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

Percentage Accurate: 98.5% → 99.6%

Time: 7.7s

Alternatives: 10

Speedup: 1.4×

• 34.3% 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: 98.5% 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: 99.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;z\_m \leq 10^{+224}:\\ \;\;\;\;\mathsf{fma}\left(z\_m, z\_m \cdot 3, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;z\_m \cdot z\_m\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m)
 :precision binary64
 (if (<= z_m 1e+224) (fma z_m (* z_m 3.0) (* x y)) (* z_m z_m)))

C

z_m = fabs(z);
double code(double x, double y, double z_m) {
	double tmp;
	if (z_m <= 1e+224) {
		tmp = fma(z_m, (z_m * 3.0), (x * y));
	} else {
		tmp = z_m * z_m;
	}
	return tmp;
}

Julia

z_m = abs(z)
function code(x, y, z_m)
	tmp = 0.0
	if (z_m <= 1e+224)
		tmp = fma(z_m, Float64(z_m * 3.0), Float64(x * y));
	else
		tmp = Float64(z_m * z_m);
	end
	return tmp
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_] := If[LessEqual[z$95$m, 1e+224], N[(z$95$m * N[(z$95$m * 3.0), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision], N[(z$95$m * z$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 9.9999999999999997e223

   1. Initial program 98.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-+l+ N/A

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

      5. lift-*.f64 N/A

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

      6. associate-+l+ N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. distribute-lft-out N/A

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

      11. lift-*.f64 N/A

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

      12. distribute-lft-out N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. lower-+.f64 99.0

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

   4. Applied egg-rr 99.0%

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

      1. count-2 N/A

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

      2. distribute-lft1-in N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 99.0

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

   6. Applied egg-rr 99.0%

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

   if 9.9999999999999997e223 < z

   1. Initial program 85.7%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 85.7

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

   5. Simplified 85.7%

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

   6. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 100.0

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

   8. Simplified 100.0%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 10^{+224}:\\ \;\;\;\;\mathsf{fma}\left(z, z \cdot 3, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;z\_m \cdot z\_m \leq 5000:\\ \;\;\;\;\mathsf{fma}\left(z\_m + z\_m, z\_m, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z\_m + z\_m, z\_m, z\_m \cdot z\_m\right)\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m)
 :precision binary64
 (if (<= (* z_m z_m) 5000.0)
   (fma (+ z_m z_m) z_m (* x y))
   (fma (+ z_m z_m) z_m (* z_m z_m))))

C

z_m = fabs(z);
double code(double x, double y, double z_m) {
	double tmp;
	if ((z_m * z_m) <= 5000.0) {
		tmp = fma((z_m + z_m), z_m, (x * y));
	} else {
		tmp = fma((z_m + z_m), z_m, (z_m * z_m));
	}
	return tmp;
}

Julia

z_m = abs(z)
function code(x, y, z_m)
	tmp = 0.0
	if (Float64(z_m * z_m) <= 5000.0)
		tmp = fma(Float64(z_m + z_m), z_m, Float64(x * y));
	else
		tmp = fma(Float64(z_m + z_m), z_m, Float64(z_m * z_m));
	end
	return tmp
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_] := If[LessEqual[N[(z$95$m * z$95$m), $MachinePrecision], 5000.0], N[(N[(z$95$m + z$95$m), $MachinePrecision] * z$95$m + N[(x * y), $MachinePrecision]), $MachinePrecision], N[(N[(z$95$m + z$95$m), $MachinePrecision] * z$95$m + N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 5e3

   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

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-+l+ N/A

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

      7. +-commutative N/A

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

      8. count-2 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. count-2 N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-+.f64 100.0

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

      14. lift-+.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lower-fma.f64 100.0

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

   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. lower-*.f64 84.9

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

   7. Simplified 84.9%

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

   if 5e3 < (*.f64 z z)

   1. Initial program 94.1%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{2 \cdot {z}^{2} + {z}^{2}} \]
   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. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 90.5

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

   5. Simplified 90.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. distribute-lft1-in N/A

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

      5. *-commutative N/A

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

   7. Applied egg-rr 90.6%

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

Alternative 3: 85.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;z\_m \cdot z\_m \leq 5000:\\ \;\;\;\;\mathsf{fma}\left(z\_m + z\_m, z\_m, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(z\_m \cdot z\_m\right)\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m)
 :precision binary64
 (if (<= (* z_m z_m) 5000.0)
   (fma (+ z_m z_m) z_m (* x y))
   (* 3.0 (* z_m z_m))))

