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

Percentage Accurate: 98.2% → 99.8%

Time: 8.3s

Alternatives: 7

Speedup: 1.3×

• 34.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \cdot x \leq -4 \cdot 10^{+118}:\\ \;\;\;\;\mathsf{fma}\left(z, 3 \cdot \frac{z}{y}, x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot z, z, \mathsf{fma}\left(z, z, y \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* y x) -4e+118)
   (* (fma z (* 3.0 (/ z y)) x) y)
   (fma (* 2.0 z) z (fma z z (* y x)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if ((y * x) <= -4e+118) {
		tmp = fma(z, (3.0 * (z / y)), x) * y;
	} else {
		tmp = fma((2.0 * z), z, fma(z, z, (y * x)));
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (Float64(y * x) <= -4e+118)
		tmp = Float64(fma(z, Float64(3.0 * Float64(z / y)), x) * y);
	else
		tmp = fma(Float64(2.0 * z), z, fma(z, z, Float64(y * x)));
	end
	return tmp
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[N[(y * x), $MachinePrecision], -4e+118], N[(N[(z * N[(3.0 * N[(z / y), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] * y), $MachinePrecision], N[(N[(2.0 * z), $MachinePrecision] * z + N[(z * z + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -3.99999999999999987e118

   1. Initial program 85.6%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 73.9

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

   5. Applied rewrites 73.9%

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

   6. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

         \[\leadsto \left(\color{blue}{3} \cdot \frac{{z}^{2}}{y} + x\right) \cdot y \]

      6. *-commutative N/A

         \[\leadsto \left(\color{blue}{\frac{{z}^{2}}{y} \cdot 3} + x\right) \cdot y \]

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 97.5

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

   8. Applied rewrites 97.5%

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

   10. Applied rewrites 100.0%

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

   if -3.99999999999999987e118 < (*.f64 x y)

   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. 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 100.0

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

      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 100.0

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

      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 100.0

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

   4. Applied rewrites 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot x \leq -4 \cdot 10^{+118}:\\ \;\;\;\;\mathsf{fma}\left(z, 3 \cdot \frac{z}{y}, x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot z, z, \mathsf{fma}\left(z, z, y \cdot x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 5 \cdot 10^{+215}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot z, z, \mathsf{fma}\left(z, z, y \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 \cdot z\right) \cdot z\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= z 5e+215) (fma (* 2.0 z) z (fma z z (* y x))) (* (* 3.0 z) z)))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (z <= 5e+215) {
		tmp = fma((2.0 * z), z, fma(z, z, (y * x)));
	} else {
		tmp = (3.0 * z) * z;
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (z <= 5e+215)
		tmp = fma(Float64(2.0 * z), z, fma(z, z, Float64(y * x)));
	else
		tmp = Float64(Float64(3.0 * z) * z);
	end
	return tmp
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[z, 5e+215], N[(N[(2.0 * z), $MachinePrecision] * z + N[(z * z + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(3.0 * z), $MachinePrecision] * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 5.0000000000000001e215

   1. Initial program 97.7%

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

      1. 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 97.8

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

      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 98.2

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

      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 98.2

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

   4. Applied rewrites 98.2%

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

   if 5.0000000000000001e215 < z

   1. Initial program 95.8%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 18.3

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

   5. Applied rewrites 18.3%

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

   6. Taylor expanded in z around inf

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

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

Alternative 3: 99.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 5 \cdot 10^{+215}:\\ \;\;\;\;\mathsf{fma}\left(3 \cdot z, z, y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 \cdot z\right) \cdot z\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= z 5e+215) (fma (* 3.0 z) z (* y x)) (* (* 3.0 z) z)))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (z <= 5e+215) {
		tmp = fma((3.0 * z), z, (y * x));
	} else {
		tmp = (3.0 * z) * z;
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (z <= 5e+215)
		tmp = fma(Float64(3.0 * z), z, Float64(y * x));
	else
		tmp = Float64(Float64(3.0 * z) * z);
	end
	return tmp
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[z, 5e+215], N[(N[(3.0 * z), $MachinePrecision] * z + N[(y * x), $MachinePrecision]), $MachinePrecision], N[(N[(3.0 * z), $MachinePrecision] * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 5.0000000000000001e215

