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

Percentage Accurate: 97.8% → 98.5%

Time: 4.4s

Alternatives: 9

Speedup: 0.5×

• Unsound rule application detected in e-graph. Results may not be sound.

• 32.6% 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: 97.8% 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, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot b + \left(z \cdot t + x \cdot y\right)\\ \mathbf{if}\;t_1 \leq \infty:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;z \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (* a b) (+ (* z t) (* x y)))))
   (if (<= t_1 INFINITY) t_1 (* z t))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a * b) + ((z * t) + (x * y));
	double tmp;
	if (t_1 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = z * t;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a * b) + ((z * t) + (x * y));
	double tmp;
	if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else {
		tmp = z * t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (a * b) + ((z * t) + (x * y))
	tmp = 0
	if t_1 <= math.inf:
		tmp = t_1
	else:
		tmp = z * t
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a * b) + Float64(Float64(z * t) + Float64(x * y)))
	tmp = 0.0
	if (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(z * t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (a * b) + ((z * t) + (x * y));
	tmp = 0.0;
	if (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = z * t;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (*.f64 x y) (*.f64 z t)) (*.f64 a b)) < +inf.0

   1. Initial program 100.0%

      \[\left(x \cdot y + z \cdot t\right) + a \cdot b \]

   if +inf.0 < (+.f64 (+.f64 (*.f64 x y) (*.f64 z t)) (*.f64 a b))

   1. Initial program 0.0%

      \[\left(x \cdot y + z \cdot t\right) + a \cdot b \]

   2. Taylor expanded in z around inf 66.8%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b + \left(z \cdot t + x \cdot y\right) \leq \infty:\\ \;\;\;\;a \cdot b + \left(z \cdot t + x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot t\\ \end{array} \]

