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

Percentage Accurate: 97.6% → 98.9%

Time: 9.9s

Alternatives: 10

Speedup: 0.5×

• 33.3% 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: 98.9% 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 97.3%

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

   1. associate-+l+ 97.3%

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

   2. fma-def 99.2%

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

   3. fma-def 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. Final simplification 99.2%

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

Alternative 2: 98.8% 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 97.3%

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

   1. +-commutative 97.3%

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

   2. fma-def 98.0%

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

   3. fma-def 99.2%

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

3. Simplified 99.2%

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

4. Final simplification 99.2%

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

Alternative 3: 98.0% 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 97.3%

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

   1. fma-def 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. Final simplification 98.4%

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

Alternative 4: 54.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -4.5 \cdot 10^{+132}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \cdot y \leq -2.15 \cdot 10^{-99}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;x \cdot y \leq -3.3 \cdot 10^{-195}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{-313}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;x \cdot y \leq 9500000000000:\\ \;\;\;\;a \cdot b\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* x y) -4.5e+132)
   (* x y)
   (if (<= (* x y) -2.15e-99)
     (* z t)
     (if (<= (* x y) -3.3e-195)
       (* a b)
       (if (<= (* x y) 2e-313)
         (* z t)
         (if (<= (* x y) 9500000000000.0) (* a b) (* x y)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((x * y) <= -4.5e+132) {
		tmp = x * y;
	} else if ((x * y) <= -2.15e-99) {
		tmp = z * t;
	} else if ((x * y) <= -3.3e-195) {
		tmp = a * b;
	} else if ((x * y) <= 2e-313) {
		tmp = z * t;
	} else if ((x * y) <= 9500000000000.0) {
		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) <= (-4.5d+132)) then
        tmp = x * y
    else if ((x * y) <= (-2.15d-99)) then
        tmp = z * t
    else if ((x * y) <= (-3.3d-195)) then
        tmp = a * b
    else if ((x * y) <= 2d-313) then
        tmp = z * t
    else if ((x * y) <= 9500000000000.0d0) 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) <= -4.5e+132) {
		tmp = x * y;
	} else if ((x * y) <= -2.15e-99) {
		tmp = z * t;
	} else if ((x * y) <= -3.3e-195) {
		tmp = a * b;
	} else if ((x * y) <= 2e-313) {
		tmp = z * t;
	} else if ((x * y) <= 9500000000000.0) {
		tmp = a * b;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (x * y) <= -4.5e+132:
		tmp = x * y
	elif (x * y) <= -2.15e-99:
		tmp = z * t
	elif (x * y) <= -3.3e-195:
		tmp = a * b
	elif (x * y) <= 2e-313:
		tmp = z * t
	elif (x * y) <= 9500000000000.0:
		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) <= -4.5e+132)
		tmp = Float64(x * y);
	elseif (Float64(x * y) <= -2.15e-99)
		tmp = Float64(z * t);
	elseif (Float64(x * y) <= -3.3e-195)
		tmp = Float64(a * b);
	elseif (Float64(x * y) <= 2e-313)
		tmp = Float64(z * t);
	elseif (Float64(x * y) <= 9500000000000.0)
		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) <= -4.5e+132)
		tmp = x * y;
	elseif ((x * y) <= -2.15e-99)
		tmp = z * t;
	elseif ((x * y) <= -3.3e-195)
		tmp = a * b;
	elseif ((x * y) <= 2e-313)
		tmp = z * t;
	elseif ((x * y) <= 9500000000000.0)
		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], -4.5e+132], N[(x * y), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], -2.15e-99], N[(z * t), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], -3.3e-195], N[(a * b), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 2e-313], N[(z * t), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 9500000000000.0], N[(a * b), $MachinePrecision], N[(x * y), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -4.49999999999999972e132 or 9.5e12 < (*.f64 x y)

   1. Initial program 93.2%

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

   2. Taylor expanded in x around inf 70.4%

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

   if -4.49999999999999972e132 < (*.f64 x y) < -2.1499999999999999e-99 or -3.3e-195 < (*.f64 x y) < 1.99999999998e-313

