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

Percentage Accurate: 98.2% → 99.3%

Time: 10.5s

Alternatives: 9

Speedup: 1.8×

• 34.0% 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: 99.3% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_] := N[(N[(z + z), $MachinePrecision] * z + N[(x * y + N[(z * z), $MachinePrecision]), $MachinePrecision]), $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. Step-by-step derivation

   1. associate-+l+ N/A

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

   2. +-commutative N/A

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

   3. count-2 N/A

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

   4. associate-*r* N/A

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

   5. count-2 N/A

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

   6. accelerator-lowering-fma.f64 N/A

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

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

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

   8. accelerator-lowering-fma.f64 N/A

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

   9. *-lowering-*.f64 99.5

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

4. Applied egg-rr 99.5%

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

Alternative 2: 98.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 1.5 \cdot 10^{+273}:\\ \;\;\;\;\mathsf{fma}\left(3, z \cdot z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 1.5e+273) (fma 3.0 (* z z) (* x y)) (* z (* z 3.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 1.5e+273) {
		tmp = fma(3.0, (z * z), (x * y));
	} else {
		tmp = z * (z * 3.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 1.5e+273)
		tmp = fma(3.0, Float64(z * z), Float64(x * y));
	else
		tmp = Float64(z * Float64(z * 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 1.5e+273], N[(3.0 * N[(z * z), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision], N[(z * N[(z * 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.5e273

   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. associate-+l+ N/A

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

      2. associate-+l+ N/A

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

      3. +-commutative N/A

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

      4. count-2 N/A

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

      5. distribute-lft1-in N/A

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

      6. metadata-eval N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

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

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

      9. *-lowering-*.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   if 1.5e273 < (*.f64 z z)

   1. Initial program 91.8%

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

   3. Taylor expanded in x around 0

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

      1. *-rgt-identity N/A

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

      2. *-inverses N/A

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

      3. associate-/l* N/A

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

      4. associate-*l/ N/A

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

      5. +-rgt-identity N/A

         \[\leadsto \color{blue}{\frac{2 \cdot {z}^{2} + {z}^{2}}{y} \cdot y + 0} \]

      6. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\left(2 \cdot {z}^{2} + {z}^{2}\right) \cdot y}{y}} + 0 \]

      7. associate-/l* N/A

         \[\leadsto \color{blue}{\left(2 \cdot {z}^{2} + {z}^{2}\right) \cdot \frac{y}{y}} + 0 \]

      8. *-inverses N/A

         \[\leadsto \left(2 \cdot {z}^{2} + {z}^{2}\right) \cdot \color{blue}{1} + 0 \]

      9. *-rgt-identity N/A

         \[\leadsto \color{blue}{\left(2 \cdot {z}^{2} + {z}^{2}\right)} + 0 \]

      10. distribute-lft1-in N/A

          \[\leadsto \color{blue}{\left(2 + 1\right) \cdot {z}^{2}} + 0 \]

      11. metadata-eval N/A

          \[\leadsto \color{blue}{3} \cdot {z}^{2} + 0 \]

      12. *-commutative N/A

          \[\leadsto \color{blue}{{z}^{2} \cdot 3} + 0 \]

      13. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left({z}^{2}, 3, 0\right)} \]

      14. +-rgt-identity N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{{z}^{2} + 0}, 3, 0\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot z} + 0, 3, 0\right) \]

      16. accelerator-lowering-fma.f64 98.6

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(z, z, 0\right)}, 3, 0\right) \]

   5. Simplified 98.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(z, z, 0\right), 3, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{\left(z \cdot z + 0\right) \cdot 3} \]

      2. +-rgt-identity N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 98.6

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

   7. Applied egg-rr 98.6%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 1.5 \cdot 10^{+273}:\\ \;\;\;\;\mathsf{fma}\left(3, z \cdot z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{-62}:\\ \;\;\;\;\mathsf{fma}\left(z + z, z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z + z, z, z \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 1e-62) (fma (+ z z) z (* x y)) (fma (+ z z) z (* z z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 1e-62) {
		tmp = fma((z + z), z, (x * y));
	} else {
		tmp = fma((z + z), z, (z * z));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 1e-62)
		tmp = fma(Float64(z + z), z, Float64(x * y));
	else
		tmp = fma(Float64(z + z), z, Float64(z * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 1e-62], N[(N[(z + z), $MachinePrecision] * z + N[(x * y), $MachinePrecision]), $MachinePrecision], N[(N[(z + z), $MachinePrecision] * z + N[(z * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1e-62

