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

Percentage Accurate: 97.6% → 98.7%

Time: 5.9s

Alternatives: 10

Speedup: 1.0×

• 33.4% 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: 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: 98.7% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.4%

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

   1. associate-+l+ 98.4%

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

   2. fma-define 98.8%

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

   3. fma-define 99.2%

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

3. Simplified 99.2%

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

Alternative 2: 98.7% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.4%

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

   1. +-commutative 98.4%

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

   2. fma-define 98.8%

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

   3. fma-define 98.8%

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

3. Simplified 98.8%

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

Alternative 3: 97.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.4%

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

   1. fma-define 98.4%

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

3. Simplified 98.4%

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

5. Final simplification 98.4%

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

Alternative 4: 53.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -8.5 \cdot 10^{+136}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \cdot y \leq -4.9 \cdot 10^{-116}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;x \cdot y \leq 0:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;x \cdot y \leq 1.15 \cdot 10^{-123}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;x \cdot y \leq 2.8 \cdot 10^{+21}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;x \cdot y \leq 5.8 \cdot 10^{+143}:\\ \;\;\;\;a \cdot b\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* x y) -8.5e+136)
   (* x y)
   (if (<= (* x y) -4.9e-116)
     (* a b)
     (if (<= (* x y) 0.0)
       (* z t)
       (if (<= (* x y) 1.15e-123)
         (* a b)
         (if (<= (* x y) 2.8e+21)
           (* z t)
           (if (<= (* x y) 5.8e+143) (* a b) (* x y))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((x * y) <= -8.5e+136) {
		tmp = x * y;
	} else if ((x * y) <= -4.9e-116) {
		tmp = a * b;
	} else if ((x * y) <= 0.0) {
		tmp = z * t;
	} else if ((x * y) <= 1.15e-123) {
		tmp = a * b;
	} else if ((x * y) <= 2.8e+21) {
		tmp = z * t;
	} else if ((x * y) <= 5.8e+143) {
		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) <= (-8.5d+136)) then
        tmp = x * y
    else if ((x * y) <= (-4.9d-116)) then
        tmp = a * b
    else if ((x * y) <= 0.0d0) then
        tmp = z * t
    else if ((x * y) <= 1.15d-123) then
        tmp = a * b
    else if ((x * y) <= 2.8d+21) then
        tmp = z * t
    else if ((x * y) <= 5.8d+143) 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) <= -8.5e+136) {
		tmp = x * y;
	} else if ((x * y) <= -4.9e-116) {
		tmp = a * b;
	} else if ((x * y) <= 0.0) {
		tmp = z * t;
	} else if ((x * y) <= 1.15e-123) {
		tmp = a * b;
	} else if ((x * y) <= 2.8e+21) {
		tmp = z * t;
	} else if ((x * y) <= 5.8e+143) {
		tmp = a * b;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (x * y) <= -8.5e+136:
		tmp = x * y
	elif (x * y) <= -4.9e-116:
		tmp = a * b
	elif (x * y) <= 0.0:
		tmp = z * t
	elif (x * y) <= 1.15e-123:
		tmp = a * b
	elif (x * y) <= 2.8e+21:
		tmp = z * t
	elif (x * y) <= 5.8e+143:
		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) <= -8.5e+136)
		tmp = Float64(x * y);
	elseif (Float64(x * y) <= -4.9e-116)
		tmp = Float64(a * b);
	elseif (Float64(x * y) <= 0.0)
		tmp = Float64(z * t);
	elseif (Float64(x * y) <= 1.15e-123)
		tmp = Float64(a * b);
	elseif (Float64(x * y) <= 2.8e+21)
		tmp = Float64(z * t);
	elseif (Float64(x * y) <= 5.8e+143)
		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) <= -8.5e+136)
		tmp = x * y;
	elseif ((x * y) <= -4.9e-116)
		tmp = a * b;
	elseif ((x * y) <= 0.0)
		tmp = z * t;
	elseif ((x * y) <= 1.15e-123)
		tmp = a * b;
	elseif ((x * y) <= 2.8e+21)
		tmp = z * t;
	elseif ((x * y) <= 5.8e+143)
		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], -8.5e+136], N[(x * y), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], -4.9e-116], N[(a * b), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 0.0], N[(z * t), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 1.15e-123], N[(a * b), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 2.8e+21], N[(z * t), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 5.8e+143], N[(a * b), $MachinePrecision], N[(x * y), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -8.49999999999999966e136 or 5.7999999999999996e143 < (*.f64 x y)

   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 inf 83.5%

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

   if -8.49999999999999966e136 < (*.f64 x y) < -4.89999999999999977e-116 or 0.0 < (*.f64 x y) < 1.14999999999999993e-123 or 2.8e21 < (*.f64 x y) < 5.7999999999999996e143

   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 63.4%

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

   if -4.89999999999999977e-116 < (*.f64 x y) < 0.0 or 1.14999999999999993e-123 < (*.f64 x y) < 2.8e21

   1. Initial program 97.9%

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

   3. Taylor expanded in z around inf 64.5%

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

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -8.5 \cdot 10^{+136}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \cdot y \leq -4.9 \cdot 10^{-116}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;x \cdot y \leq 0:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;x \cdot y \leq 1.15 \cdot 10^{-123}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;x \cdot y \leq 2.8 \cdot 10^{+21}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;x \cdot y \leq 5.8 \cdot 10^{+143}:\\ \;\;\;\;a \cdot b\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+83} \lor \neg \left(x \cdot y \leq 10^{+58}\right):\\ \;\;\;\;a \cdot b + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= (* x y) -2e+83) (not (<= (* x y) 1e+58)))
   (+ (* 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) <= -2e+83) || !((x * y) <= 1e+58)) {
		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) <= (-2d+83)) .or. (.not. ((x * y) <= 1d+58))) 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) <= -2e+83) || !((x * y) <= 1e+58)) {
		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) <= -2e+83) or not ((x * y) <= 1e+58):
		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) <= -2e+83) || !(Float64(x * y) <= 1e+58))
		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) <= -2e+83) || ~(((x * y) <= 1e+58)))
		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], -2e+83], N[Not[LessEqual[N[(x * y), $MachinePrecision], 1e+58]], $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 3 regimes