Alternative 2: 98.9% 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 97.6%

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

   1. associate-+l+ 97.6%

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

   2. fma-def 98.0%

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

   3. fma-def 98.4%

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

3. Simplified 98.4%

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

4. Final simplification 98.4%

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

Alternative 3: 98.1% 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 97.6%

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

   1. fma-def 98.0%

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

3. Simplified 98.0%

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

4. Final simplification 98.0%

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

Alternative 4: 53.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -5.1 \cdot 10^{+108}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq -1.45 \cdot 10^{+62}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;a \cdot b \leq -3.4 \cdot 10^{+23}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq -0.0155:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;a \cdot b \leq -1 \cdot 10^{-8}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq -2.3 \cdot 10^{-55}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;a \cdot b \leq -8.6 \cdot 10^{-109}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;a \cdot b \leq -7.5 \cdot 10^{-246}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;a \cdot b \leq 4.3 \cdot 10^{+27}:\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* a b) -5.1e+108)
   (* a b)
   (if (<= (* a b) -1.45e+62)
     (* x y)
     (if (<= (* a b) -3.4e+23)
       (* a b)
       (if (<= (* a b) -0.0155)
         (* z t)
         (if (<= (* a b) -1e-8)
           (* a b)
           (if (<= (* a b) -2.3e-55)
             (* x y)
             (if (<= (* a b) -8.6e-109)
               (* z t)
               (if (<= (* a b) -7.5e-246)
                 (* x y)
                 (if (<= (* a b) 4.3e+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) <= -5.1e+108) {
		tmp = a * b;
	} else if ((a * b) <= -1.45e+62) {
		tmp = x * y;
	} else if ((a * b) <= -3.4e+23) {
		tmp = a * b;
	} else if ((a * b) <= -0.0155) {
		tmp = z * t;
	} else if ((a * b) <= -1e-8) {
		tmp = a * b;
	} else if ((a * b) <= -2.3e-55) {
		tmp = x * y;
	} else if ((a * b) <= -8.6e-109) {
		tmp = z * t;
	} else if ((a * b) <= -7.5e-246) {
		tmp = x * y;
	} else if ((a * b) <= 4.3e+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) <= (-5.1d+108)) then
        tmp = a * b
    else if ((a * b) <= (-1.45d+62)) then
        tmp = x * y
    else if ((a * b) <= (-3.4d+23)) then
        tmp = a * b
    else if ((a * b) <= (-0.0155d0)) then
        tmp = z * t
    else if ((a * b) <= (-1d-8)) then
        tmp = a * b
    else if ((a * b) <= (-2.3d-55)) then
        tmp = x * y
    else if ((a * b) <= (-8.6d-109)) then
        tmp = z * t
    else if ((a * b) <= (-7.5d-246)) then
        tmp = x * y
    else if ((a * b) <= 4.3d+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) <= -5.1e+108) {
		tmp = a * b;
	} else if ((a * b) <= -1.45e+62) {
		tmp = x * y;
	} else if ((a * b) <= -3.4e+23) {
		tmp = a * b;
	} else if ((a * b) <= -0.0155) {
		tmp = z * t;
	} else if ((a * b) <= -1e-8) {
		tmp = a * b;
	} else if ((a * b) <= -2.3e-55) {
		tmp = x * y;
	} else if ((a * b) <= -8.6e-109) {
		tmp = z * t;
	} else if ((a * b) <= -7.5e-246) {
		tmp = x * y;
	} else if ((a * b) <= 4.3e+27) {
		tmp = z * t;
	} else {
		tmp = a * b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (a * b) <= -5.1e+108:
		tmp = a * b
	elif (a * b) <= -1.45e+62:
		tmp = x * y
	elif (a * b) <= -3.4e+23:
		tmp = a * b
	elif (a * b) <= -0.0155:
		tmp = z * t
	elif (a * b) <= -1e-8:
		tmp = a * b
	elif (a * b) <= -2.3e-55:
		tmp = x * y
	elif (a * b) <= -8.6e-109:
		tmp = z * t
	elif (a * b) <= -7.5e-246:
		tmp = x * y
	elif (a * b) <= 4.3e+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) <= -5.1e+108)
		tmp = Float64(a * b);
	elseif (Float64(a * b) <= -1.45e+62)
		tmp = Float64(x * y);
	elseif (Float64(a * b) <= -3.4e+23)
		tmp = Float64(a * b);
	elseif (Float64(a * b) <= -0.0155)
		tmp = Float64(z * t);
	elseif (Float64(a * b) <= -1e-8)
		tmp = Float64(a * b);
	elseif (Float64(a * b) <= -2.3e-55)
		tmp = Float64(x * y);
	elseif (Float64(a * b) <= -8.6e-109)
		tmp = Float64(z * t);
	elseif (Float64(a * b) <= -7.5e-246)
		tmp = Float64(x * y);
	elseif (Float64(a * b) <= 4.3e+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) <= -5.1e+108)
		tmp = a * b;
	elseif ((a * b) <= -1.45e+62)
		tmp = x * y;
	elseif ((a * b) <= -3.4e+23)
		tmp = a * b;
	elseif ((a * b) <= -0.0155)
		tmp = z * t;
	elseif ((a * b) <= -1e-8)
		tmp = a * b;
	elseif ((a * b) <= -2.3e-55)
		tmp = x * y;
	elseif ((a * b) <= -8.6e-109)
		tmp = z * t;
	elseif ((a * b) <= -7.5e-246)
		tmp = x * y;
	elseif ((a * b) <= 4.3e+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], -5.1e+108], N[(a * b), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], -1.45e+62], N[(x * y), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], -3.4e+23], N[(a * b), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], -0.0155], N[(z * t), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], -1e-8], N[(a * b), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], -2.3e-55], N[(x * y), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], -8.6e-109], N[(z * t), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], -7.5e-246], N[(x * y), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 4.3e+27], N[(z * t), $MachinePrecision], N[(a * b), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 a b) < -5.09999999999999968e108 or -1.44999999999999992e62 < (*.f64 a b) < -3.39999999999999992e23 or -0.0155 < (*.f64 a b) < -1e-8 or 4.30000000000000008e27 < (*.f64 a b)

   1. Initial program 95.8%

      \[\left(x \cdot y + z \cdot t\right) + a \cdot b \]

   2. Taylor expanded in a around inf 64.7%

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

   if -5.09999999999999968e108 < (*.f64 a b) < -1.44999999999999992e62 or -1e-8 < (*.f64 a b) < -2.30000000000000011e-55 or -8.5999999999999993e-109 < (*.f64 a b) < -7.50000000000000049e-246

   1. Initial program 100.0%

      \[\left(x \cdot y + z \cdot t\right) + a \cdot b \]

   2. Taylor expanded in x around inf 68.3%

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

   if -3.39999999999999992e23 < (*.f64 a b) < -0.0155 or -2.30000000000000011e-55 < (*.f64 a b) < -8.5999999999999993e-109 or -7.50000000000000049e-246 < (*.f64 a b) < 4.30000000000000008e27

   1. Initial program 99.0%

      \[\left(x \cdot y + z \cdot t\right) + a \cdot b \]