   1. Initial program 100.0%

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

   2. Taylor expanded in z around inf 56.7%

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

   if -2.1499999999999999e-99 < (*.f64 x y) < -3.3e-195 or 1.99999999998e-313 < (*.f64 x y) < 9.5e12

   1. Initial program 100.0%

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

   2. Taylor expanded in a around inf 61.9%

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

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -4.5 \cdot 10^{+132}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \cdot y \leq -2.15 \cdot 10^{-99}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;x \cdot y \leq -3.3 \cdot 10^{-195}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{-313}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;x \cdot y \leq 9500000000000:\\ \;\;\;\;a \cdot b\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 5: 98.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (a * b) + ((x * y) + (z * t));
	tmp = 0.0;
	if (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = (a * b) + (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[(x * y), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, Infinity], t$95$1, N[(N[(a * b), $MachinePrecision] + N[(z * t), $MachinePrecision]), $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 x around 0 71.4%

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

4. Final simplification 99.2%

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

Alternative 6: 84.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot y + z \cdot t\\ \mathbf{if}\;x \cdot y \leq -42000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \cdot y \leq 1.85 \cdot 10^{-22}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \mathbf{elif}\;x \cdot y \leq 6.8 \cdot 10^{+56}:\\ \;\;\;\;a \cdot b + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (* x y) (* z t))))
   (if (<= (* x y) -42000000.0)
     t_1
     (if (<= (* x y) 1.85e-22)
       (+ (* a b) (* z t))
       (if (<= (* x y) 6.8e+56) (+ (* a b) (* x y)) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x * y) + (z * t);
	double tmp;
	if ((x * y) <= -42000000.0) {
		tmp = t_1;
	} else if ((x * y) <= 1.85e-22) {
		tmp = (a * b) + (z * t);
	} else if ((x * y) <= 6.8e+56) {
		tmp = (a * b) + (x * y);
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = (x * y) + (z * t)
    if ((x * y) <= (-42000000.0d0)) then
        tmp = t_1
    else if ((x * y) <= 1.85d-22) then
        tmp = (a * b) + (z * t)
    else if ((x * y) <= 6.8d+56) then
        tmp = (a * b) + (x * y)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x * y) + (z * t);
	double tmp;
	if ((x * y) <= -42000000.0) {
		tmp = t_1;
	} else if ((x * y) <= 1.85e-22) {
		tmp = (a * b) + (z * t);
	} else if ((x * y) <= 6.8e+56) {
		tmp = (a * b) + (x * y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (x * y) + (z * t)
	tmp = 0
	if (x * y) <= -42000000.0:
		tmp = t_1
	elif (x * y) <= 1.85e-22:
		tmp = (a * b) + (z * t)
	elif (x * y) <= 6.8e+56:
		tmp = (a * b) + (x * y)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x * y) + Float64(z * t))
	tmp = 0.0
	if (Float64(x * y) <= -42000000.0)
		tmp = t_1;
	elseif (Float64(x * y) <= 1.85e-22)
		tmp = Float64(Float64(a * b) + Float64(z * t));
	elseif (Float64(x * y) <= 6.8e+56)
		tmp = Float64(Float64(a * b) + Float64(x * y));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (x * y) + (z * t);
	tmp = 0.0;
	if ((x * y) <= -42000000.0)
		tmp = t_1;
	elseif ((x * y) <= 1.85e-22)
		tmp = (a * b) + (z * t);
	elseif ((x * y) <= 6.8e+56)
		tmp = (a * b) + (x * y);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x * y), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * y), $MachinePrecision], -42000000.0], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], 1.85e-22], N[(N[(a * b), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 6.8e+56], N[(N[(a * b), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -4.2e7 or 6.80000000000000002e56 < (*.f64 x y)

   1. Initial program 93.9%

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

   2. Taylor expanded in a around 0 87.4%

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

   if -4.2e7 < (*.f64 x y) < 1.85e-22

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 91.6%

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

   if 1.85e-22 < (*.f64 x y) < 6.80000000000000002e56

   1. Initial program 100.0%

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

   2. Taylor expanded in z around 0 94.4%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -42000000:\\ \;\;\;\;x \cdot y + z \cdot t\\ \mathbf{elif}\;x \cdot y \leq 1.85 \cdot 10^{-22}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \mathbf{elif}\;x \cdot y \leq 6.8 \cdot 10^{+56}:\\ \;\;\;\;a \cdot b + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot y + z \cdot t\\ \end{array} \]

