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

Percentage Accurate: 97.7% → 98.9%

Time: 9.0s

Alternatives: 8

Speedup: 1.2×

• 34.0% 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: 97.7% 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: 98.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. lift-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. associate-+l+ N/A

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

   6. lift-*.f64 N/A

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

   7. lower-fma.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lower-fma.f64 98.8

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

4. Applied rewrites 98.8%

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

Alternative 2: 54.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+134}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;z \cdot t \leq -1 \cdot 10^{-154}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+86}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* z t) -1e+134)
   (* z t)
   (if (<= (* z t) -1e-154) (* a b) (if (<= (* z t) 5e+86) (* x y) (* z t)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z * t) <= -1e+134) {
		tmp = z * t;
	} else if ((z * t) <= -1e-154) {
		tmp = a * b;
	} else if ((z * t) <= 5e+86) {
		tmp = x * y;
	} else {
		tmp = z * t;
	}
	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) <= (-1d+134)) then
        tmp = z * t
    else if ((z * t) <= (-1d-154)) then
        tmp = a * b
    else if ((z * t) <= 5d+86) then
        tmp = x * y
    else
        tmp = z * t
    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) <= -1e+134) {
		tmp = z * t;
	} else if ((z * t) <= -1e-154) {
		tmp = a * b;
	} else if ((z * t) <= 5e+86) {
		tmp = x * y;
	} else {
		tmp = z * t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (z * t) <= -1e+134:
		tmp = z * t
	elif (z * t) <= -1e-154:
		tmp = a * b
	elif (z * t) <= 5e+86:
		tmp = x * y
	else:
		tmp = z * t
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(z * t) <= -1e+134)
		tmp = Float64(z * t);
	elseif (Float64(z * t) <= -1e-154)
		tmp = Float64(a * b);
	elseif (Float64(z * t) <= 5e+86)
		tmp = Float64(x * y);
	else
		tmp = Float64(z * t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((z * t) <= -1e+134)
		tmp = z * t;
	elseif ((z * t) <= -1e-154)
		tmp = a * b;
	elseif ((z * t) <= 5e+86)
		tmp = x * y;
	else
		tmp = z * t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(z * t), $MachinePrecision], -1e+134], N[(z * t), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], -1e-154], N[(a * b), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 5e+86], N[(x * y), $MachinePrecision], N[(z * t), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -9.99999999999999921e133 or 4.9999999999999998e86 < (*.f64 z t)

   1. Initial program 95.6%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 82.1

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

   5. Applied rewrites 82.1%

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

   if -9.99999999999999921e133 < (*.f64 z t) < -9.9999999999999997e-155

   1. Initial program 100.0%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 56.6

         \[\leadsto \color{blue}{a \cdot b} \]

   5. Applied rewrites 56.6%

      \[\leadsto \color{blue}{a \cdot b} \]

   if -9.9999999999999997e-155 < (*.f64 z t) < 4.9999999999999998e86

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 52.4

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

   5. Applied rewrites 52.4%

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

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+134}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;z \cdot t \leq -1 \cdot 10^{-154}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+86}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+134}:\\ \;\;\;\;\mathsf{fma}\left(t, z, x \cdot y\right)\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+86}:\\ \;\;\;\;\mathsf{fma}\left(y, x, a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, b, z \cdot t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* z t) -1e+134)
   (fma t z (* x y))
   (if (<= (* z t) 5e+86) (fma y x (* a b)) (fma a b (* z t)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z * t) <= -1e+134) {
		tmp = fma(t, z, (x * y));
	} else if ((z * t) <= 5e+86) {
		tmp = fma(y, x, (a * b));
	} else {
		tmp = fma(a, b, (z * t));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(z * t) <= -1e+134)
		tmp = fma(t, z, Float64(x * y));
	elseif (Float64(z * t) <= 5e+86)
		tmp = fma(y, x, Float64(a * b));
	else
		tmp = fma(a, b, Float64(z * t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(z * t), $MachinePrecision], -1e+134], N[(t * z + N[(x * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 5e+86], N[(y * x + N[(a * b), $MachinePrecision]), $MachinePrecision], N[(a * b + N[(z * t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -9.99999999999999921e133