   1. Initial program 100.0%

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

      1. associate-+l+ N/A

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

      2. +-commutative N/A

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

      3. count-2 N/A

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

      4. associate-*r* N/A

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

      5. count-2 N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

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

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. *-lowering-*.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf

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

      1. +-rgt-identity N/A

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

      2. accelerator-lowering-fma.f64 91.5

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

   7. Simplified 91.5%

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

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 91.5

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

   9. Applied egg-rr 91.5%

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

   if 1e-62 < (*.f64 z 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. Step-by-step derivation

      1. associate-+l+ N/A

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

      2. +-commutative N/A

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

      3. count-2 N/A

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

      4. associate-*r* N/A

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

      5. count-2 N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

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

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. *-lowering-*.f64 99.1

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

   4. Applied egg-rr 99.1%

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

   5. Taylor expanded in x around 0

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

      1. +-rgt-identity N/A

         \[\leadsto \mathsf{fma}\left(z + z, z, \color{blue}{{z}^{2} + 0}\right) \]

      2. unpow2 N/A

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

      3. accelerator-lowering-fma.f64 84.3

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

   7. Simplified 84.3%

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

      1. +-rgt-identity N/A

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

      2. *-lowering-*.f64 84.3

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

   9. Applied egg-rr 84.3%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{-62}:\\ \;\;\;\;\mathsf{fma}\left(z + z, z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z + z, z, z \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{-62}:\\ \;\;\;\;\mathsf{fma}\left(z + z, z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 1e-62) (fma (+ z z) z (* x y)) (* z (* z 3.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 1e-62) {
		tmp = fma((z + z), z, (x * y));
	} else {
		tmp = z * (z * 3.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 1e-62)
		tmp = fma(Float64(z + z), z, Float64(x * y));
	else
		tmp = Float64(z * Float64(z * 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 1e-62], N[(N[(z + z), $MachinePrecision] * z + N[(x * y), $MachinePrecision]), $MachinePrecision], N[(z * N[(z * 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1e-62

   1. Initial program 100.0%

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

      1. associate-+l+ N/A

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

      2. +-commutative N/A

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

      3. count-2 N/A

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

      4. associate-*r* N/A

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

      5. count-2 N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

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

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. *-lowering-*.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf

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

      1. +-rgt-identity N/A

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

      2. accelerator-lowering-fma.f64 91.5

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

   7. Simplified 91.5%

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

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 91.5

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

   9. Applied egg-rr 91.5%

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

   if 1e-62 < (*.f64 z 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. *-rgt-identity N/A

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

      2. *-inverses N/A

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

      3. associate-/l* N/A

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

      4. associate-*l/ N/A

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

      5. +-rgt-identity N/A

         \[\leadsto \color{blue}{\frac{2 \cdot {z}^{2} + {z}^{2}}{y} \cdot y + 0} \]

      6. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\left(2 \cdot {z}^{2} + {z}^{2}\right) \cdot y}{y}} + 0 \]

      7. associate-/l* N/A

         \[\leadsto \color{blue}{\left(2 \cdot {z}^{2} + {z}^{2}\right) \cdot \frac{y}{y}} + 0 \]

      8. *-inverses N/A

         \[\leadsto \left(2 \cdot {z}^{2} + {z}^{2}\right) \cdot \color{blue}{1} + 0 \]

      9. *-rgt-identity N/A

         \[\leadsto \color{blue}{\left(2 \cdot {z}^{2} + {z}^{2}\right)} + 0 \]

      10. distribute-lft1-in N/A

          \[\leadsto \color{blue}{\left(2 + 1\right) \cdot {z}^{2}} + 0 \]

      11. metadata-eval N/A

          \[\leadsto \color{blue}{3} \cdot {z}^{2} + 0 \]

      12. *-commutative N/A

          \[\leadsto \color{blue}{{z}^{2} \cdot 3} + 0 \]

      13. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left({z}^{2}, 3, 0\right)} \]

      14. +-rgt-identity N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{{z}^{2} + 0}, 3, 0\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot z} + 0, 3, 0\right) \]