C

z_m = fabs(z);
double code(double x, double y, double z_m) {
	double tmp;
	if ((z_m * z_m) <= 5000.0) {
		tmp = fma((z_m + z_m), z_m, (x * y));
	} else {
		tmp = 3.0 * (z_m * z_m);
	}
	return tmp;
}

Julia

z_m = abs(z)
function code(x, y, z_m)
	tmp = 0.0
	if (Float64(z_m * z_m) <= 5000.0)
		tmp = fma(Float64(z_m + z_m), z_m, Float64(x * y));
	else
		tmp = Float64(3.0 * Float64(z_m * z_m));
	end
	return tmp
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_] := If[LessEqual[N[(z$95$m * z$95$m), $MachinePrecision], 5000.0], N[(N[(z$95$m + z$95$m), $MachinePrecision] * z$95$m + N[(x * y), $MachinePrecision]), $MachinePrecision], N[(3.0 * N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 5e3

   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

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-+l+ N/A

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

      7. +-commutative N/A

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

      8. count-2 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. count-2 N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-+.f64 100.0

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

      14. lift-+.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lower-fma.f64 100.0

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

   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. lower-*.f64 84.9

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

   7. Simplified 84.9%

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

   if 5e3 < (*.f64 z z)

   1. Initial program 94.1%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{2 \cdot {z}^{2} + {z}^{2}} \]
   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. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 90.5

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

   5. Simplified 90.5%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5000:\\ \;\;\;\;\mathsf{fma}\left(z + z, z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(z \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;z\_m \leq 4.7 \cdot 10^{+116}:\\ \;\;\;\;\mathsf{fma}\left(3, z\_m \cdot z\_m, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z\_m + z\_m, z\_m, z\_m \cdot z\_m\right)\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m)
 :precision binary64
 (if (<= z_m 4.7e+116)
   (fma 3.0 (* z_m z_m) (* x y))
   (fma (+ z_m z_m) z_m (* z_m z_m))))

C

z_m = fabs(z);
double code(double x, double y, double z_m) {
	double tmp;
	if (z_m <= 4.7e+116) {
		tmp = fma(3.0, (z_m * z_m), (x * y));
	} else {
		tmp = fma((z_m + z_m), z_m, (z_m * z_m));
	}
	return tmp;
}

Julia

z_m = abs(z)
function code(x, y, z_m)
	tmp = 0.0
	if (z_m <= 4.7e+116)
		tmp = fma(3.0, Float64(z_m * z_m), Float64(x * y));
	else
		tmp = fma(Float64(z_m + z_m), z_m, Float64(z_m * z_m));
	end
	return tmp
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_] := If[LessEqual[z$95$m, 4.7e+116], N[(3.0 * N[(z$95$m * z$95$m), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision], N[(N[(z$95$m + z$95$m), $MachinePrecision] * z$95$m + N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 4.7000000000000003e116

   1. Initial program 98.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-+l+ N/A

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

      5. lift-*.f64 N/A

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

      6. associate-+l+ N/A

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

      7. +-commutative N/A

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

      8. count-2 N/A

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

      9. distribute-lft1-in N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 98.0

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

   4. Applied egg-rr 98.0%

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

   if 4.7000000000000003e116 < 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. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 99.9

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

   5. Simplified 99.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. distribute-lft1-in N/A

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

      5. *-commutative N/A

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

   7. Applied egg-rr 100.0%

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

Alternative 5: 85.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;z\_m \cdot z\_m \leq 5000:\\ \;\;\;\;\mathsf{fma}\left(z\_m, z\_m, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(z\_m \cdot z\_m\right)\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m)
 :precision binary64
 (if (<= (* z_m z_m) 5000.0) (fma z_m z_m (* x y)) (* 3.0 (* z_m z_m))))

C

z_m = fabs(z);
double code(double x, double y, double z_m) {
	double tmp;
	if ((z_m * z_m) <= 5000.0) {
		tmp = fma(z_m, z_m, (x * y));
	} else {
		tmp = 3.0 * (z_m * z_m);
	}
	return tmp;
}

Julia

z_m = abs(z)
function code(x, y, z_m)
	tmp = 0.0
	if (Float64(z_m * z_m) <= 5000.0)
		tmp = fma(z_m, z_m, Float64(x * y));
	else
		tmp = Float64(3.0 * Float64(z_m * z_m));
	end
	return tmp
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_] := If[LessEqual[N[(z$95$m * z$95$m), $MachinePrecision], 5000.0], N[(z$95$m * z$95$m + N[(x * y), $MachinePrecision]), $MachinePrecision], N[(3.0 * N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 5e3

   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

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 100.0

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

      10. lift-+.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. associate-+l+ N/A

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

      13. lift-*.f64 N/A

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

      14. lower-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. distribute-lft-out N/A

