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

Percentage Accurate: 98.4% → 99.4%

Time: 10.7s

Alternatives: 7

Speedup: 1.8×

• 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: 98.4% 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: 99.4% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 99.8

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

5. Applied rewrites 99.8%

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

6. Taylor expanded in x around 0

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

   1. associate-+r+ N/A

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

   2. +-commutative N/A

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

   3. associate-+r+ N/A

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

   4. distribute-lft1-in N/A

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

   5. metadata-eval N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, 3 \cdot {z}^{2}\right)} \]

   8. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(y, x, \color{blue}{3 \cdot {z}^{2}}\right) \]

   9. unpow2 N/A

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

   10. lower-*.f64 99.8

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

8. Applied rewrites 99.8%

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

Alternative 2: 81.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 2.4 \cdot 10^{+31}:\\ \;\;\;\;\mathsf{fma}\left(z, z, y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, z, z \cdot \left(z + z\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 2.4e+31) (fma z z (* y x)) (fma z z (* z (+ z z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 2.4e+31) {
		tmp = fma(z, z, (y * x));
	} else {
		tmp = fma(z, z, (z * (z + z)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 2.4e+31)
		tmp = fma(z, z, Float64(y * x));
	else
		tmp = fma(z, z, Float64(z * Float64(z + z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 2.4e+31], N[(z * z + N[(y * x), $MachinePrecision]), $MachinePrecision], N[(z * z + N[(z * N[(z + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 2.39999999999999982e31

   1. Initial program 98.8%

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

   4. Applied rewrites 79.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, z, \mathsf{fma}\left(y, x, 0\right)\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(z, z, \color{blue}{y \cdot x + 0}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(z, z, \color{blue}{y \cdot x} + 0\right) \]

      3. +-rgt-identity 79.5

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

   6. Applied rewrites 79.5%

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

   if 2.39999999999999982e31 < z

   1. Initial program 94.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 99.8

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

   5. Applied rewrites 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.9

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

   7. Applied rewrites 94.7%

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

   8. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 92.2

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

   10. Applied rewrites 92.2%

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

   12. Applied rewrites 92.2%

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

Alternative 3: 81.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 2.4 \cdot 10^{+31}:\\ \;\;\;\;\mathsf{fma}\left(z, z, y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 \cdot z\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 2.4e+31) (fma z z (* y x)) (* (* 3.0 z) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 2.4e+31) {
		tmp = fma(z, z, (y * x));
	} else {
		tmp = (3.0 * z) * z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 2.4e+31)
		tmp = fma(z, z, Float64(y * x));
	else
		tmp = Float64(Float64(3.0 * z) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 2.4e+31], N[(z * z + N[(y * x), $MachinePrecision]), $MachinePrecision], N[(N[(3.0 * z), $MachinePrecision] * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 2.39999999999999982e31

   1. Initial program 98.8%

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

   4. Applied rewrites 79.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, z, \mathsf{fma}\left(y, x, 0\right)\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(z, z, \color{blue}{y \cdot x + 0}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(z, z, \color{blue}{y \cdot x} + 0\right) \]

      3. +-rgt-identity 79.5

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

   6. Applied rewrites 79.5%

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

   if 2.39999999999999982e31 < z

   1. Initial program 94.6%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{2 \cdot {z}^{2} + {z}^{2}} \]
   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 92.1

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

   5. Applied rewrites 92.1%

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

   7. Applied rewrites 92.1%

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

Alternative 4: 68.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 3.4 \cdot 10^{-29}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(3 \cdot z\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 3.4e-29) (* y x) (* (* 3.0 z) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 3.4e-29) {
		tmp = y * x;
	} 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 <= 3.4d-29) then
        tmp = y * x
    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 <= 3.4e-29) {
		tmp = y * x;
	} else {
		tmp = (3.0 * z) * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 3.4e-29:
		tmp = y * x
	else:
		tmp = (3.0 * z) * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 3.4e-29)
		tmp = Float64(y * x);
	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 <= 3.4e-29)
		tmp = y * x;
	else
		tmp = (3.0 * z) * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 3.4e-29], N[(y * x), $MachinePrecision], N[(N[(3.0 * z), $MachinePrecision] * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 3.39999999999999972e-29

