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

Percentage Accurate: 98.0% → 99.0%

Time: 8.7s

Alternatives: 9

Speedup: 1.0×

• 32.4% 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: 98.0% 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, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. associate-+l+ 99.6%

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

   2. fma-define 99.6%

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

   3. +-commutative 99.6%

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

   4. fma-define 99.6%

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

3. Simplified 99.6%

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

Alternative 2: 98.4% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. fma-define 99.6%

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

3. Simplified 99.6%

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

Alternative 3: 51.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2.65 \cdot 10^{+160}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \cdot y \leq -2.4 \cdot 10^{+50}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;x \cdot y \leq 5.9 \cdot 10^{-46}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;x \cdot y \leq 1.52 \cdot 10^{+162}:\\ \;\;\;\;a \cdot b\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* x y) -2.65e+160)
   (* x y)
   (if (<= (* x y) -2.4e+50)
     (* a b)
     (if (<= (* x y) 5.9e-46)
       (* z t)
       (if (<= (* x y) 1.52e+162) (* a b) (* x y))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((x * y) <= -2.65e+160) {
		tmp = x * y;
	} else if ((x * y) <= -2.4e+50) {
		tmp = a * b;
	} else if ((x * y) <= 5.9e-46) {
		tmp = z * t;
	} else if ((x * y) <= 1.52e+162) {
		tmp = a * b;
	} else {
		tmp = x * y;
	}
	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 ((x * y) <= (-2.65d+160)) then
        tmp = x * y
    else if ((x * y) <= (-2.4d+50)) then
        tmp = a * b
    else if ((x * y) <= 5.9d-46) then
        tmp = z * t
    else if ((x * y) <= 1.52d+162) then
        tmp = a * b
    else
        tmp = x * y
    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 ((x * y) <= -2.65e+160) {
		tmp = x * y;
	} else if ((x * y) <= -2.4e+50) {
		tmp = a * b;
	} else if ((x * y) <= 5.9e-46) {
		tmp = z * t;
	} else if ((x * y) <= 1.52e+162) {
		tmp = a * b;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (x * y) <= -2.65e+160:
		tmp = x * y
	elif (x * y) <= -2.4e+50:
		tmp = a * b
	elif (x * y) <= 5.9e-46:
		tmp = z * t
	elif (x * y) <= 1.52e+162:
		tmp = a * b
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(x * y) <= -2.65e+160)
		tmp = Float64(x * y);
	elseif (Float64(x * y) <= -2.4e+50)
		tmp = Float64(a * b);
	elseif (Float64(x * y) <= 5.9e-46)
		tmp = Float64(z * t);
	elseif (Float64(x * y) <= 1.52e+162)
		tmp = Float64(a * b);
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((x * y) <= -2.65e+160)
		tmp = x * y;
	elseif ((x * y) <= -2.4e+50)
		tmp = a * b;
	elseif ((x * y) <= 5.9e-46)
		tmp = z * t;
	elseif ((x * y) <= 1.52e+162)
		tmp = a * b;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(x * y), $MachinePrecision], -2.65e+160], N[(x * y), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], -2.4e+50], N[(a * b), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 5.9e-46], N[(z * t), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 1.52e+162], N[(a * b), $MachinePrecision], N[(x * y), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -2.65e160 or 1.5200000000000001e162 < (*.f64 x y)

   1. Initial program 98.8%

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

   3. Taylor expanded in x around inf 82.5%

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

   if -2.65e160 < (*.f64 x y) < -2.4000000000000002e50 or 5.8999999999999999e-46 < (*.f64 x y) < 1.5200000000000001e162

