Linear.V3:$cdot from linear-1.19.1.3, B

Percentage Accurate: 97.9% → 99.0%

Time: 5.4s

Alternatives: 7

Speedup: 1.2×

• 32.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

(FPCore (x y z t a b) :precision binary64 (+ (+ (* x y) (* z t)) (* a b)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(N[(x * y), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision] + N[(a * b), $MachinePrecision]), $MachinePrecision]

Initial Program: 97.9% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z t a b) :precision binary64 (+ (+ (* x y) (* z t)) (* a b)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(N[(x * y), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision] + N[(a * b), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.0% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (x y z t a b) :precision binary64 (fma z t (fma b a (* y x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return fma(z, t, fma(b, a, (y * x)));
}

Julia

function code(x, y, z, t, a, b)
	return fma(z, t, fma(b, a, Float64(y * x)))
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(z * t + N[(b * a + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.4%

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

   1. lift-+.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. associate-+l+ N/A

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

   5. lift-*.f64 N/A

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

   6. lower-fma.f64 N/A

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

   7. +-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 99.2

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 99.2

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

4. Applied rewrites 99.2%

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

Alternative 2: 85.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+134} \lor \neg \left(z \cdot t \leq 10^{+83}\right):\\ \;\;\;\;\mathsf{fma}\left(t, z, b \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, y \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= (* z t) -2e+134) (not (<= (* z t) 1e+83)))
   (fma t z (* b a))
   (fma b a (* y x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((z * t) <= -2e+134) || !((z * t) <= 1e+83)) {
		tmp = fma(t, z, (b * a));
	} else {
		tmp = fma(b, a, (y * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((Float64(z * t) <= -2e+134) || !(Float64(z * t) <= 1e+83))
		tmp = fma(t, z, Float64(b * a));
	else
		tmp = fma(b, a, Float64(y * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[N[(z * t), $MachinePrecision], -2e+134], N[Not[LessEqual[N[(z * t), $MachinePrecision], 1e+83]], $MachinePrecision]], N[(t * z + N[(b * a), $MachinePrecision]), $MachinePrecision], N[(b * a + N[(y * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -1.99999999999999984e134 or 1.00000000000000003e83 < (*.f64 z t)

   1. Initial program 96.8%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{a \cdot b + t \cdot z} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{t \cdot z + a \cdot b} \]

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 90.1

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

   5. Applied rewrites 90.1%

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

   if -1.99999999999999984e134 < (*.f64 z t) < 1.00000000000000003e83

   1. Initial program 99.4%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 89.9

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

   5. Applied rewrites 89.9%

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

   7. Applied rewrites 89.3%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+134} \lor \neg \left(z \cdot t \leq 10^{+83}\right):\\ \;\;\;\;\mathsf{fma}\left(t, z, b \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, y \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+182} \lor \neg \left(z \cdot t \leq 10^{+83}\right):\\ \;\;\;\;t \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, y \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= (* z t) -2e+182) (not (<= (* z t) 1e+83)))
   (* t z)
   (fma b a (* y x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((z * t) <= -2e+182) || !((z * t) <= 1e+83)) {
		tmp = t * z;
	} else {
		tmp = fma(b, a, (y * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((Float64(z * t) <= -2e+182) || !(Float64(z * t) <= 1e+83))
		tmp = Float64(t * z);
	else
		tmp = fma(b, a, Float64(y * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[N[(z * t), $MachinePrecision], -2e+182], N[Not[LessEqual[N[(z * t), $MachinePrecision], 1e+83]], $MachinePrecision]], N[(t * z), $MachinePrecision], N[(b * a + N[(y * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -2.0000000000000001e182 or 1.00000000000000003e83 < (*.f64 z t)

   1. Initial program 96.4%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{a \cdot b + t \cdot z} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{t \cdot z + a \cdot b} \]