2. if (*.f64 x y) < -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 z around 0 92.3%

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

   if -2.00000000000000006e83 < (*.f64 x y) < 9.99999999999999944e57

   1. Initial program 98.7%

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

   3. Taylor expanded in x around 0 90.6%

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

   if 9.99999999999999944e57 < (*.f64 x y)

   1. Initial program 96.1%

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

   3. Taylor expanded in x around inf 96.1%

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

   4. Taylor expanded in x around inf 90.4%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+83} \lor \neg \left(x \cdot y \leq 10^{+58}\right):\\ \;\;\;\;a \cdot b + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2.5 \cdot 10^{+49} \lor \neg \left(x \cdot y \leq 7.2 \cdot 10^{+62}\right):\\ \;\;\;\;a \cdot b + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= (* x y) -2.5e+49) (not (<= (* x y) 7.2e+62)))
   (+ (* 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) <= -2.5e+49) || !((x * y) <= 7.2e+62)) {
		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) <= (-2.5d+49)) .or. (.not. ((x * y) <= 7.2d+62))) 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) <= -2.5e+49) || !((x * y) <= 7.2e+62)) {
		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) <= -2.5e+49) or not ((x * y) <= 7.2e+62):
		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) <= -2.5e+49) || !(Float64(x * y) <= 7.2e+62))
		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) <= -2.5e+49) || ~(((x * y) <= 7.2e+62)))
		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], -2.5e+49], N[Not[LessEqual[N[(x * y), $MachinePrecision], 7.2e+62]], $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) < -2.5000000000000002e49 or 7.2e62 < (*.f64 x y)

   1. Initial program 98.0%

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

   3. Taylor expanded in z around 0 91.3%

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

   if -2.5000000000000002e49 < (*.f64 x y) < 7.2e62

   1. Initial program 98.7%

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

   3. Taylor expanded in x around 0 90.6%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2.5 \cdot 10^{+49} \lor \neg \left(x \cdot y \leq 7.2 \cdot 10^{+62}\right):\\ \;\;\;\;a \cdot b + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -3.8 \cdot 10^{+205} \lor \neg \left(x \cdot y \leq 1.1 \cdot 10^{+148}\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= (* x y) -3.8e+205) (not (<= (* x y) 1.1e+148)))
   (* 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) <= -3.8e+205) || !((x * y) <= 1.1e+148)) {
		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) <= (-3.8d+205)) .or. (.not. ((x * y) <= 1.1d+148))) 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) <= -3.8e+205) || !((x * y) <= 1.1e+148)) {
		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) <= -3.8e+205) or not ((x * y) <= 1.1e+148):
		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) <= -3.8e+205) || !(Float64(x * y) <= 1.1e+148))
		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) <= -3.8e+205) || ~(((x * y) <= 1.1e+148)))
		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], -3.8e+205], N[Not[LessEqual[N[(x * y), $MachinePrecision], 1.1e+148]], $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) < -3.8e205 or 1.0999999999999999e148 < (*.f64 x y)

   1. Initial program 97.1%

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

   3. Taylor expanded in x around inf 89.1%

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

   if -3.8e205 < (*.f64 x y) < 1.0999999999999999e148

   1. Initial program 98.9%

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

   3. Taylor expanded in x around 0 85.7%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -3.8 \cdot 10^{+205} \lor \neg \left(x \cdot y \leq 1.1 \cdot 10^{+148}\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 53.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -5.8 \cdot 10^{+45} \lor \neg \left(a \cdot b \leq 1.35 \cdot 10^{-58}\right):\\ \;\;\;\;a \cdot b\\ \mathbf{else}:\\ \;\;\;\;z \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= (* a b) -5.8e+45) (not (<= (* a b) 1.35e-58))) (* a b) (* z t)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((a * b) <= -5.8e+45) || !((a * b) <= 1.35e-58)) {
		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) <= (-5.8d+45)) .or. (.not. ((a * b) <= 1.35d-58))) 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) <= -5.8e+45) || !((a * b) <= 1.35e-58)) {
		tmp = a * b;
	} else {
		tmp = z * t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if ((a * b) <= -5.8e+45) or not ((a * b) <= 1.35e-58):
		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) <= -5.8e+45) || !(Float64(a * b) <= 1.35e-58))
		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) <= -5.8e+45) || ~(((a * b) <= 1.35e-58)))
		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], -5.8e+45], N[Not[LessEqual[N[(a * b), $MachinePrecision], 1.35e-58]], $MachinePrecision]], N[(a * b), $MachinePrecision], N[(z * t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a b) < -5.7999999999999994e45 or 1.3499999999999999e-58 < (*.f64 a b)

   1. Initial program 97.0%

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

   3. Taylor expanded in a around inf 62.9%

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

   if -5.7999999999999994e45 < (*.f64 a b) < 1.3499999999999999e-58

   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 51.2%

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

4. Final simplification 57.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -5.8 \cdot 10^{+45} \lor \neg \left(a \cdot b \leq 1.35 \cdot 10^{-58}\right):\\ \;\;\;\;a \cdot b\\ \mathbf{else}:\\ \;\;\;\;z \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.6% 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 98.4%

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

3. Final simplification 98.4%

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

Alternative 10: 36.5% 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 37.2%

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

Reproduce

herbie shell --seed 2024103 
(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)))