   2. Taylor expanded in z around inf 61.0%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -5.1 \cdot 10^{+108}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq -1.45 \cdot 10^{+62}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;a \cdot b \leq -3.4 \cdot 10^{+23}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq -0.0155:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;a \cdot b \leq -1 \cdot 10^{-8}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq -2.3 \cdot 10^{-55}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;a \cdot b \leq -8.6 \cdot 10^{-109}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;a \cdot b \leq -7.5 \cdot 10^{-246}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;a \cdot b \leq 4.3 \cdot 10^{+27}:\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \]

Alternative 5: 68.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{-47}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{+96} \lor \neg \left(y \leq 2.55 \cdot 10^{+110}\right) \land y \leq 1.9 \cdot 10^{+209}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -7.2e-47)
   (* x y)
   (if (or (<= y 2.6e+96) (and (not (<= y 2.55e+110)) (<= y 1.9e+209)))
     (+ (* a b) (* z t))
     (* x y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -7.2e-47) {
		tmp = x * y;
	} else if ((y <= 2.6e+96) || (!(y <= 2.55e+110) && (y <= 1.9e+209))) {
		tmp = (a * b) + (z * t);
	} 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 (y <= (-7.2d-47)) then
        tmp = x * y
    else if ((y <= 2.6d+96) .or. (.not. (y <= 2.55d+110)) .and. (y <= 1.9d+209)) then
        tmp = (a * b) + (z * t)
    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 (y <= -7.2e-47) {
		tmp = x * y;
	} else if ((y <= 2.6e+96) || (!(y <= 2.55e+110) && (y <= 1.9e+209))) {
		tmp = (a * b) + (z * t);
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -7.2e-47:
		tmp = x * y
	elif (y <= 2.6e+96) or (not (y <= 2.55e+110) and (y <= 1.9e+209)):
		tmp = (a * b) + (z * t)
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -7.2e-47)
		tmp = Float64(x * y);
	elseif ((y <= 2.6e+96) || (!(y <= 2.55e+110) && (y <= 1.9e+209)))
		tmp = Float64(Float64(a * b) + Float64(z * t));
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -7.2e-47)
		tmp = x * y;
	elseif ((y <= 2.6e+96) || (~((y <= 2.55e+110)) && (y <= 1.9e+209)))
		tmp = (a * b) + (z * t);
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -7.2e-47], N[(x * y), $MachinePrecision], If[Or[LessEqual[y, 2.6e+96], And[N[Not[LessEqual[y, 2.55e+110]], $MachinePrecision], LessEqual[y, 1.9e+209]]], N[(N[(a * b), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -7.19999999999999982e-47 or 2.6e96 < y < 2.5500000000000001e110 or 1.89999999999999992e209 < y

   1. Initial program 98.0%

      \[\left(x \cdot y + z \cdot t\right) + a \cdot b \]

   2. Taylor expanded in x around inf 54.6%

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

   if -7.19999999999999982e-47 < y < 2.6e96 or 2.5500000000000001e110 < y < 1.89999999999999992e209

   1. Initial program 97.4%

      \[\left(x \cdot y + z \cdot t\right) + a \cdot b \]

   2. Taylor expanded in x around 0 80.3%

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

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{-47}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{+96} \lor \neg \left(y \leq 2.55 \cdot 10^{+110}\right) \land y \leq 1.9 \cdot 10^{+209}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 6: 84.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -8.5 \cdot 10^{+23} \lor \neg \left(a \cdot b \leq 7.5 \cdot 10^{+20}\right):\\ \;\;\;\;a \cdot b + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot t + x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= (* a b) -8.5e+23) (not (<= (* a b) 7.5e+20)))
   (+ (* a b) (* x y))
   (+ (* z t) (* x y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((a * b) <= -8.5e+23) || !((a * b) <= 7.5e+20)) {
		tmp = (a * b) + (x * y);
	} else {
		tmp = (z * t) + (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 (((a * b) <= (-8.5d+23)) .or. (.not. ((a * b) <= 7.5d+20))) then
        tmp = (a * b) + (x * y)
    else
        tmp = (z * t) + (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 (((a * b) <= -8.5e+23) || !((a * b) <= 7.5e+20)) {
		tmp = (a * b) + (x * y);
	} else {
		tmp = (z * t) + (x * y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if ((a * b) <= -8.5e+23) or not ((a * b) <= 7.5e+20):
		tmp = (a * b) + (x * y)
	else:
		tmp = (z * t) + (x * y)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((Float64(a * b) <= -8.5e+23) || !(Float64(a * b) <= 7.5e+20))
		tmp = Float64(Float64(a * b) + Float64(x * y));
	else
		tmp = Float64(Float64(z * t) + Float64(x * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (((a * b) <= -8.5e+23) || ~(((a * b) <= 7.5e+20)))
		tmp = (a * b) + (x * y);
	else
		tmp = (z * t) + (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[N[(a * b), $MachinePrecision], -8.5e+23], N[Not[LessEqual[N[(a * b), $MachinePrecision], 7.5e+20]], $MachinePrecision]], N[(N[(a * b), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision], N[(N[(z * t), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a b) < -8.5000000000000001e23 or 7.5e20 < (*.f64 a b)