   1. Initial program 98.8%

      \[\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z \]
   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. remove-double-neg N/A

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

      8. mul-1-neg N/A

         \[\leadsto \left(3 \cdot z\right) \cdot z + x \cdot \left(\mathsf{neg}\left(\color{blue}{-1 \cdot y}\right)\right) \]

      9. distribute-rgt-neg-in N/A

         \[\leadsto \left(3 \cdot z\right) \cdot z + \color{blue}{\left(\mathsf{neg}\left(x \cdot \left(-1 \cdot y\right)\right)\right)} \]

      10. distribute-lft-neg-in N/A

          \[\leadsto \left(3 \cdot z\right) \cdot z + \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \left(-1 \cdot y\right)} \]

      11. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(3 \cdot z, z, \left(\mathsf{neg}\left(x\right)\right) \cdot \left(-1 \cdot y\right)\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{3 \cdot z}, z, \left(\mathsf{neg}\left(x\right)\right) \cdot \left(-1 \cdot y\right)\right) \]

      13. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{fma}\left(3 \cdot z, z, \color{blue}{\mathsf{neg}\left(x \cdot \left(-1 \cdot y\right)\right)}\right) \]

      14. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{fma}\left(3 \cdot z, z, \color{blue}{x \cdot \left(\mathsf{neg}\left(-1 \cdot y\right)\right)}\right) \]

      15. mul-1-neg N/A

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

      16. remove-double-neg N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 98.8

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

   5. Applied rewrites 98.8%

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

   7. Applied rewrites 40.9%

      \[\leadsto \frac{9 \cdot {z}^{4}}{\mathsf{fma}\left(3 \cdot z, z, \left(-x\right) \cdot y\right)} + \color{blue}{\frac{\left(\left(-y\right) \cdot x\right) \cdot \left(y \cdot x\right)}{\mathsf{fma}\left(3 \cdot z, z, \left(-x\right) \cdot y\right)}} \]

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 59.5%

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

   if 3.39999999999999972e-29 < 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 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 88.2

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

   5. Applied rewrites 88.2%

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

   7. Applied rewrites 88.2%

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

Alternative 5: 68.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 3.4 \cdot 10^{-29}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(z \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 3.4e-29) (* y x) (* 3.0 (* z z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 3.4e-29) {
		tmp = y * x;
	} 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 <= 3.4d-29) then
        tmp = y * x
    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 <= 3.4e-29) {
		tmp = y * x;
	} else {
		tmp = 3.0 * (z * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 3.4e-29:
		tmp = y * x
	else:
		tmp = 3.0 * (z * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 3.4e-29)
		tmp = Float64(y * x);
	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 <= 3.4e-29)
		tmp = y * x;
	else
		tmp = 3.0 * (z * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 3.4e-29], N[(y * x), $MachinePrecision], N[(3.0 * N[(z * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 3.39999999999999972e-29

   1. Initial program 98.8%

      \[\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z \]
   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. remove-double-neg N/A

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

      8. mul-1-neg N/A

         \[\leadsto \left(3 \cdot z\right) \cdot z + x \cdot \left(\mathsf{neg}\left(\color{blue}{-1 \cdot y}\right)\right) \]

      9. distribute-rgt-neg-in N/A

         \[\leadsto \left(3 \cdot z\right) \cdot z + \color{blue}{\left(\mathsf{neg}\left(x \cdot \left(-1 \cdot y\right)\right)\right)} \]

      10. distribute-lft-neg-in N/A

          \[\leadsto \left(3 \cdot z\right) \cdot z + \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \left(-1 \cdot y\right)} \]

      11. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(3 \cdot z, z, \left(\mathsf{neg}\left(x\right)\right) \cdot \left(-1 \cdot y\right)\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{3 \cdot z}, z, \left(\mathsf{neg}\left(x\right)\right) \cdot \left(-1 \cdot y\right)\right) \]

      13. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{fma}\left(3 \cdot z, z, \color{blue}{\mathsf{neg}\left(x \cdot \left(-1 \cdot y\right)\right)}\right) \]