   1. Initial program 95.7%

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

   3. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 97.6

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

   5. Applied rewrites 97.6%

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

   if -9.99999999999999921e133 < (*.f64 z t) < 4.9999999999999998e86

   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. lower-fma.f64 N/A

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

      2. lower-*.f64 90.6

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

   5. Applied rewrites 90.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 90.6

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

   7. Applied rewrites 90.6%

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

   if 4.9999999999999998e86 < (*.f64 z t)

   1. Initial program 95.5%

      \[\left(x \cdot y + z \cdot t\right) + a \cdot b \]
   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. lower-fma.f64 N/A

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

      2. lower-*.f64 90.4

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

   5. Applied rewrites 90.4%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+134}:\\ \;\;\;\;\mathsf{fma}\left(t, z, x \cdot y\right)\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+86}:\\ \;\;\;\;\mathsf{fma}\left(y, x, a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, b, z \cdot t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+134}:\\ \;\;\;\;\mathsf{fma}\left(t, z, x \cdot y\right)\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+86}:\\ \;\;\;\;\mathsf{fma}\left(a, b, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, b, z \cdot t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* z t) -1e+134)
   (fma t z (* x y))
   (if (<= (* z t) 5e+86) (fma a b (* x y)) (fma a b (* z t)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z * t) <= -1e+134) {
		tmp = fma(t, z, (x * y));
	} else if ((z * t) <= 5e+86) {
		tmp = fma(a, b, (x * y));
	} else {
		tmp = fma(a, b, (z * t));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(z * t) <= -1e+134)
		tmp = fma(t, z, Float64(x * y));
	elseif (Float64(z * t) <= 5e+86)
		tmp = fma(a, b, Float64(x * y));
	else
		tmp = fma(a, b, Float64(z * t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(z * t), $MachinePrecision], -1e+134], N[(t * z + N[(x * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 5e+86], N[(a * b + N[(x * y), $MachinePrecision]), $MachinePrecision], N[(a * b + N[(z * t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -9.99999999999999921e133

   1. Initial program 95.7%

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

   3. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 97.6

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

   5. Applied rewrites 97.6%

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

   if -9.99999999999999921e133 < (*.f64 z t) < 4.9999999999999998e86

   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. lower-fma.f64 N/A

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

      2. lower-*.f64 90.6

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

   5. Applied rewrites 90.6%

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

   if 4.9999999999999998e86 < (*.f64 z t)

   1. Initial program 95.5%

      \[\left(x \cdot y + z \cdot t\right) + a \cdot b \]
   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. lower-fma.f64 N/A

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

      2. lower-*.f64 90.4

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

   5. Applied rewrites 90.4%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+134}:\\ \;\;\;\;\mathsf{fma}\left(t, z, x \cdot y\right)\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+86}:\\ \;\;\;\;\mathsf{fma}\left(a, b, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, b, z \cdot t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(a, b, z \cdot t\right)\\ \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+62}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+86}:\\ \;\;\;\;\mathsf{fma}\left(a, b, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma a b (* z t))))
   (if (<= (* z t) -5e+62) t_1 (if (<= (* z t) 5e+86) (fma a b (* x y)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(a, b, (z * t));
	double tmp;
	if ((z * t) <= -5e+62) {
		tmp = t_1;
	} else if ((z * t) <= 5e+86) {
		tmp = fma(a, b, (x * y));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(a, b, Float64(z * t))
	tmp = 0.0
	if (Float64(z * t) <= -5e+62)
		tmp = t_1;
	elseif (Float64(z * t) <= 5e+86)
		tmp = fma(a, b, Float64(x * y));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(a * b + N[(z * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(z * t), $MachinePrecision], -5e+62], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], 5e+86], N[(a * b + N[(x * y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -5.00000000000000029e62 or 4.9999999999999998e86 < (*.f64 z t)