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 89.9

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

   5. Applied rewrites 89.9%

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

   6. Taylor expanded in z around inf

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

      1. lower-*.f64 80.0

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

   8. Applied rewrites 80.0%

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

   if -2.0000000000000001e182 < (*.f64 z t) < 1.00000000000000003e83

   1. Initial program 99.4%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 88.9

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

   5. Applied rewrites 88.9%

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

   7. Applied rewrites 88.3%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+182} \lor \neg \left(z \cdot t \leq 10^{+83}\right):\\ \;\;\;\;t \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, y \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 53.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+182} \lor \neg \left(z \cdot t \leq 2 \cdot 10^{+27}\right):\\ \;\;\;\;t \cdot z\\ \mathbf{else}:\\ \;\;\;\;b \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= (* z t) -2e+182) (not (<= (* z t) 2e+27))) (* t z) (* b a)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((z * t) <= -2e+182) || !((z * t) <= 2e+27)) {
		tmp = t * z;
	} else {
		tmp = b * a;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (((z * t) <= (-2d+182)) .or. (.not. ((z * t) <= 2d+27))) then
        tmp = t * z
    else
        tmp = b * a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((z * t) <= -2e+182) || !((z * t) <= 2e+27)) {
		tmp = t * z;
	} else {
		tmp = b * a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if ((z * t) <= -2e+182) or not ((z * t) <= 2e+27):
		tmp = t * z
	else:
		tmp = b * a
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((Float64(z * t) <= -2e+182) || !(Float64(z * t) <= 2e+27))
		tmp = Float64(t * z);
	else
		tmp = Float64(b * a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (((z * t) <= -2e+182) || ~(((z * t) <= 2e+27)))
		tmp = t * z;
	else
		tmp = b * a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[N[(z * t), $MachinePrecision], -2e+182], N[Not[LessEqual[N[(z * t), $MachinePrecision], 2e+27]], $MachinePrecision]], N[(t * z), $MachinePrecision], N[(b * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -2.0000000000000001e182 or 2e27 < (*.f64 z t)

   1. Initial program 96.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{a \cdot b + t \cdot z} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{t \cdot z + a \cdot b} \]

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 86.7

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

   5. Applied rewrites 86.7%

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

   6. Taylor expanded in z around inf

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

      1. lower-*.f64 76.7

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

   8. Applied rewrites 76.7%

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

   if -2.0000000000000001e182 < (*.f64 z t) < 2e27

   1. Initial program 99.4%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 90.6

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

   5. Applied rewrites 90.6%

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

   6. Taylor expanded in x around 0

      \[\leadsto a \cdot \color{blue}{b} \]
   7. Step-by-step derivation

   8. Applied rewrites 48.5%

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

4. Final simplification 58.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+182} \lor \neg \left(z \cdot t \leq 2 \cdot 10^{+27}\right):\\ \;\;\;\;t \cdot z\\ \mathbf{else}:\\ \;\;\;\;b \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -15000000000000:\\ \;\;\;\;\mathsf{fma}\left(t, z, b \cdot a\right)\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{-98}:\\ \;\;\;\;\mathsf{fma}\left(y, x, b \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, z, y \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -15000000000000.0)
   (fma t z (* b a))
   (if (<= z 3.1e-98) (fma y x (* b a)) (fma t z (* y x)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -15000000000000.0) {
		tmp = fma(t, z, (b * a));
	} else if (z <= 3.1e-98) {
		tmp = fma(y, x, (b * a));
	} else {
		tmp = fma(t, z, (y * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -15000000000000.0)
		tmp = fma(t, z, Float64(b * a));
	elseif (z <= 3.1e-98)
		tmp = fma(y, x, Float64(b * a));
	else
		tmp = fma(t, z, Float64(y * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -15000000000000.0], N[(t * z + N[(b * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3.1e-98], N[(y * x + N[(b * a), $MachinePrecision]), $MachinePrecision], N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.5e13

   1. Initial program 96.6%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{a \cdot b + t \cdot z} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{t \cdot z + a \cdot b} \]