      16. accelerator-lowering-fma.f64 84.2

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(z, z, 0\right)}, 3, 0\right) \]

   5. Simplified 84.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(z, z, 0\right), 3, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{\left(z \cdot z + 0\right) \cdot 3} \]

      2. +-rgt-identity N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 84.3

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

   7. Applied egg-rr 84.3%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{-62}:\\ \;\;\;\;\mathsf{fma}\left(z + z, z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{-62}:\\ \;\;\;\;\mathsf{fma}\left(z, z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 1e-62) (fma z z (* x y)) (* z (* z 3.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 1e-62) {
		tmp = fma(z, z, (x * y));
	} else {
		tmp = z * (z * 3.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 1e-62)
		tmp = fma(z, z, Float64(x * y));
	else
		tmp = Float64(z * Float64(z * 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 1e-62], N[(z * z + N[(x * y), $MachinePrecision]), $MachinePrecision], N[(z * N[(z * 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1e-62

   1. Initial program 100.0%

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

      1. associate-+l+ N/A

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

      2. +-commutative N/A

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

      3. count-2 N/A

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

      4. associate-*r* N/A

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

      5. count-2 N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

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

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. *-lowering-*.f64 100.0

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

   4. Applied egg-rr 100.0%

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. flip-+ N/A

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

      6. +-inverses N/A

         \[\leadsto z \cdot z + \left(x \cdot y + z \cdot \frac{\color{blue}{0}}{z - z}\right) \]

      7. metadata-eval N/A

         \[\leadsto z \cdot z + \left(x \cdot y + z \cdot \frac{\color{blue}{0 - 0}}{z - z}\right) \]

      8. metadata-eval N/A

         \[\leadsto z \cdot z + \left(x \cdot y + z \cdot \frac{\color{blue}{0 \cdot 0} - 0}{z - z}\right) \]

      9. metadata-eval N/A

         \[\leadsto z \cdot z + \left(x \cdot y + z \cdot \frac{0 \cdot 0 - \color{blue}{0 \cdot 0}}{z - z}\right) \]

      10. +-inverses N/A

          \[\leadsto z \cdot z + \left(x \cdot y + z \cdot \frac{0 \cdot 0 - 0 \cdot 0}{\color{blue}{0}}\right) \]

      11. metadata-eval N/A

          \[\leadsto z \cdot z + \left(x \cdot y + z \cdot \frac{0 \cdot 0 - 0 \cdot 0}{\color{blue}{0 + 0}}\right) \]

      12. flip-- N/A

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

      13. metadata-eval N/A

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

      14. +-inverses N/A

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

      15. distribute-lft-out-- N/A

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

      16. +-inverses N/A

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

      17. +-rgt-identity N/A

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

      18. accelerator-lowering-fma.f64 N/A

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

      19. +-rgt-identity N/A

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

      20. accelerator-lowering-fma.f64 91.3

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

   6. Applied egg-rr 91.3%

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

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 91.3

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

   8. Applied egg-rr 91.3%

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

   if 1e-62 < (*.f64 z 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. *-rgt-identity N/A

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

      2. *-inverses N/A

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

      3. associate-/l* N/A

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

      4. associate-*l/ N/A

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

      5. +-rgt-identity N/A

         \[\leadsto \color{blue}{\frac{2 \cdot {z}^{2} + {z}^{2}}{y} \cdot y + 0} \]

      6. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\left(2 \cdot {z}^{2} + {z}^{2}\right) \cdot y}{y}} + 0 \]

      7. associate-/l* N/A

         \[\leadsto \color{blue}{\left(2 \cdot {z}^{2} + {z}^{2}\right) \cdot \frac{y}{y}} + 0 \]

      8. *-inverses N/A

         \[\leadsto \left(2 \cdot {z}^{2} + {z}^{2}\right) \cdot \color{blue}{1} + 0 \]

      9. *-rgt-identity N/A

         \[\leadsto \color{blue}{\left(2 \cdot {z}^{2} + {z}^{2}\right)} + 0 \]

      10. distribute-lft1-in N/A

          \[\leadsto \color{blue}{\left(2 + 1\right) \cdot {z}^{2}} + 0 \]