   1. Initial program 96.1%

      \[\left(x \cdot y + z \cdot t\right) + a \cdot b \]

   2. Taylor expanded in z around 0 81.6%

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

   if -8.5000000000000001e23 < (*.f64 a b) < 7.5e20

   1. Initial program 99.2%

      \[\left(x \cdot y + z \cdot t\right) + a \cdot b \]

   2. Taylor expanded in a around 0 92.1%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -8.5 \cdot 10^{+23} \lor \neg \left(a \cdot b \leq 7.5 \cdot 10^{+20}\right):\\ \;\;\;\;a \cdot b + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot t + x \cdot y\\ \end{array} \]

Alternative 7: 78.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.3 \cdot 10^{-47} \lor \neg \left(y \leq 4.3 \cdot 10^{+94}\right):\\ \;\;\;\;a \cdot b + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -4.3e-47) (not (<= y 4.3e+94)))
   (+ (* 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 ((y <= -4.3e-47) || !(y <= 4.3e+94)) {
		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 ((y <= (-4.3d-47)) .or. (.not. (y <= 4.3d+94))) 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 ((y <= -4.3e-47) || !(y <= 4.3e+94)) {
		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 (y <= -4.3e-47) or not (y <= 4.3e+94):
		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 ((y <= -4.3e-47) || !(y <= 4.3e+94))
		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 ((y <= -4.3e-47) || ~((y <= 4.3e+94)))
		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[y, -4.3e-47], N[Not[LessEqual[y, 4.3e+94]], $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 y < -4.2999999999999998e-47 or 4.3e94 < y

   1. Initial program 97.5%

      \[\left(x \cdot y + z \cdot t\right) + a \cdot b \]

   2. Taylor expanded in z around 0 79.8%

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

   if -4.2999999999999998e-47 < y < 4.3e94

   1. Initial program 97.8%

      \[\left(x \cdot y + z \cdot t\right) + a \cdot b \]

   2. Taylor expanded in x around 0 83.7%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.3 \cdot 10^{-47} \lor \neg \left(y \leq 4.3 \cdot 10^{+94}\right):\\ \;\;\;\;a \cdot b + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \end{array} \]

Alternative 8: 54.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -8.5 \cdot 10^{+23}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq 1.4 \cdot 10^{+26}:\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* a b) -8.5e+23) (* a b) (if (<= (* a b) 1.4e+26) (* z t) (* a b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a * b) <= -8.5e+23) {
		tmp = a * b;
	} else if ((a * b) <= 1.4e+26) {
		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) <= (-8.5d+23)) then
        tmp = a * b
    else if ((a * b) <= 1.4d+26) 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) <= -8.5e+23) {
		tmp = a * b;
	} else if ((a * b) <= 1.4e+26) {
		tmp = z * t;
	} else {
		tmp = a * b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (a * b) <= -8.5e+23:
		tmp = a * b
	elif (a * b) <= 1.4e+26:
		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) <= -8.5e+23)
		tmp = Float64(a * b);
	elseif (Float64(a * b) <= 1.4e+26)
		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) <= -8.5e+23)
		tmp = a * b;
	elseif ((a * b) <= 1.4e+26)
		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], -8.5e+23], N[(a * b), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 1.4e+26], N[(z * t), $MachinePrecision], N[(a * b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a b) < -8.5000000000000001e23 or 1.4e26 < (*.f64 a b)

   1. Initial program 96.0%

      \[\left(x \cdot y + z \cdot t\right) + a \cdot b \]

   2. Taylor expanded in a around inf 60.6%

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

   if -8.5000000000000001e23 < (*.f64 a b) < 1.4e26

   1. Initial program 99.2%

      \[\left(x \cdot y + z \cdot t\right) + a \cdot b \]

   2. Taylor expanded in z around inf 53.4%

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

4. Final simplification 56.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -8.5 \cdot 10^{+23}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq 1.4 \cdot 10^{+26}:\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \]

Alternative 9: 35.5% 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 97.6%

   \[\left(x \cdot y + z \cdot t\right) + a \cdot b \]

2. Taylor expanded in a around inf 34.5%

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

3. Final simplification 34.5%

   \[\leadsto a \cdot b \]

Reproduce

herbie shell --seed 2023171 
(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)))