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

Percentage Accurate: 98.5% → 99.5%

Time: 7.3s

Alternatives: 6

Speedup: 1.8×

• 33.5% 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: 99.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;z\_m \leq 10^{+175}:\\ \;\;\;\;\mathsf{fma}\left(z\_m \cdot 3, z\_m, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z\_m \cdot z\_m\right) \cdot 3\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m)
 :precision binary64
 (if (<= z_m 1e+175) (fma (* z_m 3.0) z_m (* 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 <= 1e+175) {
		tmp = fma((z_m * 3.0), z_m, (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 (z_m <= 1e+175)
		tmp = fma(Float64(z_m * 3.0), z_m, Float64(x * y));
	else
		tmp = Float64(Float64(z_m * z_m) * 3.0);
	end
	return tmp
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_] := If[LessEqual[z$95$m, 1e+175], N[(N[(z$95$m * 3.0), $MachinePrecision] * z$95$m + N[(x * y), $MachinePrecision]), $MachinePrecision], N[(N[(z$95$m * z$95$m), $MachinePrecision] * 3.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 9.9999999999999994e174

   1. Initial program 97.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 98.1

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

   5. Applied rewrites 98.1%

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

   if 9.9999999999999994e174 < z

   1. Initial program 92.3%

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

   3. Taylor expanded in x around 0

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

      1. distribute-lft1-in N/A

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

      2. metadata-eval N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 10^{+175}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot 3, z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot z\right) \cdot 3\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;z\_m \cdot z\_m \leq 2 \cdot 10^{-24}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(z\_m \cdot 3\right) \cdot z\_m\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m)
 :precision binary64
 (if (<= (* z_m z_m) 2e-24) (* x y) (* (* z_m 3.0) z_m)))

C

z_m = fabs(z);
double code(double x, double y, double z_m) {
	double tmp;
	if ((z_m * z_m) <= 2e-24) {
		tmp = x * y;
	} else {
		tmp = (z_m * 3.0) * 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) <= 2d-24) then
        tmp = x * y
    else
        tmp = (z_m * 3.0d0) * 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) <= 2e-24) {
		tmp = x * y;
	} else {
		tmp = (z_m * 3.0) * z_m;
	}
	return tmp;
}

Python

z_m = math.fabs(z)
def code(x, y, z_m):
	tmp = 0
	if (z_m * z_m) <= 2e-24:
		tmp = x * y
	else:
		tmp = (z_m * 3.0) * z_m
	return tmp

Julia

z_m = abs(z)
function code(x, y, z_m)
	tmp = 0.0
	if (Float64(z_m * z_m) <= 2e-24)
		tmp = Float64(x * y);
	else
		tmp = Float64(Float64(z_m * 3.0) * 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) <= 2e-24)
		tmp = x * y;
	else
		tmp = (z_m * 3.0) * 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], 2e-24], N[(x * y), $MachinePrecision], N[(N[(z$95$m * 3.0), $MachinePrecision] * z$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.99999999999999985e-24

   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 0

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

      1. distribute-lft1-in N/A

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

      2. metadata-eval N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 19.2

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

   5. Applied rewrites 19.2%

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

   7. Applied rewrites 6.3%

      \[\leadsto 3 \cdot e^{\log z \cdot 2} \]

   8. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 85.4

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

   10. Applied rewrites 85.4%

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

   if 1.99999999999999985e-24 < (*.f64 z z)

   1. Initial program 94.9%

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

   3. Taylor expanded in x around 0

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

      1. distribute-lft1-in N/A

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

      2. metadata-eval N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 85.6

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

   5. Applied rewrites 85.6%

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

   7. Applied rewrites 85.7%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{-24}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot 3\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;z\_m \cdot z\_m \leq 1.2 \cdot 10^{-21}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(z\_m \cdot z\_m\right) \cdot 3\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m)
 :precision binary64
 (if (<= (* z_m z_m) 1.2e-21) (* 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) <= 1.2e-21) {
		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) <= 1.2d-21) 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) <= 1.2e-21) {
		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) <= 1.2e-21:
		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) <= 1.2e-21)
		tmp = Float64(x * y);
	else
		tmp = Float64(Float64(z_m * 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) <= 1.2e-21)
		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], 1.2e-21], N[(x * y), $MachinePrecision], N[(N[(z$95$m * z$95$m), $MachinePrecision] * 3.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.2e-21

   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 0

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

      1. distribute-lft1-in N/A

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

      2. metadata-eval N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 19.2

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

   5. Applied rewrites 19.2%

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

   7. Applied rewrites 6.3%

      \[\leadsto 3 \cdot e^{\log z \cdot 2} \]

   8. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 85.4

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

   10. Applied rewrites 85.4%

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

   if 1.2e-21 < (*.f64 z z)

   1. Initial program 94.9%

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

   3. Taylor expanded in x around 0

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

      1. distribute-lft1-in N/A

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

      2. metadata-eval N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 85.6

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

   5. Applied rewrites 85.6%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 1.2 \cdot 10^{-21}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot z\right) \cdot 3\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;z\_m \leq 4.6 \cdot 10^{-10}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z\_m, z\_m + z\_m, z\_m \cdot z\_m\right)\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m)
 :precision binary64
 (if (<= z_m 4.6e-10) (* 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 <= 4.6e-10) {
		tmp = 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 <= 4.6e-10)
		tmp = Float64(x * y);
	else
		tmp = fma(z_m, Float64(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, 4.6e-10], N[(x * y), $MachinePrecision], N[(z$95$m * N[(z$95$m + z$95$m), $MachinePrecision] + N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 4.60000000000000014e-10

   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 0

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

      1. distribute-lft1-in N/A

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

      2. metadata-eval N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 45.2

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

   5. Applied rewrites 45.2%

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

   7. Applied rewrites 3.8%

      \[\leadsto 3 \cdot e^{\log z \cdot 2} \]

   8. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 58.5

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

   10. Applied rewrites 58.5%

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

   if 4.60000000000000014e-10 < z

   1. Initial program 96.9%

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

   3. Taylor expanded in x around 0

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

      1. distribute-lft1-in N/A

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

      2. metadata-eval N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 87.4

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

   5. Applied rewrites 87.4%

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

   7. Applied rewrites 87.4%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 4.6 \cdot 10^{-10}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, z + z, z \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.4% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 97.1%

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

   1. lift-+.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-+.f64 N/A

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

   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. associate-+l+ N/A

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

   6. lift-*.f64 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

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

   8. lower-fma.f64 N/A

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

   9. count-2 N/A

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

   10. distribute-lft1-in N/A

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

   11. metadata-eval N/A

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

   12. lower-*.f64 99.4

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

4. Applied rewrites 99.4%

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

5. Final simplification 99.4%

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

Alternative 6: 53.0% 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 97.1%

   \[\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z \]
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. lower-*.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 56.6

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

5. Applied rewrites 56.6%

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

7. Applied rewrites 25.4%

   \[\leadsto 3 \cdot e^{\log z \cdot 2} \]

8. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 47.7

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

10. Applied rewrites 47.7%

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

11. Final simplification 47.7%

    \[\leadsto x \cdot y \]
12. 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 2024288 
(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)))