      11. metadata-eval N/A

          \[\leadsto \color{blue}{3} \cdot {z}^{2} + 0 \]

      12. *-commutative N/A

          \[\leadsto \color{blue}{{z}^{2} \cdot 3} + 0 \]

      13. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left({z}^{2}, 3, 0\right)} \]

      14. +-rgt-identity N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{{z}^{2} + 0}, 3, 0\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot z} + 0, 3, 0\right) \]

      16. accelerator-lowering-fma.f64 84.2

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(z, z, 0\right)}, 3, 0\right) \]

   5. Simplified 84.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(z, z, 0\right), 3, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{\left(z \cdot z + 0\right) \cdot 3} \]

      2. +-rgt-identity N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 84.3

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

   7. Applied egg-rr 84.3%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{-62}:\\ \;\;\;\;\mathsf{fma}\left(z, z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{-62}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 1e-62) (* x y) (* z (* z 3.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 1e-62) {
		tmp = x * y;
	} else {
		tmp = z * (z * 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z * z) <= 1d-62) then
        tmp = x * y
    else
        tmp = z * (z * 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 1e-62) {
		tmp = x * y;
	} else {
		tmp = z * (z * 3.0);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z * z) <= 1e-62:
		tmp = x * y
	else:
		tmp = z * (z * 3.0)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 1e-62)
		tmp = Float64(x * y);
	else
		tmp = Float64(z * Float64(z * 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 1e-62)
		tmp = x * y;
	else
		tmp = z * (z * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 1e-62], N[(x * y), $MachinePrecision], N[(z * N[(z * 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1e-62

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{x \cdot y + 0} \]

      2. accelerator-lowering-fma.f64 90.0

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

   5. Simplified 90.0%

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

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 90.0

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

   7. Applied egg-rr 90.0%

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

   if 1e-62 < (*.f64 z 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. *-rgt-identity N/A

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

      2. *-inverses N/A

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

      3. associate-/l* N/A

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

      4. associate-*l/ N/A

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

      5. +-rgt-identity N/A

         \[\leadsto \color{blue}{\frac{2 \cdot {z}^{2} + {z}^{2}}{y} \cdot y + 0} \]

      6. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\left(2 \cdot {z}^{2} + {z}^{2}\right) \cdot y}{y}} + 0 \]

      7. associate-/l* N/A

         \[\leadsto \color{blue}{\left(2 \cdot {z}^{2} + {z}^{2}\right) \cdot \frac{y}{y}} + 0 \]

      8. *-inverses N/A

         \[\leadsto \left(2 \cdot {z}^{2} + {z}^{2}\right) \cdot \color{blue}{1} + 0 \]

      9. *-rgt-identity N/A

         \[\leadsto \color{blue}{\left(2 \cdot {z}^{2} + {z}^{2}\right)} + 0 \]

      10. distribute-lft1-in N/A

          \[\leadsto \color{blue}{\left(2 + 1\right) \cdot {z}^{2}} + 0 \]

      11. metadata-eval N/A

          \[\leadsto \color{blue}{3} \cdot {z}^{2} + 0 \]

      12. *-commutative N/A

          \[\leadsto \color{blue}{{z}^{2} \cdot 3} + 0 \]

      13. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left({z}^{2}, 3, 0\right)} \]

      14. +-rgt-identity N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{{z}^{2} + 0}, 3, 0\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot z} + 0, 3, 0\right) \]

      16. accelerator-lowering-fma.f64 84.2

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(z, z, 0\right)}, 3, 0\right) \]

   5. Simplified 84.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(z, z, 0\right), 3, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{\left(z \cdot z + 0\right) \cdot 3} \]

      2. +-rgt-identity N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 84.3

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

   7. Applied egg-rr 84.3%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{-62}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+271}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= (* z z) 2e+271) (* x y) (* z z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e+271) {
		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) <= 2d+271) 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) <= 2e+271) {
		tmp = x * y;
	} else {
		tmp = z * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z * z) <= 2e+271:
		tmp = x * y
	else:
		tmp = z * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 2e+271)
		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) <= 2e+271)
		tmp = x * y;
	else
		tmp = z * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 2e+271], N[(x * y), $MachinePrecision], N[(z * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.99999999999999991e271

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

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

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{x \cdot y + 0} \]