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 87.1

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

   5. Applied rewrites 87.1%

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

   if -1.5e13 < z < 3.1e-98

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 91.7

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

   5. Applied rewrites 91.7%

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

   if 3.1e-98 < z

   1. Initial program 97.5%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{a \cdot b + t \cdot z} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{t \cdot z + a \cdot b} \]

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 69.7

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

   5. Applied rewrites 69.7%

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

   6. Taylor expanded in a around 0

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

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

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

      2. distribute-lft-neg-out N/A

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

      3. distribute-rgt-neg-out N/A

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

      4. mul-1-neg N/A

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

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

         \[\leadsto \color{blue}{t \cdot z + \left(\mathsf{neg}\left(x\right)\right) \cdot \left(-1 \cdot y\right)} \]

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. mul-1-neg N/A

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

      10. distribute-lft-neg-in N/A

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

      11. distribute-rgt-neg-in N/A

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

      12. mul-1-neg N/A

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

      13. remove-double-neg N/A

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

      14. lower-*.f64 79.3

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

   8. Applied rewrites 79.3%

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

Alternative 6: 79.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -15000000000000:\\ \;\;\;\;\mathsf{fma}\left(t, z, b \cdot a\right)\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{-98}:\\ \;\;\;\;\mathsf{fma}\left(b, a, y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, z, y \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -15000000000000.0)
   (fma t z (* b a))
   (if (<= z 3.1e-98) (fma b a (* y x)) (fma t z (* y x)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -15000000000000.0) {
		tmp = fma(t, z, (b * a));
	} else if (z <= 3.1e-98) {
		tmp = fma(b, a, (y * x));
	} else {
		tmp = fma(t, z, (y * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -15000000000000.0)
		tmp = fma(t, z, Float64(b * a));
	elseif (z <= 3.1e-98)
		tmp = fma(b, a, Float64(y * x));
	else
		tmp = fma(t, z, Float64(y * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -15000000000000.0], N[(t * z + N[(b * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3.1e-98], N[(b * a + N[(y * x), $MachinePrecision]), $MachinePrecision], N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.5e13

   1. Initial program 96.6%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{a \cdot b + t \cdot z} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{t \cdot z + a \cdot b} \]

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 87.1

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

   5. Applied rewrites 87.1%

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

   if -1.5e13 < z < 3.1e-98

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 91.7

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

   5. Applied rewrites 91.7%

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

   7. Applied rewrites 91.7%

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

   if 3.1e-98 < z

   1. Initial program 97.5%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{a \cdot b + t \cdot z} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{t \cdot z + a \cdot b} \]

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 69.7

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

   5. Applied rewrites 69.7%

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

   6. Taylor expanded in a around 0

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

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

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

      2. distribute-lft-neg-out N/A

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

      3. distribute-rgt-neg-out N/A

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

      4. mul-1-neg N/A

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

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

         \[\leadsto \color{blue}{t \cdot z + \left(\mathsf{neg}\left(x\right)\right) \cdot \left(-1 \cdot y\right)} \]

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. mul-1-neg N/A

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

      10. distribute-lft-neg-in N/A

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

      11. distribute-rgt-neg-in N/A

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

      12. mul-1-neg N/A

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

      13. remove-double-neg N/A

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

      14. lower-*.f64 79.3

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

   8. Applied rewrites 79.3%

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

Alternative 7: 35.5% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ b \cdot a \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (* b a))

C

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

Fortran

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

Java

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

Python

def code(x, y, z, t, a, b):
	return b * a

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(b * a), $MachinePrecision]

Derivation

1. Initial program 98.4%

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

3. Taylor expanded in z around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 68.0

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

5. Applied rewrites 68.0%

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

6. Taylor expanded in x around 0

   \[\leadsto a \cdot \color{blue}{b} \]
7. Step-by-step derivation

8. Applied rewrites 35.9%

   \[\leadsto b \cdot \color{blue}{a} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024337 
(FPCore (x y z t a b)
  :name "Linear.V3:$cdot from linear-1.19.1.3, B"
  :precision binary64
  (+ (+ (* x y) (* z t)) (* a b)))