   1. Initial program 95.9%

      \[\left(x \cdot y + z \cdot t\right) + a \cdot b \]
   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. lower-fma.f64 N/A

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

      2. lower-*.f64 89.8

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

   5. Applied rewrites 89.8%

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

   if -5.00000000000000029e62 < (*.f64 z t) < 4.9999999999999998e86

   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. lower-fma.f64 N/A

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

      2. lower-*.f64 92.0

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

   5. Applied rewrites 92.0%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+62}:\\ \;\;\;\;\mathsf{fma}\left(a, b, z \cdot t\right)\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+86}:\\ \;\;\;\;\mathsf{fma}\left(a, b, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, b, z \cdot t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -7.8 \cdot 10^{+204}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \cdot y \leq 7.2 \cdot 10^{+186}:\\ \;\;\;\;\mathsf{fma}\left(a, b, z \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* x y) -7.8e+204)
   (* x y)
   (if (<= (* x y) 7.2e+186) (fma a b (* z t)) (* x y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((x * y) <= -7.8e+204) {
		tmp = x * y;
	} else if ((x * y) <= 7.2e+186) {
		tmp = fma(a, b, (z * t));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(x * y) <= -7.8e+204)
		tmp = Float64(x * y);
	elseif (Float64(x * y) <= 7.2e+186)
		tmp = fma(a, b, Float64(z * t));
	else
		tmp = Float64(x * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(x * y), $MachinePrecision], -7.8e+204], N[(x * y), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 7.2e+186], N[(a * b + N[(z * t), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -7.80000000000000033e204 or 7.2000000000000003e186 < (*.f64 x y)

   1. Initial program 98.2%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 82.1

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

   5. Applied rewrites 82.1%

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

   if -7.80000000000000033e204 < (*.f64 x y) < 7.2000000000000003e186

   1. Initial program 98.5%

      \[\left(x \cdot y + z \cdot t\right) + a \cdot b \]
   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. lower-fma.f64 N/A

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

      2. lower-*.f64 81.2

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

   5. Applied rewrites 81.2%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -7.8 \cdot 10^{+204}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \cdot y \leq 7.2 \cdot 10^{+186}:\\ \;\;\;\;\mathsf{fma}\left(a, b, z \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 52.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+134}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+83}:\\ \;\;\;\;a \cdot b\\ \mathbf{else}:\\ \;\;\;\;z \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* z t) -1e+134) (* z t) (if (<= (* z t) 2e+83) (* a b) (* z t))))

C

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

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (z * t) <= -1e+134:
		tmp = z * t
	elif (z * t) <= 2e+83:
		tmp = a * b
	else:
		tmp = z * t
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((z * t) <= -1e+134)
		tmp = z * t;
	elseif ((z * t) <= 2e+83)
		tmp = a * b;
	else
		tmp = z * t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(z * t), $MachinePrecision], -1e+134], N[(z * t), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 2e+83], N[(a * b), $MachinePrecision], N[(z * t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -9.99999999999999921e133 or 2.00000000000000006e83 < (*.f64 z t)

   1. Initial program 95.7%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 79.6

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

   5. Applied rewrites 79.6%

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

   if -9.99999999999999921e133 < (*.f64 z t) < 2.00000000000000006e83

   1. Initial program 100.0%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 48.1

         \[\leadsto \color{blue}{a \cdot b} \]

   5. Applied rewrites 48.1%

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

4. Final simplification 59.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+134}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+83}:\\ \;\;\;\;a \cdot b\\ \mathbf{else}:\\ \;\;\;\;z \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 35.3% accurate, 3.7× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t, double a, double b) {
	return 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 = a * b
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. lower-*.f64 36.5

      \[\leadsto \color{blue}{a \cdot b} \]

5. Applied rewrites 36.5%

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

Reproduce

herbie shell --seed 2024214 
(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)))