   1. Initial program 97.7%

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

   3. Taylor expanded in z around 0

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

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 98.1

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

   5. Applied rewrites 98.1%

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

   if 5.0000000000000001e215 < z

   1. Initial program 95.8%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 18.3

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

   5. Applied rewrites 18.3%

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

   6. Taylor expanded in z around inf

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

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

Alternative 4: 83.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+95}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot z\right) \cdot 3\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 2e+95) (* y x) (* (* z z) 3.0)))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e+95) {
		tmp = y * x;
	} else {
		tmp = (z * z) * 3.0;
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
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) <= 2d+95) then
        tmp = y * x
    else
        tmp = (z * z) * 3.0d0
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e+95) {
		tmp = y * x;
	} else {
		tmp = (z * z) * 3.0;
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if (z * z) <= 2e+95:
		tmp = y * x
	else:
		tmp = (z * z) * 3.0
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 2e+95)
		tmp = Float64(y * x);
	else
		tmp = Float64(Float64(z * z) * 3.0);
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 2e+95)
		tmp = y * x;
	else
		tmp = (z * z) * 3.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 2e+95], N[(y * x), $MachinePrecision], N[(N[(z * z), $MachinePrecision] * 3.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 2.00000000000000004e95

   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 z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 80.6

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

   5. Applied rewrites 80.6%

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

   if 2.00000000000000004e95 < (*.f64 z z)

   1. Initial program 94.7%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 94.4

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

   5. Applied rewrites 94.4%

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

Alternative 5: 83.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+95}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(3 \cdot z\right) \cdot z\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 2e+95) (* y x) (* (* 3.0 z) z)))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e+95) {
		tmp = y * x;
	} else {
		tmp = (3.0 * z) * z;
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
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) <= 2d+95) then
        tmp = y * x
    else
        tmp = (3.0d0 * z) * z
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e+95) {
		tmp = y * x;
	} else {
		tmp = (3.0 * z) * z;
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if (z * z) <= 2e+95:
		tmp = y * x
	else:
		tmp = (3.0 * z) * z
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 2e+95)
		tmp = Float64(y * x);
	else
		tmp = Float64(Float64(3.0 * z) * z);
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 2e+95)
		tmp = y * x;
	else
		tmp = (3.0 * z) * z;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 2e+95], N[(y * x), $MachinePrecision], N[(N[(3.0 * z), $MachinePrecision] * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 2.00000000000000004e95

   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 z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 80.6

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

   5. Applied rewrites 80.6%

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

   if 2.00000000000000004e95 < (*.f64 z z)

   1. Initial program 94.7%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 12.3

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

   5. Applied rewrites 12.3%

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

   6. Taylor expanded in z around inf

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

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 94.3

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

   8. Applied rewrites 94.3%

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

Alternative 6: 67.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 3.6 \cdot 10^{+47}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, z + z, z \cdot z\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= z 3.6e+47) (* y x) (fma z (+ z z) (* z z))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (z <= 3.6e+47) {
		tmp = y * x;
	} else {
		tmp = fma(z, (z + z), (z * z));
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (z <= 3.6e+47)
		tmp = Float64(y * x);
	else
		tmp = fma(z, Float64(z + z), Float64(z * z));
	end
	return tmp
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[z, 3.6e+47], N[(y * x), $MachinePrecision], N[(z * N[(z + z), $MachinePrecision] + N[(z * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 3.60000000000000008e47

   1. Initial program 97.4%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 60.5

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

   5. Applied rewrites 60.5%

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

   if 3.60000000000000008e47 < z

   1. Initial program 98.2%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 96.5

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

   5. Applied rewrites 96.5%

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

   7. Applied rewrites 96.6%

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

Alternative 7: 52.3% accurate, 5.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ y \cdot x \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z) :precision binary64 (* y x))

C

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

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
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

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

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	return y * x

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	return Float64(y * x)
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp = code(x, y, z)
	tmp = y * x;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := N[(y * x), $MachinePrecision]

Derivation

1. Initial program 97.5%

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

3. Taylor expanded in z around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 49.4

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

5. Applied rewrites 49.4%

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

Developer Target 1: 98.3% 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 2024276 
(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)))