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

Percentage Accurate: 98.4% → 99.0%

Time: 7.4s

Alternatives: 9

Speedup: 1.8×

• 34.9% 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: 98.4% 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: 99.0% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. lift-+.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. associate-+l+ N/A

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

   4. +-commutative N/A

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

   5. count-2 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*r* N/A

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

   8. count-2 N/A

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

   9. lower-fma.f64 N/A

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

   10. count-2 N/A

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

   11. lower-*.f64 99.6

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

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. lift-*.f64 N/A

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

   15. lower-fma.f64 99.6

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

   16. lift-*.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 99.6

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

4. Applied rewrites 99.6%

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

5. Final simplification 99.6%

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

Alternative 2: 84.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-20}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, z + z, z \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 5e-20) (* x y) (fma z (+ z z) (* z z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 5e-20) {
		tmp = x * y;
	} else {
		tmp = fma(z, (z + z), (z * z));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 5e-20)
		tmp = Float64(x * y);
	else
		tmp = fma(z, Float64(z + z), Float64(z * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 5e-20], N[(x * y), $MachinePrecision], N[(z * N[(z + z), $MachinePrecision] + N[(z * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 4.9999999999999999e-20

   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 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in x around 0

      \[\leadsto \left(3 \cdot \frac{{z}^{2}}{y}\right) \cdot y \]
   10. Step-by-step derivation

   11. Applied rewrites 20.0%

       \[\leadsto \left(\frac{z \cdot z}{y} \cdot 3\right) \cdot y \]

   12. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 87.0

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

   14. Applied rewrites 87.0%

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

   if 4.9999999999999999e-20 < (*.f64 z z)

   1. Initial program 98.9%

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

   3. Taylor expanded in x around 0

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

      1. distribute-lft1-in N/A

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

      2. metadata-eval N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 82.3

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

   5. Applied rewrites 82.3%

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

   7. Applied rewrites 82.4%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-20}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, z + z, z \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-20}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(3 \cdot z\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 5e-20) (* x y) (* (* 3.0 z) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 5e-20) {
		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) <= 5d-20) 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) <= 5e-20) {
		tmp = x * y;
	} else {
		tmp = (3.0 * z) * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z * z) <= 5e-20:
		tmp = x * y
	else:
		tmp = (3.0 * z) * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 5e-20)
		tmp = Float64(x * y);
	else
		tmp = Float64(Float64(3.0 * z) * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 5e-20)
		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], 5e-20], N[(x * y), $MachinePrecision], N[(N[(3.0 * z), $MachinePrecision] * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 4.9999999999999999e-20

   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 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in x around 0

      \[\leadsto \left(3 \cdot \frac{{z}^{2}}{y}\right) \cdot y \]
   10. Step-by-step derivation

   11. Applied rewrites 20.0%

       \[\leadsto \left(\frac{z \cdot z}{y} \cdot 3\right) \cdot y \]

   12. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 87.0

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

   14. Applied rewrites 87.0%

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

   if 4.9999999999999999e-20 < (*.f64 z z)

   1. Initial program 98.9%

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

   3. Taylor expanded in x around 0

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

      1. distribute-lft1-in N/A

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

      2. metadata-eval N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 82.3

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

   5. Applied rewrites 82.3%

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

   7. Applied rewrites 82.3%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-20}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(3 \cdot z\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-20}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot z\right) \cdot 3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 5e-20) (* x y) (* (* z z) 3.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 5e-20) {
		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-20) 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-20) {
		tmp = x * y;
	} else {
		tmp = (z * z) * 3.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z * z) <= 5e-20:
		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-20)
		tmp = Float64(x * y);
	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) <= 5e-20)
		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-20], N[(x * y), $MachinePrecision], N[(N[(z * z), $MachinePrecision] * 3.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 4.9999999999999999e-20

   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 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in x around 0

      \[\leadsto \left(3 \cdot \frac{{z}^{2}}{y}\right) \cdot y \]
   10. Step-by-step derivation

   11. Applied rewrites 20.0%

       \[\leadsto \left(\frac{z \cdot z}{y} \cdot 3\right) \cdot y \]