   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 50.0%

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

   if -2.4000000000000002e50 < (*.f64 x y) < 5.8999999999999999e-46

   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 inf 58.3%

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

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2.65 \cdot 10^{+160}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \cdot y \leq -2.4 \cdot 10^{+50}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;x \cdot y \leq 5.9 \cdot 10^{-46}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;x \cdot y \leq 1.52 \cdot 10^{+162}:\\ \;\;\;\;a \cdot b\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -4.6 \cdot 10^{+98} \lor \neg \left(x \cdot y \leq 2.4 \cdot 10^{-55}\right):\\ \;\;\;\;a \cdot b + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= (* x y) -4.6e+98) (not (<= (* x y) 2.4e-55)))
   (+ (* a b) (* x y))
   (+ (* a b) (* z t))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((x * y) <= -4.6e+98) || !((x * y) <= 2.4e-55)) {
		tmp = (a * b) + (x * y);
	} else {
		tmp = (a * b) + (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 (((x * y) <= (-4.6d+98)) .or. (.not. ((x * y) <= 2.4d-55))) then
        tmp = (a * b) + (x * y)
    else
        tmp = (a * b) + (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 (((x * y) <= -4.6e+98) || !((x * y) <= 2.4e-55)) {
		tmp = (a * b) + (x * y);
	} else {
		tmp = (a * b) + (z * t);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if ((x * y) <= -4.6e+98) or not ((x * y) <= 2.4e-55):
		tmp = (a * b) + (x * y)
	else:
		tmp = (a * b) + (z * t)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((Float64(x * y) <= -4.6e+98) || !(Float64(x * y) <= 2.4e-55))
		tmp = Float64(Float64(a * b) + Float64(x * y));
	else
		tmp = Float64(Float64(a * b) + Float64(z * t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (((x * y) <= -4.6e+98) || ~(((x * y) <= 2.4e-55)))
		tmp = (a * b) + (x * y);
	else
		tmp = (a * b) + (z * t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[N[(x * y), $MachinePrecision], -4.6e+98], N[Not[LessEqual[N[(x * y), $MachinePrecision], 2.4e-55]], $MachinePrecision]], N[(N[(a * b), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision], N[(N[(a * b), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -4.60000000000000026e98 or 2.39999999999999991e-55 < (*.f64 x y)

   1. Initial program 99.2%

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

   3. Taylor expanded in z around 0 83.5%

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

   if -4.60000000000000026e98 < (*.f64 x y) < 2.39999999999999991e-55

   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 0 90.8%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -4.6 \cdot 10^{+98} \lor \neg \left(x \cdot y \leq 2.4 \cdot 10^{-55}\right):\\ \;\;\;\;a \cdot b + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -7.7 \cdot 10^{+164} \lor \neg \left(x \cdot y \leq 2.2 \cdot 10^{+162}\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= (* x y) -7.7e+164) (not (<= (* x y) 2.2e+162)))
   (* x y)
   (+ (* a b) (* z t))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((x * y) <= -7.7e+164) || !((x * y) <= 2.2e+162)) {
		tmp = x * y;
	} else {
		tmp = (a * b) + (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 (((x * y) <= (-7.7d+164)) .or. (.not. ((x * y) <= 2.2d+162))) then
        tmp = x * y
    else
        tmp = (a * b) + (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 (((x * y) <= -7.7e+164) || !((x * y) <= 2.2e+162)) {
		tmp = x * y;
	} else {
		tmp = (a * b) + (z * t);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if ((x * y) <= -7.7e+164) or not ((x * y) <= 2.2e+162):
		tmp = x * y
	else:
		tmp = (a * b) + (z * t)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((Float64(x * y) <= -7.7e+164) || !(Float64(x * y) <= 2.2e+162))
		tmp = Float64(x * y);
	else
		tmp = Float64(Float64(a * b) + Float64(z * t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (((x * y) <= -7.7e+164) || ~(((x * y) <= 2.2e+162)))
		tmp = x * y;
	else
		tmp = (a * b) + (z * t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[N[(x * y), $MachinePrecision], -7.7e+164], N[Not[LessEqual[N[(x * y), $MachinePrecision], 2.2e+162]], $MachinePrecision]], N[(x * y), $MachinePrecision], N[(N[(a * b), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -7.7000000000000001e164 or 2.2000000000000002e162 < (*.f64 x y)

