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

Percentage Accurate: 97.6% → 98.5%

Time: 6.9s

Alternatives: 8

Speedup: 1.0×

• 33.0% 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: 97.6% 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: 98.5% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 97.6%

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

   1. associate-+l+ N/A

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

   2. *-commutative N/A

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

   3. fma-define N/A

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

   4. fma-lowering-fma.f64 N/A

      \[\leadsto \mathsf{fma.f64}\left(y, \color{blue}{x}, \left(z \cdot t + a \cdot b\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{fma.f64}\left(y, x, \mathsf{+.f64}\left(\left(z \cdot t\right), \left(a \cdot b\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{fma.f64}\left(y, x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, t\right), \left(a \cdot b\right)\right)\right) \]

   7. *-lowering-*.f64 98.0%

      \[\leadsto \mathsf{fma.f64}\left(y, x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, t\right), \mathsf{*.f64}\left(a, b\right)\right)\right) \]

4. Applied egg-rr 98.0%

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

5. Final simplification 98.0%

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

Alternative 2: 53.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -4.2 \cdot 10^{+158}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq -2.5 \cdot 10^{-79}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;a \cdot b \leq -1.8 \cdot 10^{-201}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;a \cdot b \leq 5 \cdot 10^{+49}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* a b) -4.2e+158)
   (* a b)
   (if (<= (* a b) -2.5e-79)
     (* y x)
     (if (<= (* a b) -1.8e-201)
       (* z t)
       (if (<= (* a b) 5e+49) (* y x) (* a b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a * b) <= -4.2e+158) {
		tmp = a * b;
	} else if ((a * b) <= -2.5e-79) {
		tmp = y * x;
	} else if ((a * b) <= -1.8e-201) {
		tmp = z * t;
	} else if ((a * b) <= 5e+49) {
		tmp = y * x;
	} else {
		tmp = a * b;
	}
	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) <= (-4.2d+158)) then
        tmp = a * b
    else if ((a * b) <= (-2.5d-79)) then
        tmp = y * x
    else if ((a * b) <= (-1.8d-201)) then
        tmp = z * t
    else if ((a * b) <= 5d+49) then
        tmp = y * x
    else
        tmp = a * b
    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) <= -4.2e+158) {
		tmp = a * b;
	} else if ((a * b) <= -2.5e-79) {
		tmp = y * x;
	} else if ((a * b) <= -1.8e-201) {
		tmp = z * t;
	} else if ((a * b) <= 5e+49) {
		tmp = y * x;
	} else {
		tmp = a * b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (a * b) <= -4.2e+158:
		tmp = a * b
	elif (a * b) <= -2.5e-79:
		tmp = y * x
	elif (a * b) <= -1.8e-201:
		tmp = z * t
	elif (a * b) <= 5e+49:
		tmp = y * x
	else:
		tmp = a * b
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(a * b) <= -4.2e+158)
		tmp = Float64(a * b);
	elseif (Float64(a * b) <= -2.5e-79)
		tmp = Float64(y * x);
	elseif (Float64(a * b) <= -1.8e-201)
		tmp = Float64(z * t);
	elseif (Float64(a * b) <= 5e+49)
		tmp = Float64(y * x);
	else
		tmp = Float64(a * b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((a * b) <= -4.2e+158)
		tmp = a * b;
	elseif ((a * b) <= -2.5e-79)
		tmp = y * x;
	elseif ((a * b) <= -1.8e-201)
		tmp = z * t;
	elseif ((a * b) <= 5e+49)
		tmp = y * x;
	else
		tmp = a * b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(a * b), $MachinePrecision], -4.2e+158], N[(a * b), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], -2.5e-79], N[(y * x), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], -1.8e-201], N[(z * t), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 5e+49], N[(y * x), $MachinePrecision], N[(a * b), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 a b) < -4.1999999999999998e158 or 5.0000000000000004e49 < (*.f64 a b)

   1. Initial program 95.0%

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

   3. Taylor expanded in a around inf

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

      1. *-lowering-*.f64 84.2%

         \[\leadsto \mathsf{*.f64}\left(a, \color{blue}{b}\right) \]

   5. Simplified 84.2%

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

   if -4.1999999999999998e158 < (*.f64 a b) < -2.5e-79 or -1.80000000000000016e-201 < (*.f64 a b) < 5.0000000000000004e49

   1. Initial program 99.3%

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

   3. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 51.8%

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{y}\right) \]

   5. Simplified 51.8%

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

   if -2.5e-79 < (*.f64 a b) < -1.80000000000000016e-201

   1. Initial program 95.8%

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

   3. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 60.0%

         \[\leadsto \mathsf{*.f64}\left(t, \color{blue}{z}\right) \]