      14. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{fma}\left(3 \cdot z, z, \color{blue}{x \cdot \left(\mathsf{neg}\left(-1 \cdot y\right)\right)}\right) \]

      15. mul-1-neg N/A

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

      16. remove-double-neg N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 98.8

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

   5. Applied rewrites 98.8%

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

   7. Applied rewrites 40.9%

      \[\leadsto \frac{9 \cdot {z}^{4}}{\mathsf{fma}\left(3 \cdot z, z, \left(-x\right) \cdot y\right)} + \color{blue}{\frac{\left(\left(-y\right) \cdot x\right) \cdot \left(y \cdot x\right)}{\mathsf{fma}\left(3 \cdot z, z, \left(-x\right) \cdot y\right)}} \]

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 59.5%

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

   if 3.39999999999999972e-29 < 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 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 88.2

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

   5. Applied rewrites 88.2%

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

Alternative 6: 63.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 2.35 \cdot 10^{+123}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;z \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= z 2.35e+123) (* y x) (* z z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 2.35e+123) {
		tmp = y * x;
	} 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 <= 2.35d+123) then
        tmp = y * x
    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 <= 2.35e+123) {
		tmp = y * x;
	} else {
		tmp = z * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 2.35e+123:
		tmp = y * x
	else:
		tmp = z * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 2.35e+123)
		tmp = Float64(y * x);
	else
		tmp = Float64(z * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 2.35e+123)
		tmp = y * x;
	else
		tmp = z * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 2.35e+123], N[(y * x), $MachinePrecision], N[(z * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 2.3499999999999999e123

   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

      \[\leadsto \color{blue}{2 \cdot {z}^{2} + \left(x \cdot y + {z}^{2}\right)} \]
   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. remove-double-neg N/A

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

      8. mul-1-neg N/A

         \[\leadsto \left(3 \cdot z\right) \cdot z + x \cdot \left(\mathsf{neg}\left(\color{blue}{-1 \cdot y}\right)\right) \]

      9. distribute-rgt-neg-in N/A

         \[\leadsto \left(3 \cdot z\right) \cdot z + \color{blue}{\left(\mathsf{neg}\left(x \cdot \left(-1 \cdot y\right)\right)\right)} \]

      10. distribute-lft-neg-in N/A

          \[\leadsto \left(3 \cdot z\right) \cdot z + \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \left(-1 \cdot y\right)} \]

      11. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(3 \cdot z, z, \left(\mathsf{neg}\left(x\right)\right) \cdot \left(-1 \cdot y\right)\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{3 \cdot z}, z, \left(\mathsf{neg}\left(x\right)\right) \cdot \left(-1 \cdot y\right)\right) \]

      13. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{fma}\left(3 \cdot z, z, \color{blue}{\mathsf{neg}\left(x \cdot \left(-1 \cdot y\right)\right)}\right) \]

      14. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{fma}\left(3 \cdot z, z, \color{blue}{x \cdot \left(\mathsf{neg}\left(-1 \cdot y\right)\right)}\right) \]

      15. mul-1-neg N/A

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

      16. remove-double-neg N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 98.9

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

   5. Applied rewrites 98.9%

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

   7. Applied rewrites 43.6%

      \[\leadsto \frac{9 \cdot {z}^{4}}{\mathsf{fma}\left(3 \cdot z, z, \left(-x\right) \cdot y\right)} + \color{blue}{\frac{\left(\left(-y\right) \cdot x\right) \cdot \left(y \cdot x\right)}{\mathsf{fma}\left(3 \cdot z, z, \left(-x\right) \cdot y\right)}} \]

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 54.2%

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

   if 2.3499999999999999e123 < z

   1. Initial program 92.6%

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

   4. Applied rewrites 70.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, z, \mathsf{fma}\left(y, x, 0\right)\right)} \]

   5. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 73.0

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

   7. Applied rewrites 73.0%

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

Alternative 7: 33.4% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.9%

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

4. Applied rewrites 74.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(z, z, \mathsf{fma}\left(y, x, 0\right)\right)} \]

5. Taylor expanded in x around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 36.0

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

7. Applied rewrites 36.0%

   \[\leadsto \color{blue}{z \cdot z} \]
8. 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 2024338 
(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)))