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

Percentage Accurate: 98.3% → 99.3%

Time: 7.8s

Alternatives: 5

Speedup: 0.1×

• 34.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_] := N[(y * x + 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+ N/A

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

   2. associate-+l+ N/A

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

   3. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(x \cdot y\right), \color{blue}{\left(z \cdot z + \left(z \cdot z + z \cdot z\right)\right)}\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{z \cdot z} + \left(z \cdot z + z \cdot z\right)\right)\right) \]

   5. count-2 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot z + 2 \cdot \color{blue}{\left(z \cdot z\right)}\right)\right) \]

   6. distribute-rgt1-in N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(2 + 1\right) \cdot \color{blue}{\left(z \cdot z\right)}\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(z \cdot z\right) \cdot \color{blue}{\left(2 + 1\right)}\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(z \cdot z\right), \color{blue}{\left(2 + 1\right)}\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(\color{blue}{2} + 1\right)\right)\right) \]

   10. metadata-eval 98.7%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, z\right), 3\right)\right) \]

3. Simplified 98.7%

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

   1. *-commutative N/A

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

   2. fma-define N/A

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

   3. fma-lowering-fma.f64 N/A

      \[\leadsto \mathsf{fma.f64}\left(y, \color{blue}{x}, \left(\left(z \cdot z\right) \cdot 3\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{fma.f64}\left(y, x, \mathsf{*.f64}\left(\left(z \cdot z\right), 3\right)\right) \]

   5. *-lowering-*.f64 99.9%

      \[\leadsto \mathsf{fma.f64}\left(y, x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, z\right), 3\right)\right) \]

6. Applied egg-rr 99.9%

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

Alternative 2: 84.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 4.5 \cdot 10^{-37}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot z\right) \cdot 3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 4.5e-37) (* y x) (* (* z z) 3.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 4.5e-37) {
		tmp = y * x;
	} 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) <= 4.5d-37) then
        tmp = y * x
    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) <= 4.5e-37) {
		tmp = y * x;
	} else {
		tmp = (z * z) * 3.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z * z) <= 4.5e-37:
		tmp = y * x
	else:
		tmp = (z * z) * 3.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 4.5e-37)
		tmp = Float64(y * x);
	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) <= 4.5e-37)
		tmp = y * x;
	else
		tmp = (z * z) * 3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 4.5e-37], N[(y * x), $MachinePrecision], N[(N[(z * z), $MachinePrecision] * 3.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 4.5000000000000004e-37

   1. Initial program 99.9%

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

      1. associate-+l+ N/A

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

      2. associate-+l+ N/A

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

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(x \cdot y\right), \color{blue}{\left(z \cdot z + \left(z \cdot z + z \cdot z\right)\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{z \cdot z} + \left(z \cdot z + z \cdot z\right)\right)\right) \]

      5. count-2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot z + 2 \cdot \color{blue}{\left(z \cdot z\right)}\right)\right) \]

      6. distribute-rgt1-in N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(2 + 1\right) \cdot \color{blue}{\left(z \cdot z\right)}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(z \cdot z\right) \cdot \color{blue}{\left(2 + 1\right)}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(z \cdot z\right), \color{blue}{\left(2 + 1\right)}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(\color{blue}{2} + 1\right)\right)\right) \]

      10. metadata-eval 100.0%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, z\right), 3\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot y} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 88.7%

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{y}\right) \]

   7. Simplified 88.7%

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

   if 4.5000000000000004e-37 < (*.f64 z z)

   1. Initial program 97.7%

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

      1. associate-+l+ N/A

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

      2. associate-+l+ N/A

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

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(x \cdot y\right), \color{blue}{\left(z \cdot z + \left(z \cdot z + z \cdot z\right)\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{z \cdot z} + \left(z \cdot z + z \cdot z\right)\right)\right) \]

      5. count-2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot z + 2 \cdot \color{blue}{\left(z \cdot z\right)}\right)\right) \]

      6. distribute-rgt1-in N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(2 + 1\right) \cdot \color{blue}{\left(z \cdot z\right)}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(z \cdot z\right) \cdot \color{blue}{\left(2 + 1\right)}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(z \cdot z\right), \color{blue}{\left(2 + 1\right)}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(\color{blue}{2} + 1\right)\right)\right) \]

      10. metadata-eval 97.7%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, z\right), 3\right)\right) \]

   3. Simplified 97.7%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

         \[\leadsto {z}^{2} \cdot \color{blue}{3} \]

      2. unpow2 N/A

         \[\leadsto \left(z \cdot z\right) \cdot 3 \]

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \color{blue}{\left(z \cdot 3\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(z, \left(3 \cdot \color{blue}{z}\right)\right) \]

      6. *-lowering-*.f64 84.0%

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(3, \color{blue}{z}\right)\right) \]

   7. Simplified 84.0%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(z \cdot z\right), \color{blue}{3}\right) \]

      4. *-lowering-*.f64 84.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, z\right), 3\right) \]