   5. Simplified 60.0%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -4.2 \cdot 10^{+158}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq -2.5 \cdot 10^{-79}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;a \cdot b \leq -1.8 \cdot 10^{-201}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;a \cdot b \leq 5 \cdot 10^{+49}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -9000000000:\\ \;\;\;\;a \cdot b + y \cdot x\\ \mathbf{elif}\;a \cdot b \leq 3.9 \cdot 10^{+49}:\\ \;\;\;\;y \cdot x + z \cdot t\\ \mathbf{else}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* a b) -9000000000.0)
   (+ (* a b) (* y x))
   (if (<= (* a b) 3.9e+49) (+ (* y x) (* 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) <= -9000000000.0) {
		tmp = (a * b) + (y * x);
	} else if ((a * b) <= 3.9e+49) {
		tmp = (y * x) + (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) <= (-9000000000.0d0)) then
        tmp = (a * b) + (y * x)
    else if ((a * b) <= 3.9d+49) then
        tmp = (y * x) + (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) <= -9000000000.0) {
		tmp = (a * b) + (y * x);
	} else if ((a * b) <= 3.9e+49) {
		tmp = (y * x) + (z * t);
	} else {
		tmp = (a * b) + (z * t);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (a * b) <= -9000000000.0:
		tmp = (a * b) + (y * x)
	elif (a * b) <= 3.9e+49:
		tmp = (y * x) + (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) <= -9000000000.0)
		tmp = Float64(Float64(a * b) + Float64(y * x));
	elseif (Float64(a * b) <= 3.9e+49)
		tmp = Float64(Float64(y * x) + 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) <= -9000000000.0)
		tmp = (a * b) + (y * x);
	elseif ((a * b) <= 3.9e+49)
		tmp = (y * x) + (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], -9000000000.0], N[(N[(a * b), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 3.9e+49], N[(N[(y * x), $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) < -9e9

   1. Initial program 93.7%

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

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(x \cdot y\right)}, \mathsf{*.f64}\left(a, b\right)\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 89.3%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\color{blue}{a}, b\right)\right) \]

   5. Simplified 89.3%

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

   if -9e9 < (*.f64 a b) < 3.9000000000000001e49

   1. Initial program 99.3%

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

         \[\leadsto \mathsf{+.f64}\left(\left(t \cdot z\right), \color{blue}{\left(x \cdot y\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(t, z\right), \left(\color{blue}{x} \cdot y\right)\right) \]

      3. *-lowering-*.f64 87.5%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(t, z\right), \mathsf{*.f64}\left(x, \color{blue}{y}\right)\right) \]

   5. Simplified 87.5%

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

   if 3.9000000000000001e49 < (*.f64 a b)

   1. Initial program 97.6%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(t \cdot z\right)}, \mathsf{*.f64}\left(a, b\right)\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 93.1%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(t, z\right), \mathsf{*.f64}\left(\color{blue}{a}, b\right)\right) \]

   5. Simplified 93.1%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -9000000000:\\ \;\;\;\;a \cdot b + y \cdot x\\ \mathbf{elif}\;a \cdot b \leq 3.9 \cdot 10^{+49}:\\ \;\;\;\;y \cdot x + z \cdot t\\ \mathbf{else}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -9.5 \cdot 10^{+155}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq 4.5 \cdot 10^{+49}:\\ \;\;\;\;y \cdot x + z \cdot t\\ \mathbf{else}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* a b) -9.5e+155)
   (* a b)
   (if (<= (* a b) 4.5e+49) (+ (* y x) (* 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) <= -9.5e+155) {
		tmp = a * b;
	} else if ((a * b) <= 4.5e+49) {
		tmp = (y * x) + (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) <= (-9.5d+155)) then
        tmp = a * b
    else if ((a * b) <= 4.5d+49) then
        tmp = (y * x) + (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) <= -9.5e+155) {
		tmp = a * b;
	} else if ((a * b) <= 4.5e+49) {
		tmp = (y * x) + (z * t);
	} else {
		tmp = (a * b) + (z * t);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (a * b) <= -9.5e+155:
		tmp = a * b
	elif (a * b) <= 4.5e+49:
		tmp = (y * x) + (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) <= -9.5e+155)
		tmp = Float64(a * b);
	elseif (Float64(a * b) <= 4.5e+49)
		tmp = Float64(Float64(y * x) + 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) <= -9.5e+155)
		tmp = a * b;
	elseif ((a * b) <= 4.5e+49)
		tmp = (y * x) + (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], -9.5e+155], N[(a * b), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 4.5e+49], N[(N[(y * x), $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) < -9.5000000000000006e155

   1. Initial program 92.1%

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

   3. Taylor expanded in a around inf

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

      1. *-lowering-*.f64 94.5%

         \[\leadsto \mathsf{*.f64}\left(a, \color{blue}{b}\right) \]

   5. Simplified 94.5%

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

   if -9.5000000000000006e155 < (*.f64 a b) < 4.49999999999999982e49

   1. Initial program 98.8%

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

         \[\leadsto \mathsf{+.f64}\left(\left(t \cdot z\right), \color{blue}{\left(x \cdot y\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(t, z\right), \left(\color{blue}{x} \cdot y\right)\right) \]

      3. *-lowering-*.f64 84.6%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(t, z\right), \mathsf{*.f64}\left(x, \color{blue}{y}\right)\right) \]

   5. Simplified 84.6%

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

   if 4.49999999999999982e49 < (*.f64 a b)