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

      18. lower-*.f64 N/A

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

      19. lower-+.f64 100.0

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

   4. Applied egg-rr 100.0%

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

      1. lift-*.f64 N/A

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

      2. distribute-lft-in N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. distribute-lft-in N/A

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

      8. flip-+ N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. +-inverses N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. +-inverses N/A

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

      16. metadata-eval N/A

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

      17. flip-- N/A

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

      18. metadata-eval N/A

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

      19. mul0-rgt N/A

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

      20. +-commutative N/A

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

      21. +-lft-identity 84.5

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

      22. lift-*.f64 N/A

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

      23. *-commutative N/A

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

      24. lower-*.f64 84.5

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

   6. Applied egg-rr 84.5%

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

   if 5e3 < (*.f64 z z)

   1. Initial program 94.1%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{2 \cdot {z}^{2} + {z}^{2}} \]
   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. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 90.5

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

   5. Simplified 90.5%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5000:\\ \;\;\;\;\mathsf{fma}\left(z, z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(z \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;z\_m \cdot z\_m \leq 5000:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(z\_m \cdot z\_m\right)\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m)
 :precision binary64
 (if (<= (* z_m z_m) 5000.0) (* x y) (* 3.0 (* z_m z_m))))

C

z_m = fabs(z);
double code(double x, double y, double z_m) {
	double tmp;
	if ((z_m * z_m) <= 5000.0) {
		tmp = x * y;
	} else {
		tmp = 3.0 * (z_m * z_m);
	}
	return tmp;
}

Fortran

z_m = abs(z)
real(8) function code(x, y, z_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if ((z_m * z_m) <= 5000.0d0) then
        tmp = x * y
    else
        tmp = 3.0d0 * (z_m * z_m)
    end if
    code = tmp
end function

Java

z_m = Math.abs(z);
public static double code(double x, double y, double z_m) {
	double tmp;
	if ((z_m * z_m) <= 5000.0) {
		tmp = x * y;
	} else {
		tmp = 3.0 * (z_m * z_m);
	}
	return tmp;
}

Python

z_m = math.fabs(z)
def code(x, y, z_m):
	tmp = 0
	if (z_m * z_m) <= 5000.0:
		tmp = x * y
	else:
		tmp = 3.0 * (z_m * z_m)
	return tmp

Julia

z_m = abs(z)
function code(x, y, z_m)
	tmp = 0.0
	if (Float64(z_m * z_m) <= 5000.0)
		tmp = Float64(x * y);
	else
		tmp = Float64(3.0 * Float64(z_m * z_m));
	end
	return tmp
end

MATLAB

z_m = abs(z);
function tmp_2 = code(x, y, z_m)
	tmp = 0.0;
	if ((z_m * z_m) <= 5000.0)
		tmp = x * y;
	else
		tmp = 3.0 * (z_m * z_m);
	end
	tmp_2 = tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_] := If[LessEqual[N[(z$95$m * z$95$m), $MachinePrecision], 5000.0], N[(x * y), $MachinePrecision], N[(3.0 * N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 5e3

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 82.3

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

   5. Simplified 82.3%

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

   if 5e3 < (*.f64 z z)

   1. Initial program 94.1%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{2 \cdot {z}^{2} + {z}^{2}} \]
   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. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 90.5

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

   5. Simplified 90.5%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5000:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(z \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;z\_m \cdot z\_m \leq 5000:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z\_m \cdot \left(z\_m \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m)
 :precision binary64
 (if (<= (* z_m z_m) 5000.0) (* x y) (* z_m (* z_m 3.0))))

C

z_m = fabs(z);
double code(double x, double y, double z_m) {
	double tmp;
	if ((z_m * z_m) <= 5000.0) {
		tmp = x * y;
	} else {
		tmp = z_m * (z_m * 3.0);
	}
	return tmp;
}

Fortran

z_m = abs(z)
real(8) function code(x, y, z_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if ((z_m * z_m) <= 5000.0d0) then
        tmp = x * y
    else
        tmp = z_m * (z_m * 3.0d0)
    end if
    code = tmp
end function

Java

z_m = Math.abs(z);
public static double code(double x, double y, double z_m) {
	double tmp;
	if ((z_m * z_m) <= 5000.0) {
		tmp = x * y;
	} else {
		tmp = z_m * (z_m * 3.0);
	}
	return tmp;
}

Python

z_m = math.fabs(z)
def code(x, y, z_m):
	tmp = 0
	if (z_m * z_m) <= 5000.0:
		tmp = x * y
	else:
		tmp = z_m * (z_m * 3.0)
	return tmp

