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

Percentage Accurate: 98.5% → 98.5%

Time: 8.4s

Alternatives: 8

Speedup: 1.7×

• 34.0% 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: 98.5% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.7%

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

   1. +-commutative 98.7%

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

   2. fma-define 98.7%

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

   3. associate-+l+ 98.7%

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

   4. *-un-lft-identity 98.7%

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

   5. *-un-lft-identity 98.7%

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

   6. distribute-rgt-out 98.7%

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

   7. metadata-eval 98.7%

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

4. Applied egg-rr 98.7%

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

Alternative 2: 98.5% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.7%

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

   1. associate-+l+ 98.7%

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

   2. associate-+l+ 98.7%

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

   3. +-commutative 98.7%

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

   4. count-2 98.7%

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

   5. distribute-lft1-in 98.7%

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

   6. metadata-eval 98.7%

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

   7. metadata-eval 98.7%

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

   8. metadata-eval 98.7%

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

   9. *-commutative 98.7%

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

   10. fma-define 98.7%

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

   11. metadata-eval 98.7%

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

   12. metadata-eval 98.7%

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

3. Simplified 98.7%

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

Alternative 3: 82.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{-104} \lor \neg \left(z \cdot z \leq 2 \cdot 10^{+30}\right) \land z \cdot z \leq 5 \cdot 10^{+129}:\\ \;\;\;\;x \cdot y + z \cdot z\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot z\right) \cdot 3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= (* z z) 1e-104)
         (and (not (<= (* z z) 2e+30)) (<= (* z z) 5e+129)))
   (+ (* x y) (* z z))
   (* (* z z) 3.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if (((z * z) <= 1e-104) || (!((z * z) <= 2e+30) && ((z * z) <= 5e+129))) {
		tmp = (x * y) + (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) <= 1d-104) .or. (.not. ((z * z) <= 2d+30)) .and. ((z * z) <= 5d+129)) then
        tmp = (x * y) + (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) <= 1e-104) || (!((z * z) <= 2e+30) && ((z * z) <= 5e+129))) {
		tmp = (x * y) + (z * z);
	} else {
		tmp = (z * z) * 3.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if ((z * z) <= 1e-104) or (not ((z * z) <= 2e+30) and ((z * z) <= 5e+129)):
		tmp = (x * y) + (z * z)
	else:
		tmp = (z * z) * 3.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((Float64(z * z) <= 1e-104) || (!(Float64(z * z) <= 2e+30) && (Float64(z * z) <= 5e+129)))
		tmp = Float64(Float64(x * y) + Float64(z * z));
	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) <= 1e-104) || (~(((z * z) <= 2e+30)) && ((z * z) <= 5e+129)))
		tmp = (x * y) + (z * z);
	else
		tmp = (z * z) * 3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[N[(z * z), $MachinePrecision], 1e-104], And[N[Not[LessEqual[N[(z * z), $MachinePrecision], 2e+30]], $MachinePrecision], LessEqual[N[(z * z), $MachinePrecision], 5e+129]]], N[(N[(x * y), $MachinePrecision] + N[(z * z), $MachinePrecision]), $MachinePrecision], N[(N[(z * z), $MachinePrecision] * 3.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 9.99999999999999927e-105 or 2e30 < (*.f64 z z) < 5.0000000000000003e129

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 89.1%

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

   if 9.99999999999999927e-105 < (*.f64 z z) < 2e30 or 5.0000000000000003e129 < (*.f64 z z)

   1. Initial program 97.6%

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

   3. Taylor expanded in x around 0 84.5%

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

      1. unpow2 84.5%

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

      2. unpow2 84.5%

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

      3. distribute-lft1-in 84.5%

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

      4. metadata-eval 84.5%

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

      5. *-commutative 84.5%

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

      6. associate-*r* 84.5%

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

   5. Simplified 84.5%

      \[\leadsto \color{blue}{z \cdot \left(z \cdot 3\right)} \]
   6. Step-by-step derivation