   9. Applied egg-rr 84.0%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 4.5 \cdot 10^{-37}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot z\right) \cdot 3\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 68.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 6.5 \cdot 10^{-19}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 6.5e-19) (* y x) (* z (* z 3.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 6.5e-19) {
		tmp = y * x;
	} 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 <= 6.5d-19) then
        tmp = y * x
    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 <= 6.5e-19) {
		tmp = y * x;
	} else {
		tmp = z * (z * 3.0);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 6.5e-19:
		tmp = y * x
	else:
		tmp = z * (z * 3.0)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 6.5e-19)
		tmp = Float64(y * x);
	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 <= 6.5e-19)
		tmp = y * x;
	else
		tmp = z * (z * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 6.5e-19], N[(y * x), $MachinePrecision], N[(z * N[(z * 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 6.5000000000000001e-19

   1. Initial program 98.8%

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

      1. associate-+l+ N/A

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

      2. associate-+l+ N/A

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

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(x \cdot y\right), \color{blue}{\left(z \cdot z + \left(z \cdot z + z \cdot z\right)\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{z \cdot z} + \left(z \cdot z + z \cdot z\right)\right)\right) \]

      5. count-2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot z + 2 \cdot \color{blue}{\left(z \cdot z\right)}\right)\right) \]

      6. distribute-rgt1-in N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(2 + 1\right) \cdot \color{blue}{\left(z \cdot z\right)}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(z \cdot z\right) \cdot \color{blue}{\left(2 + 1\right)}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(z \cdot z\right), \color{blue}{\left(2 + 1\right)}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(\color{blue}{2} + 1\right)\right)\right) \]

      10. metadata-eval 98.8%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, z\right), 3\right)\right) \]

   3. Simplified 98.8%

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

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot y} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 63.0%

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{y}\right) \]

   7. Simplified 63.0%

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

   if 6.5000000000000001e-19 < z

   1. Initial program 98.5%

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

      1. associate-+l+ N/A

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

      2. associate-+l+ N/A

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

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(x \cdot y\right), \color{blue}{\left(z \cdot z + \left(z \cdot z + z \cdot z\right)\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{z \cdot z} + \left(z \cdot z + z \cdot z\right)\right)\right) \]

      5. count-2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot z + 2 \cdot \color{blue}{\left(z \cdot z\right)}\right)\right) \]

      6. distribute-rgt1-in N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(2 + 1\right) \cdot \color{blue}{\left(z \cdot z\right)}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(z \cdot z\right) \cdot \color{blue}{\left(2 + 1\right)}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(z \cdot z\right), \color{blue}{\left(2 + 1\right)}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(\color{blue}{2} + 1\right)\right)\right) \]

      10. metadata-eval 98.5%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, z\right), 3\right)\right) \]

   3. Simplified 98.5%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

         \[\leadsto {z}^{2} \cdot \color{blue}{3} \]

      2. unpow2 N/A

         \[\leadsto \left(z \cdot z\right) \cdot 3 \]

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \color{blue}{\left(z \cdot 3\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(z, \left(3 \cdot \color{blue}{z}\right)\right) \]

      6. *-lowering-*.f64 81.1%

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(3, \color{blue}{z}\right)\right) \]

   7. Simplified 81.1%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 6.5 \cdot 10^{-19}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.3% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. associate-+l+ N/A

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

   3. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(x \cdot y\right), \color{blue}{\left(z \cdot z + \left(z \cdot z + z \cdot z\right)\right)}\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{z \cdot z} + \left(z \cdot z + z \cdot z\right)\right)\right) \]

   5. count-2 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot z + 2 \cdot \color{blue}{\left(z \cdot z\right)}\right)\right) \]

   6. distribute-rgt1-in N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(2 + 1\right) \cdot \color{blue}{\left(z \cdot z\right)}\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(z \cdot z\right) \cdot \color{blue}{\left(2 + 1\right)}\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(z \cdot z\right), \color{blue}{\left(2 + 1\right)}\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(\color{blue}{2} + 1\right)\right)\right) \]

   10. metadata-eval 98.7%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, z\right), 3\right)\right) \]

3. Simplified 98.7%

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

5. Final simplification 98.7%

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

Alternative 5: 53.3% accurate, 5.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return 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 = y * x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. associate-+l+ N/A

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

   3. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(x \cdot y\right), \color{blue}{\left(z \cdot z + \left(z \cdot z + z \cdot z\right)\right)}\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{z \cdot z} + \left(z \cdot z + z \cdot z\right)\right)\right) \]

   5. count-2 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot z + 2 \cdot \color{blue}{\left(z \cdot z\right)}\right)\right) \]

   6. distribute-rgt1-in N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(2 + 1\right) \cdot \color{blue}{\left(z \cdot z\right)}\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(z \cdot z\right) \cdot \color{blue}{\left(2 + 1\right)}\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(z \cdot z\right), \color{blue}{\left(2 + 1\right)}\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(\color{blue}{2} + 1\right)\right)\right) \]

   10. metadata-eval 98.7%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, z\right), 3\right)\right) \]

3. Simplified 98.7%

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

5. Taylor expanded in x around inf

   \[\leadsto \color{blue}{x \cdot y} \]
6. Step-by-step derivation

   1. *-lowering-*.f64 52.2%

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{y}\right) \]

7. Simplified 52.2%

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

8. Final simplification 52.2%

   \[\leadsto y \cdot x \]
9. Add Preprocessing

Developer Target 1: 98.3% 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 2024145 
(FPCore (x y z)
  :name "Linear.Quaternion:$c/ from linear-1.19.1.3, A"
  :precision binary64

  :alt
  (! :herbie-platform default (+ (* (* 3 z) z) (* y x)))

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