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

Percentage Accurate: 98.3% → 99.4%

Time: 7.7s

Alternatives: 7

Speedup: 1.8×

• 34.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 97.1%

   \[\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z \]
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 97.2

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

5. Applied rewrites 97.2%

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

7. Applied rewrites 99.9%

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

Alternative 2: 85.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+129}:\\ \;\;\;\;\mathsf{fma}\left(z, z, y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, z + z, z \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 1e+129) (fma z z (* y x)) (fma z (+ z z) (* z z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 1e+129) {
		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 (Float64(z * z) <= 1e+129)
		tmp = fma(z, z, Float64(y * x));
	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], 1e+129], N[(z * z + N[(y * x), $MachinePrecision]), $MachinePrecision], N[(z * N[(z + z), $MachinePrecision] + N[(z * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1e129

   1. Initial program 99.9%

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

   4. Applied rewrites 89.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, z, \mathsf{fma}\left(y, x, 0\right)\right)} \]
   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 89.3

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

   6. Applied rewrites 89.3%

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

   if 1e129 < (*.f64 z z)

   1. Initial program 93.4%

      \[\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z \]
   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. count-2 N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. lower--.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval 93.4

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

   4. Applied rewrites 93.4%

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

   6. Applied rewrites 93.5%

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

      1. +-rgt-identity N/A

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

      2. +-inverses N/A

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

      3. associate--l+ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. fp-cancel-sign-sub-inv N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate--l- N/A

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

      10. lower--.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lift-*.f64 42.7

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

   8. Applied rewrites 42.7%

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

   9. Taylor expanded in x around 0

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

       1. unpow2 N/A

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

       2. lower-*.f64 92.0

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

   11. Applied rewrites 92.0%

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

Alternative 3: 85.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+129}:\\ \;\;\;\;\mathsf{fma}\left(z, z, y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 \cdot z\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 1e+129) (fma z z (* y x)) (* (* 3.0 z) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 1e+129) {
		tmp = fma(z, z, (y * x));
	} else {
		tmp = (3.0 * z) * z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 1e+129)
		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[N[(z * z), $MachinePrecision], 1e+129], 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 (*.f64 z z) < 1e129

   1. Initial program 99.9%

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

   4. Applied rewrites 89.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, z, \mathsf{fma}\left(y, x, 0\right)\right)} \]
   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 89.3

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

   6. Applied rewrites 89.3%

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

   if 1e129 < (*.f64 z z)

   1. Initial program 93.4%

      \[\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z \]
   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 91.9

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

   5. Applied rewrites 91.9%

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

   7. Applied rewrites 92.0%

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

Alternative 4: 83.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+129}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(3 \cdot z\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 1e+129) (* x y) (* (* 3.0 z) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 1e+129) {
		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) <= 1d+129) 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) <= 1e+129) {
		tmp = x * y;
	} else {
		tmp = (3.0 * z) * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z * z) <= 1e+129:
		tmp = x * y
	else:
		tmp = (3.0 * z) * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 1e+129)
		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) <= 1e+129)
		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], 1e+129], N[(x * y), $MachinePrecision], N[(N[(3.0 * z), $MachinePrecision] * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1e129

   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. count-2 N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. lower--.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval 99.9

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

   4. Applied rewrites 99.9%

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

   6. Applied rewrites 100.0%

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

      1. +-rgt-identity N/A

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

      2. +-inverses N/A

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

      3. associate--l+ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. fp-cancel-sign-sub-inv N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate--l- N/A

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

      10. lower--.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lift-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in x around inf

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

       1. lower-*.f64 87.6

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

   11. Applied rewrites 87.6%

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

   if 1e129 < (*.f64 z z)

   1. Initial program 93.4%

      \[\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z \]
   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 91.9

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

   5. Applied rewrites 91.9%

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

   7. Applied rewrites 92.0%

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

4. Final simplification 89.5%

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

Alternative 5: 83.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 9 \cdot 10^{+128}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(z \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 9e+128) (* x y) (* 3.0 (* z z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 9e+128) {
		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) <= 9d+128) 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) <= 9e+128) {
		tmp = x * y;
	} else {
		tmp = 3.0 * (z * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z * z) <= 9e+128:
		tmp = x * y
	else:
		tmp = 3.0 * (z * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 9e+128)
		tmp = Float64(x * y);
	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 * z) <= 9e+128)
		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], 9e+128], N[(x * y), $MachinePrecision], N[(3.0 * N[(z * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 9.0000000000000003e128

   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. count-2 N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. lower--.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval 99.9

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

   4. Applied rewrites 99.9%

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

   6. Applied rewrites 100.0%

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

      1. +-rgt-identity N/A

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

      2. +-inverses N/A

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

      3. associate--l+ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. fp-cancel-sign-sub-inv N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate--l- N/A

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

      10. lower--.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lift-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in x around inf

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

       1. lower-*.f64 87.6

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

   11. Applied rewrites 87.6%

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

   if 9.0000000000000003e128 < (*.f64 z z)

   1. Initial program 93.4%

      \[\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z \]
   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 91.9

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

   5. Applied rewrites 91.9%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 9 \cdot 10^{+128}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(z \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2.6 \cdot 10^{+226}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= (* z z) 2.6e+226) (* x y) (* z z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2.6e+226) {
		tmp = x * y;
	} else {
		tmp = z * z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z * z) <= 2.6d+226) then
        tmp = x * y
    else
        tmp = z * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2.6e+226) {
		tmp = x * y;
	} else {
		tmp = z * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z * z) <= 2.6e+226:
		tmp = x * y
	else:
		tmp = z * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 2.6e+226)
		tmp = Float64(x * y);
	else
		tmp = Float64(z * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 2.6e+226)
		tmp = x * y;
	else
		tmp = z * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 2.6e+226], N[(x * y), $MachinePrecision], N[(z * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 2.6000000000000002e226

   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. 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. count-2 N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. lower--.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval 99.9

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

   4. Applied rewrites 99.9%

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

   6. Applied rewrites 99.9%

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

      1. +-rgt-identity N/A

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

      2. +-inverses N/A

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

      3. associate--l+ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. fp-cancel-sign-sub-inv N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate--l- N/A

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

      10. lower--.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lift-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in x around inf

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

       1. lower-*.f64 77.2

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

   11. Applied rewrites 77.2%

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

   if 2.6000000000000002e226 < (*.f64 z z)

   1. Initial program 91.2%

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

   4. Applied rewrites 75.1%

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

   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 81.5

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

   7. Applied rewrites 81.5%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2.6 \cdot 10^{+226}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 32.6% 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 97.1%

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

4. Applied rewrites 78.8%

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

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 34.5

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

7. Applied rewrites 34.5%

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

8. Final simplification 34.5%

   \[\leadsto z \cdot z \]
9. 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 2024326 
(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)))