      1. associate-*r* 97.7%

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

   7. Applied egg-rr 84.5%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{-104} \lor \neg \left(z \cdot z \leq 2 \cdot 10^{+30}\right) \land z \cdot z \leq 5 \cdot 10^{+129}:\\ \;\;\;\;x \cdot y + z \cdot z\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot z\right) \cdot 3\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-115}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 5e-115) (* x y) (* z (* z 3.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 5e-115) {
		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-115) 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-115) {
		tmp = x * y;
	} else {
		tmp = z * (z * 3.0);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z * z) <= 5e-115:
		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-115)
		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-115)
		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-115], N[(x * y), $MachinePrecision], N[(z * N[(z * 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 5.0000000000000003e-115

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 95.0%

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

   if 5.0000000000000003e-115 < (*.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. Add Preprocessing

   3. Taylor expanded in x around 0 78.9%

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

      1. unpow2 78.9%

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

      2. unpow2 78.9%

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

      3. distribute-lft1-in 78.9%

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

      4. metadata-eval 78.9%

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

      5. *-commutative 78.9%

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

      6. associate-*r* 78.9%

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

   5. Simplified 78.9%

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

Alternative 5: 73.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 8 \cdot 10^{+294}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= (* z z) 8e+294) (* x y) (* z z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 8e+294) {
		tmp = x * y;
	} else {
		tmp = 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) <= 8d+294) then
        tmp = x * y
    else
        tmp = z * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 8e+294) {
		tmp = x * y;
	} else {
		tmp = z * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z * z) <= 8e+294:
		tmp = x * y
	else:
		tmp = z * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 8e+294)
		tmp = Float64(x * y);
	else
		tmp = Float64(z * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 8e+294)
		tmp = x * y;
	else
		tmp = z * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 8e+294], N[(x * y), $MachinePrecision], N[(z * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 8.00000000000000053e294

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf 65.0%

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

   if 8.00000000000000053e294 < (*.f64 z z)

   1. Initial program 95.4%

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

   3. Taylor expanded in x around inf 91.6%

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

   4. Taylor expanded in x around 0 96.2%

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

      1. unpow2 96.2%

         \[\leadsto \color{blue}{z \cdot z} \]

   6. Simplified 96.2%

      \[\leadsto \color{blue}{z \cdot z} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 98.5% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

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) * 3.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.7%

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

   1. associate-+l+ 98.7%

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

   2. associate-+l+ 98.7%

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

   3. +-commutative 98.7%

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

   4. count-2 98.7%

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

   5. distribute-lft1-in 98.7%

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

   6. metadata-eval 98.7%

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

   7. metadata-eval 98.7%

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

   8. metadata-eval 98.7%

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

   9. *-commutative 98.7%

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

   10. fma-define 98.7%

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

   11. metadata-eval 98.7%

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

   12. metadata-eval 98.7%

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

3. Simplified 98.7%

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

   1. fma-undefine 98.7%

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

   2. associate-*l* 98.7%

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

6. Applied egg-rr 98.7%

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

   1. associate-*r* 98.7%

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

8. Applied egg-rr 98.7%

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

9. Final simplification 98.7%

   \[\leadsto x \cdot y + \left(z \cdot z\right) \cdot 3 \]
10. Add Preprocessing

Alternative 7: 98.5% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

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 * 3.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.7%

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

   1. associate-+l+ 98.7%

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

   2. associate-+l+ 98.7%

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

   3. +-commutative 98.7%

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

   4. count-2 98.7%

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

   5. distribute-lft1-in 98.7%

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

   6. metadata-eval 98.7%

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

   7. metadata-eval 98.7%

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

   8. metadata-eval 98.7%

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

   9. *-commutative 98.7%

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

   10. fma-define 98.7%

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

   11. metadata-eval 98.7%

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

   12. metadata-eval 98.7%

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

3. Simplified 98.7%

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

   1. fma-undefine 98.7%

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

   2. associate-*l* 98.7%

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

6. Applied egg-rr 98.7%

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

7. Final simplification 98.7%

   \[\leadsto x \cdot y + z \cdot \left(z \cdot 3\right) \]
8. Add Preprocessing

Alternative 8: 51.8% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ x \cdot y \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in x around inf 50.1%

   \[\leadsto \color{blue}{x \cdot y} \]
4. Add Preprocessing

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 2024097 
(FPCore (x y z)
  :name "Linear.Quaternion:$c/ from linear-1.19.1.3, A"
  :precision binary64

  :alt
  (+ (* (* 3.0 z) z) (* y x))

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