   12. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 87.0

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

   14. Applied rewrites 87.0%

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

   if 4.9999999999999999e-20 < (*.f64 z z)

   1. Initial program 98.9%

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

   3. Taylor expanded in x around 0

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

      1. distribute-lft1-in N/A

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

      2. metadata-eval N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 82.3

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

   5. Applied rewrites 82.3%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-20}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot z\right) \cdot 3\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+292}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, z, 0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 2e+292) (* x y) (fma z z 0.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e+292) {
		tmp = x * y;
	} else {
		tmp = fma(z, z, 0.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 2e+292)
		tmp = Float64(x * y);
	else
		tmp = fma(z, z, 0.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 2e+292], N[(x * y), $MachinePrecision], N[(z * z + 0.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 2e292

   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 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 93.1%

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

   9. Taylor expanded in x around 0

      \[\leadsto \left(3 \cdot \frac{{z}^{2}}{y}\right) \cdot y \]
   10. Step-by-step derivation

   11. Applied rewrites 28.0%

       \[\leadsto \left(\frac{z \cdot z}{y} \cdot 3\right) \cdot y \]

   12. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 70.6

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

   14. Applied rewrites 70.6%

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

   if 2e292 < (*.f64 z z)

   1. Initial program 98.0%

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

   3. Taylor expanded in x around 0

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

      1. distribute-lft1-in N/A

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

      2. metadata-eval N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   7. Applied rewrites 100.0%

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

   9. Applied rewrites 96.6%

      \[\leadsto \mathsf{fma}\left(z, \color{blue}{2}, z \cdot z\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 96.6%

       \[\leadsto \mathsf{fma}\left(z, \color{blue}{z}, 0\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+292}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, z, 0\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.4% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. lift-+.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. associate-+l+ N/A

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

   5. associate-+l+ N/A

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

   6. +-commutative N/A

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

   7. count-2 N/A

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

   8. distribute-lft1-in N/A

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

   9. metadata-eval N/A

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

   10. lower-fma.f64 99.5

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 99.5

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

4. Applied rewrites 99.5%

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

5. Final simplification 99.5%

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

Alternative 7: 99.0% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. associate-+r+ N/A

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

   3. distribute-lft1-in N/A

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

   4. metadata-eval N/A

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

   5. unpow2 N/A

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

   6. associate-*r* N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 99.5

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

5. Applied rewrites 99.5%

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

6. Final simplification 99.5%

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

Alternative 8: 53.3% 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 99.5%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. associate-+r+ N/A

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

   3. distribute-lft1-in N/A

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

   4. metadata-eval N/A

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

   5. unpow2 N/A

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

   6. associate-*r* N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 99.5

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

5. Applied rewrites 99.5%

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

6. Taylor expanded in y around inf

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

8. Applied rewrites 94.1%

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

9. Taylor expanded in x around 0

   \[\leadsto \left(3 \cdot \frac{{z}^{2}}{y}\right) \cdot y \]
10. Step-by-step derivation

11. Applied rewrites 41.5%

    \[\leadsto \left(\frac{z \cdot z}{y} \cdot 3\right) \cdot y \]

12. Taylor expanded in x around inf

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

    1. *-commutative N/A

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

    2. lower-*.f64 58.7

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

14. Applied rewrites 58.7%

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

15. Final simplification 58.7%

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

Alternative 9: 3.7% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ z \cdot 2 \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = z * 2.0d0
end function

Java

public static double code(double x, double y, double z) {
	return z * 2.0;
}

Python

def code(x, y, z):
	return z * 2.0

Julia

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

MATLAB

function tmp = code(x, y, z)
	tmp = z * 2.0;
end

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in x around 0

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

   1. distribute-lft1-in N/A

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

   2. metadata-eval N/A

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

   3. lower-*.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 47.3

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

5. Applied rewrites 47.3%

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

7. Applied rewrites 47.3%

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

9. Applied rewrites 23.5%

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

10. Taylor expanded in z around 0

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

12. Applied rewrites 3.3%

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

13. Final simplification 3.3%

    \[\leadsto z \cdot 2 \]
14. Add Preprocessing

Developer Target 1: 98.4% accurate, 1.6× 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 2024331 
(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)))