   1. Initial program 98.8%

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

   3. Taylor expanded in x around inf 82.5%

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

   if -7.7000000000000001e164 < (*.f64 x y) < 2.2000000000000002e162

   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 0 84.2%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -7.7 \cdot 10^{+164} \lor \neg \left(x \cdot y \leq 2.2 \cdot 10^{+162}\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -1.75 \cdot 10^{+162}:\\ \;\;\;\;a \cdot b + x \cdot y\\ \mathbf{elif}\;a \cdot b \leq 6400000000000:\\ \;\;\;\;x \cdot y + z \cdot t\\ \mathbf{else}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* a b) -1.75e+162)
   (+ (* a b) (* x y))
   (if (<= (* a b) 6400000000000.0) (+ (* x y) (* z t)) (+ (* a b) (* z t)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a * b) <= -1.75e+162) {
		tmp = (a * b) + (x * y);
	} else if ((a * b) <= 6400000000000.0) {
		tmp = (x * y) + (z * t);
	} else {
		tmp = (a * b) + (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 ((a * b) <= (-1.75d+162)) then
        tmp = (a * b) + (x * y)
    else if ((a * b) <= 6400000000000.0d0) then
        tmp = (x * y) + (z * t)
    else
        tmp = (a * b) + (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 ((a * b) <= -1.75e+162) {
		tmp = (a * b) + (x * y);
	} else if ((a * b) <= 6400000000000.0) {
		tmp = (x * y) + (z * t);
	} else {
		tmp = (a * b) + (z * t);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (a * b) <= -1.75e+162:
		tmp = (a * b) + (x * y)
	elif (a * b) <= 6400000000000.0:
		tmp = (x * y) + (z * t)
	else:
		tmp = (a * b) + (z * t)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(a * b) <= -1.75e+162)
		tmp = Float64(Float64(a * b) + Float64(x * y));
	elseif (Float64(a * b) <= 6400000000000.0)
		tmp = Float64(Float64(x * y) + Float64(z * t));
	else
		tmp = Float64(Float64(a * b) + Float64(z * t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((a * b) <= -1.75e+162)
		tmp = (a * b) + (x * y);
	elseif ((a * b) <= 6400000000000.0)
		tmp = (x * y) + (z * t);
	else
		tmp = (a * b) + (z * t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(a * b), $MachinePrecision], -1.75e+162], N[(N[(a * b), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 6400000000000.0], N[(N[(x * y), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision], N[(N[(a * b), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 a b) < -1.75000000000000009e162

   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 96.8%

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

   if -1.75000000000000009e162 < (*.f64 a b) < 6.4e12

   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 0 90.5%

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

   if 6.4e12 < (*.f64 a b)

   1. Initial program 98.4%

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

   3. Taylor expanded in x around 0 80.4%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -1.75 \cdot 10^{+162}:\\ \;\;\;\;a \cdot b + x \cdot y\\ \mathbf{elif}\;a \cdot b \leq 6400000000000:\\ \;\;\;\;x \cdot y + z \cdot t\\ \mathbf{else}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 51.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -1.4 \cdot 10^{+162} \lor \neg \left(a \cdot b \leq 1.4 \cdot 10^{+202}\right):\\ \;\;\;\;a \cdot b\\ \mathbf{else}:\\ \;\;\;\;z \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= (* a b) -1.4e+162) (not (<= (* a b) 1.4e+202))) (* a b) (* z t)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((a * b) <= -1.4e+162) || !((a * b) <= 1.4e+202)) {
		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 (((a * b) <= (-1.4d+162)) .or. (.not. ((a * b) <= 1.4d+202))) 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 (((a * b) <= -1.4e+162) || !((a * b) <= 1.4e+202)) {
		tmp = a * b;
	} else {
		tmp = z * t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if ((a * b) <= -1.4e+162) or not ((a * b) <= 1.4e+202):
		tmp = a * b
	else:
		tmp = z * t
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((Float64(a * b) <= -1.4e+162) || !(Float64(a * b) <= 1.4e+202))
		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 (((a * b) <= -1.4e+162) || ~(((a * b) <= 1.4e+202)))
		tmp = a * b;
	else
		tmp = z * t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[N[(a * b), $MachinePrecision], -1.4e+162], N[Not[LessEqual[N[(a * b), $MachinePrecision], 1.4e+202]], $MachinePrecision]], N[(a * b), $MachinePrecision], N[(z * t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a b) < -1.39999999999999995e162 or 1.40000000000000008e202 < (*.f64 a b)

   1. Initial program 98.2%

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

   3. Taylor expanded in a around inf 80.7%

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

   if -1.39999999999999995e162 < (*.f64 a b) < 1.40000000000000008e202

   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 inf 46.8%

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

4. Final simplification 54.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -1.4 \cdot 10^{+162} \lor \neg \left(a \cdot b \leq 1.4 \cdot 10^{+202}\right):\\ \;\;\;\;a \cdot b\\ \mathbf{else}:\\ \;\;\;\;z \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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) + ((x * y) + (z * t))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Final simplification 99.6%

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

Alternative 9: 35.9% 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 99.6%

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

3. Taylor expanded in a around inf 30.2%

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

Reproduce

herbie shell --seed 2024132 
(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)))