Julia

z_m = abs(z)
function code(x, y, z_m)
	tmp = 0.0
	if (Float64(z_m * z_m) <= 5000.0)
		tmp = Float64(x * y);
	else
		tmp = Float64(z_m * Float64(z_m * 3.0));
	end
	return tmp
end

MATLAB

z_m = abs(z);
function tmp_2 = code(x, y, z_m)
	tmp = 0.0;
	if ((z_m * z_m) <= 5000.0)
		tmp = x * y;
	else
		tmp = z_m * (z_m * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_] := If[LessEqual[N[(z$95$m * z$95$m), $MachinePrecision], 5000.0], N[(x * y), $MachinePrecision], N[(z$95$m * N[(z$95$m * 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 5e3

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 82.3

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

   5. Simplified 82.3%

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

   if 5e3 < (*.f64 z z)

   1. Initial program 94.1%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{2 \cdot {z}^{2} + {z}^{2}} \]
   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. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 90.5

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

   5. Simplified 90.5%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 90.4

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

   7. Applied egg-rr 90.4%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5000:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.5% accurate, 1.4× speedup

Math

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

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m)
 :precision binary64
 (fma z_m z_m (fma x y (* z_m (+ z_m z_m)))))

C

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

Julia

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

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_] := N[(z$95$m * z$95$m + N[(x * y + N[(z$95$m * N[(z$95$m + z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.2%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. +-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. lower-fma.f64 97.2

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

   10. lift-+.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. associate-+l+ N/A

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

   13. lift-*.f64 N/A

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

   14. lower-fma.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. lift-*.f64 N/A

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

   17. distribute-lft-out N/A

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

   18. lower-*.f64 N/A

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

   19. lower-+.f64 99.2

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

4. Applied egg-rr 99.2%

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

Alternative 9: 74.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;z\_m \cdot z\_m \leq 10^{+213}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z\_m \cdot z\_m\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m)
 :precision binary64
 (if (<= (* z_m z_m) 1e+213) (* x y) (* z_m z_m)))

C

z_m = fabs(z);
double code(double x, double y, double z_m) {
	double tmp;
	if ((z_m * z_m) <= 1e+213) {
		tmp = x * y;
	} else {
		tmp = z_m * z_m;
	}
	return tmp;
}

Fortran

z_m = abs(z)
real(8) function code(x, y, z_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if ((z_m * z_m) <= 1d+213) then
        tmp = x * y
    else
        tmp = z_m * z_m
    end if
    code = tmp
end function

Java

z_m = Math.abs(z);
public static double code(double x, double y, double z_m) {
	double tmp;
	if ((z_m * z_m) <= 1e+213) {
		tmp = x * y;
	} else {
		tmp = z_m * z_m;
	}
	return tmp;
}

Python

z_m = math.fabs(z)
def code(x, y, z_m):
	tmp = 0
	if (z_m * z_m) <= 1e+213:
		tmp = x * y
	else:
		tmp = z_m * z_m
	return tmp

Julia

z_m = abs(z)
function code(x, y, z_m)
	tmp = 0.0
	if (Float64(z_m * z_m) <= 1e+213)
		tmp = Float64(x * y);
	else
		tmp = Float64(z_m * z_m);
	end
	return tmp
end

MATLAB

z_m = abs(z);
function tmp_2 = code(x, y, z_m)
	tmp = 0.0;
	if ((z_m * z_m) <= 1e+213)
		tmp = x * y;
	else
		tmp = z_m * z_m;
	end
	tmp_2 = tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_] := If[LessEqual[N[(z$95$m * z$95$m), $MachinePrecision], 1e+213], N[(x * y), $MachinePrecision], N[(z$95$m * z$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 9.99999999999999984e212

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 71.7

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

   5. Simplified 71.7%

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

   if 9.99999999999999984e212 < (*.f64 z z)

   1. Initial program 92.1%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 80.1

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

   5. Simplified 80.1%

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

   6. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 85.7

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

   8. Simplified 85.7%

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

Alternative 10: 52.5% accurate, 5.0× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ x \cdot y \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m) :precision binary64 (* x y))

C

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

Fortran

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

Java

z_m = Math.abs(z);
public static double code(double x, double y, double z_m) {
	return x * y;
}

Python

z_m = math.fabs(z)
def code(x, y, z_m):
	return x * y

Julia

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

MATLAB

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

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_] := N[(x * y), $MachinePrecision]

Derivation

1. Initial program 97.2%

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

3. Taylor expanded in x around inf

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

   1. lower-*.f64 51.1

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

5. Simplified 51.1%

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

Developer Target 1: 98.5% 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 2024208 
(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)))