1.75d-74)) .and. ((z * z) <= 8d-12)) 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) <= 8.6e-89) || (!((z * z) <= 1.75e-74) && ((z * z) <= 8e-12))) {
		tmp = x * y;
	} else {
		tmp = 3.0 * (z * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if ((z * z) <= 8.6e-89) or (not ((z * z) <= 1.75e-74) and ((z * z) <= 8e-12)):
		tmp = x * y
	else:
		tmp = 3.0 * (z * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((Float64(z * z) <= 8.6e-89) || (!(Float64(z * z) <= 1.75e-74) && (Float64(z * z) <= 8e-12)))
		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) <= 8.6e-89) || (~(((z * z) <= 1.75e-74)) && ((z * z) <= 8e-12)))
		tmp = x * y;
	else
		tmp = 3.0 * (z * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[N[(z * z), $MachinePrecision], 8.6e-89], And[N[Not[LessEqual[N[(z * z), $MachinePrecision], 1.75e-74]], $MachinePrecision], LessEqual[N[(z * z), $MachinePrecision], 8e-12]]], N[(x * y), $MachinePrecision], N[(3.0 * N[(z * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 8.59999999999999974e-89 or 1.75000000000000007e-74 < (*.f64 z z) < 7.99999999999999984e-12

   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%

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

      1. unpow2 99.9%

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

      2. +-commutative 99.9%

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

      3. unpow2 99.9%

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

      4. *-commutative 99.9%

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

      5. associate-*r* 99.9%

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

      6. *-commutative 99.9%

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

      7. associate-+r+ 99.9%

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

      8. distribute-lft-out 100.0%

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

      9. fma-def 100.0%

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

      10. *-commutative 100.0%

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

      11. distribute-rgt1-in 100.0%

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

      12. metadata-eval 100.0%

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

      13. *-commutative 100.0%

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

      14. *-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 87.3%

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

   if 8.59999999999999974e-89 < (*.f64 z z) < 1.75000000000000007e-74 or 7.99999999999999984e-12 < (*.f64 z z)

   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 86.4%

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

      1. unpow2 86.4%

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

      2. unpow2 86.4%

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

      3. distribute-rgt1-in 86.4%

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

      4. metadata-eval 86.4%

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

   4. Simplified 86.4%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 8.6 \cdot 10^{-89} \lor \neg \left(z \cdot z \leq 1.75 \cdot 10^{-74}\right) \land z \cdot z \leq 8 \cdot 10^{-12}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(z \cdot z\right)\\ \end{array} \]

Alternative 3: 84.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-91} \lor \neg \left(z \cdot z \leq 2 \cdot 10^{-74}\right) \land z \cdot z \leq 4 \cdot 10^{-15}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= (* z z) 5e-91) (and (not (<= (* z z) 2e-74)) (<= (* z z) 4e-15)))
   (* x y)
   (* z (* z 3.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (((z * z) <= 5e-91) || (!((z * z) <= 2e-74) && ((z * z) <= 4e-15))) {
		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-91) .or. (.not. ((z * z) <= 2d-74)) .and. ((z * z) <= 4d-15)) 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-91) || (!((z * z) <= 2e-74) && ((z * z) <= 4e-15))) {
		tmp = x * y;
	} else {
		tmp = z * (z * 3.0);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if ((z * z) <= 5e-91) or (not ((z * z) <= 2e-74) and ((z * z) <= 4e-15)):
		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-91) || (!(Float64(z * z) <= 2e-74) && (Float64(z * z) <= 4e-15)))
		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 * z) <= 5e-91) || (~(((z * z) <= 2e-74)) && ((z * z) <= 4e-15)))
		tmp = x * y;
	else
		tmp = z * (z * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[N[(z * z), $MachinePrecision], 5e-91], And[N[Not[LessEqual[N[(z * z), $MachinePrecision], 2e-74]], $MachinePrecision], LessEqual[N[(z * z), $MachinePrecision], 4e-15]]], N[(x * y), $MachinePrecision], N[(z * N[(z * 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 4.99999999999999997e-91 or 1.99999999999999992e-74 < (*.f64 z z) < 4.0000000000000003e-15

   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%

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

      1. unpow2 99.9%

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

      2. +-commutative 99.9%

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

      3. unpow2 99.9%

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

      4. *-commutative 99.9%

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

      5. associate-*r* 99.9%

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

      6. *-commutative 99.9%

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

      7. associate-+r+ 99.9%

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

      8. distribute-lft-out 100.0%

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

      9. fma-def 100.0%

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

      10. *-commutative 100.0%

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

      11. distribute-rgt1-in 100.0%

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

      12. metadata-eval 100.0%

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

      13. *-commutative 100.0%

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

      14. *-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 87.3%

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

   if 4.99999999999999997e-91 < (*.f64 z z) < 1.99999999999999992e-74 or 4.0000000000000003e-15 < (*.f64 z z)

   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 86.4%

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

      1. unpow2 86.4%

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

      2. unpow2 86.4%

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

      3. distribute-rgt1-in 86.4%

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

      4. metadata-eval 86.4%

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

      5. *-commutative 86.4%

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

      6. associate-*r* 86.5%

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

   4. Simplified 86.5%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-91} \lor \neg \left(z \cdot z \leq 2 \cdot 10^{-74}\right) \land z \cdot z \leq 4 \cdot 10^{-15}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \end{array} \]

Alternative 4: 98.2% 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 99.0%

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

   1. associate-+l+ 99.0%

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

   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. metadata-eval 99.5%

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

   7. metadata-eval 99.5%

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

   8. metadata-eval 99.5%

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

   9. *-commutative 99.5%

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

   10. associate-*l* 99.6%

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

   11. metadata-eval 99.6%

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

   12. metadata-eval 99.6%

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

3. Simplified 99.6%

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

   1. fma-udef 99.2%

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

   2. +-commutative 99.2%

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

5. Applied egg-rr 99.2%

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

6. Final simplification 99.2%

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

Alternative 5: 52.5% 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 99.0%

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

2. Taylor expanded in x around 0 99.0%

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

   1. unpow2 99.0%

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

   2. +-commutative 99.0%

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

   3. unpow2 99.0%

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

   4. *-commutative 99.0%

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

   5. associate-*r* 99.0%

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

   6. *-commutative 99.0%

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

   7. associate-+r+ 99.1%

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

   8. distribute-lft-out 99.2%

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

   9. fma-def 99.6%

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

   10. *-commutative 99.6%

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

   11. distribute-rgt1-in 99.6%

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

   12. metadata-eval 99.6%

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

   13. *-commutative 99.6%

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

   14. *-commutative 99.6%

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

4. Simplified 99.6%

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

5. Taylor expanded in z around 0 50.6%

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

6. Final simplification 50.6%

   \[\leadsto x \cdot y \]

Developer target: 98.2% 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 2023268 
(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)))