      2. accelerator-lowering-fma.f64 68.9

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

   5. Simplified 68.9%

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

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 68.9

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

   7. Applied egg-rr 68.9%

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

   if 1.99999999999999991e271 < (*.f64 z z)

   1. Initial program 92.0%

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

      1. associate-+l+ N/A

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

      2. +-commutative N/A

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

      3. count-2 N/A

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

      4. associate-*r* N/A

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

      5. count-2 N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

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

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. *-lowering-*.f64 98.6

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

   4. Applied egg-rr 98.6%

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

   5. Taylor expanded in x around 0

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

      1. +-rgt-identity N/A

         \[\leadsto \mathsf{fma}\left(z + z, z, \color{blue}{{z}^{2} + 0}\right) \]

      2. unpow2 N/A

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

      3. accelerator-lowering-fma.f64 97.5

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

   7. Simplified 97.5%

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{z \cdot \left(z + z\right)} + \left(z \cdot z + 0\right) \]

      2. flip-+ N/A

         \[\leadsto z \cdot \color{blue}{\frac{z \cdot z - z \cdot z}{z - z}} + \left(z \cdot z + 0\right) \]

      3. +-inverses N/A

         \[\leadsto z \cdot \frac{\color{blue}{0}}{z - z} + \left(z \cdot z + 0\right) \]

      4. metadata-eval N/A

         \[\leadsto z \cdot \frac{\color{blue}{0 - 0}}{z - z} + \left(z \cdot z + 0\right) \]

      5. metadata-eval N/A

         \[\leadsto z \cdot \frac{\color{blue}{0 \cdot 0} - 0}{z - z} + \left(z \cdot z + 0\right) \]

      6. metadata-eval N/A

         \[\leadsto z \cdot \frac{0 \cdot 0 - \color{blue}{0 \cdot 0}}{z - z} + \left(z \cdot z + 0\right) \]

      7. +-inverses N/A

         \[\leadsto z \cdot \frac{0 \cdot 0 - 0 \cdot 0}{\color{blue}{0}} + \left(z \cdot z + 0\right) \]

      8. metadata-eval N/A

         \[\leadsto z \cdot \frac{0 \cdot 0 - 0 \cdot 0}{\color{blue}{0 + 0}} + \left(z \cdot z + 0\right) \]

      9. flip-- N/A

         \[\leadsto z \cdot \color{blue}{\left(0 - 0\right)} + \left(z \cdot z + 0\right) \]

      10. metadata-eval N/A

          \[\leadsto z \cdot \color{blue}{0} + \left(z \cdot z + 0\right) \]

      11. +-inverses N/A

          \[\leadsto z \cdot \color{blue}{\left(z - z\right)} + \left(z \cdot z + 0\right) \]

      12. distribute-lft-out-- N/A

          \[\leadsto \color{blue}{\left(z \cdot z - z \cdot z\right)} + \left(z \cdot z + 0\right) \]

      13. +-inverses N/A

          \[\leadsto \color{blue}{0} + \left(z \cdot z + 0\right) \]

      14. +-rgt-identity N/A

          \[\leadsto 0 + \color{blue}{z \cdot z} \]

      15. +-lft-identity N/A

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

      16. *-lowering-*.f64 89.9

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

   9. Applied egg-rr 89.9%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+271}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.2% 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 97.5%

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

   1. associate-+l+ N/A

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

   2. associate-+l+ N/A

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

   3. *-commutative N/A

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

   4. accelerator-lowering-fma.f64 N/A

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

   5. count-2 N/A

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

   6. distribute-lft1-in N/A

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

   7. metadata-eval N/A

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

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

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

   9. *-lowering-*.f64 99.5

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

4. Applied egg-rr 99.5%

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

5. Final simplification 99.5%

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

Alternative 9: 53.2% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 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 x around inf

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

   1. +-rgt-identity N/A

      \[\leadsto \color{blue}{x \cdot y + 0} \]

   2. accelerator-lowering-fma.f64 51.3

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

5. Simplified 51.3%

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

   1. +-rgt-identity N/A

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

   2. *-commutative N/A

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

   3. *-lowering-*.f64 51.3

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

7. Applied egg-rr 51.3%

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

8. Final simplification 51.3%

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

Developer Target 1: 98.2% 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 2024198 
(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)))