   1. Initial program 97.6%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(t \cdot z\right)}, \mathsf{*.f64}\left(a, b\right)\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 93.1%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(t, z\right), \mathsf{*.f64}\left(\color{blue}{a}, b\right)\right) \]

   5. Simplified 93.1%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -9.5 \cdot 10^{+155}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq 4.5 \cdot 10^{+49}:\\ \;\;\;\;y \cdot x + z \cdot t\\ \mathbf{else}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot x \leq -1 \cdot 10^{+254}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \cdot x \leq 10^{+156}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* y x) -1e+254)
   (* y x)
   (if (<= (* y x) 1e+156) (+ (* a b) (* z t)) (* y x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y * x) <= -1e+254) {
		tmp = y * x;
	} else if ((y * x) <= 1e+156) {
		tmp = (a * b) + (z * t);
	} else {
		tmp = y * x;
	}
	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 ((y * x) <= (-1d+254)) then
        tmp = y * x
    else if ((y * x) <= 1d+156) then
        tmp = (a * b) + (z * t)
    else
        tmp = y * x
    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 ((y * x) <= -1e+254) {
		tmp = y * x;
	} else if ((y * x) <= 1e+156) {
		tmp = (a * b) + (z * t);
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y * x) <= -1e+254:
		tmp = y * x
	elif (y * x) <= 1e+156:
		tmp = (a * b) + (z * t)
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(y * x) <= -1e+254)
		tmp = Float64(y * x);
	elseif (Float64(y * x) <= 1e+156)
		tmp = Float64(Float64(a * b) + Float64(z * t));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y * x) <= -1e+254)
		tmp = y * x;
	elseif ((y * x) <= 1e+156)
		tmp = (a * b) + (z * t);
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(y * x), $MachinePrecision], -1e+254], N[(y * x), $MachinePrecision], If[LessEqual[N[(y * x), $MachinePrecision], 1e+156], N[(N[(a * b), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -9.9999999999999994e253 or 9.9999999999999998e155 < (*.f64 x y)

   1. Initial program 93.1%

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

   3. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 80.0%

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{y}\right) \]

   5. Simplified 80.0%

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

   if -9.9999999999999994e253 < (*.f64 x y) < 9.9999999999999998e155

   1. Initial program 99.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(t \cdot z\right)}, \mathsf{*.f64}\left(a, b\right)\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 79.7%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(t, z\right), \mathsf{*.f64}\left(\color{blue}{a}, b\right)\right) \]

   5. Simplified 79.7%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot x \leq -1 \cdot 10^{+254}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \cdot x \leq 10^{+156}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 54.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -2.1 \cdot 10^{+106}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq 8 \cdot 10^{+27}:\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* a b) -2.1e+106) (* a b) (if (<= (* a b) 8e+27) (* z t) (* a b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a * b) <= -2.1e+106) {
		tmp = a * b;
	} else if ((a * b) <= 8e+27) {
		tmp = z * t;
	} else {
		tmp = a * b;
	}
	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) <= (-2.1d+106)) then
        tmp = a * b
    else if ((a * b) <= 8d+27) then
        tmp = z * t
    else
        tmp = a * b
    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) <= -2.1e+106) {
		tmp = a * b;
	} else if ((a * b) <= 8e+27) {
		tmp = z * t;
	} else {
		tmp = a * b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (a * b) <= -2.1e+106:
		tmp = a * b
	elif (a * b) <= 8e+27:
		tmp = z * t
	else:
		tmp = a * b
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(a * b) <= -2.1e+106)
		tmp = Float64(a * b);
	elseif (Float64(a * b) <= 8e+27)
		tmp = Float64(z * t);
	else
		tmp = Float64(a * b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((a * b) <= -2.1e+106)
		tmp = a * b;
	elseif ((a * b) <= 8e+27)
		tmp = z * t;
	else
		tmp = a * b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(a * b), $MachinePrecision], -2.1e+106], N[(a * b), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 8e+27], N[(z * t), $MachinePrecision], N[(a * b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a b) < -2.10000000000000005e106 or 8.0000000000000001e27 < (*.f64 a b)

   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 inf

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

      1. *-lowering-*.f64 76.1%

         \[\leadsto \mathsf{*.f64}\left(a, \color{blue}{b}\right) \]

   5. Simplified 76.1%

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

   if -2.10000000000000005e106 < (*.f64 a b) < 8.0000000000000001e27

   1. Initial program 98.7%

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

   3. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 45.1%

         \[\leadsto \mathsf{*.f64}\left(t, \color{blue}{z}\right) \]

   5. Simplified 45.1%

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

4. Final simplification 56.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -2.1 \cdot 10^{+106}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq 8 \cdot 10^{+27}:\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.6%

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

3. Final simplification 97.6%

   \[\leadsto a \cdot b + \left(y \cdot x + z \cdot t\right) \]
4. 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 97.6%

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

3. Taylor expanded in a around inf

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

   1. *-lowering-*.f64 38.3%

      \[\leadsto \mathsf{*.f64}\left(a, \color{blue}{b}\right) \]

5. Simplified 38.3%

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

Reproduce

herbie shell --seed 2024145 
(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)))