Alternative 7: 84.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -9.5 \cdot 10^{+46} \lor \neg \left(x \cdot y \leq 9.6 \cdot 10^{-23}\right):\\ \;\;\;\;a \cdot b + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= (* x y) -9.5e+46) (not (<= (* x y) 9.6e-23)))
   (+ (* 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) <= -9.5e+46) || !((x * y) <= 9.6e-23)) {
		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) <= (-9.5d+46)) .or. (.not. ((x * y) <= 9.6d-23))) 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) <= -9.5e+46) || !((x * y) <= 9.6e-23)) {
		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) <= -9.5e+46) or not ((x * y) <= 9.6e-23):
		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) <= -9.5e+46) || !(Float64(x * y) <= 9.6e-23))
		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) <= -9.5e+46) || ~(((x * y) <= 9.6e-23)))
		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], -9.5e+46], N[Not[LessEqual[N[(x * y), $MachinePrecision], 9.6e-23]], $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) < -9.5000000000000008e46 or 9.59999999999999986e-23 < (*.f64 x y)

   1. Initial program 94.5%

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

   2. Taylor expanded in z around 0 78.4%

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

   if -9.5000000000000008e46 < (*.f64 x y) < 9.59999999999999986e-23

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 91.2%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -9.5 \cdot 10^{+46} \lor \neg \left(x \cdot y \leq 9.6 \cdot 10^{-23}\right):\\ \;\;\;\;a \cdot b + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \end{array} \]

Alternative 8: 81.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2.8 \cdot 10^{+153}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \cdot y \leq 9 \cdot 10^{+119}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* x y) -2.8e+153)
   (* x y)
   (if (<= (* x y) 9e+119) (+ (* a b) (* z t)) (* x y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((x * y) <= -2.8e+153) {
		tmp = x * y;
	} else if ((x * y) <= 9e+119) {
		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 ((x * y) <= (-2.8d+153)) then
        tmp = x * y
    else if ((x * y) <= 9d+119) 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 ((x * y) <= -2.8e+153) {
		tmp = x * y;
	} else if ((x * y) <= 9e+119) {
		tmp = (a * b) + (z * t);
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (x * y) <= -2.8e+153:
		tmp = x * y
	elif (x * y) <= 9e+119:
		tmp = (a * b) + (z * t)
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(x * y) <= -2.8e+153)
		tmp = Float64(x * y);
	elseif (Float64(x * y) <= 9e+119)
		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 ((x * y) <= -2.8e+153)
		tmp = x * y;
	elseif ((x * y) <= 9e+119)
		tmp = (a * b) + (z * t);
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(x * y), $MachinePrecision], -2.8e+153], N[(x * y), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 9e+119], N[(N[(a * b), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -2.79999999999999985e153 or 9.00000000000000039e119 < (*.f64 x y)

   1. Initial program 90.8%

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

   2. Taylor expanded in x around inf 79.4%

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

   if -2.79999999999999985e153 < (*.f64 x y) < 9.00000000000000039e119

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 82.4%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2.8 \cdot 10^{+153}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \cdot y \leq 9 \cdot 10^{+119}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 9: 54.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -3.1 \cdot 10^{+73}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq 7.2 \cdot 10^{+23}:\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* a b) -3.1e+73) (* a b) (if (<= (* a b) 7.2e+23) (* z t) (* a b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a * b) <= -3.1e+73) {
		tmp = a * b;
	} else if ((a * b) <= 7.2e+23) {
		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) <= (-3.1d+73)) then
        tmp = a * b
    else if ((a * b) <= 7.2d+23) 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) <= -3.1e+73) {
		tmp = a * b;
	} else if ((a * b) <= 7.2e+23) {
		tmp = z * t;
	} else {
		tmp = a * b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (a * b) <= -3.1e+73:
		tmp = a * b
	elif (a * b) <= 7.2e+23:
		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) <= -3.1e+73)
		tmp = Float64(a * b);
	elseif (Float64(a * b) <= 7.2e+23)
		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) <= -3.1e+73)
		tmp = a * b;
	elseif ((a * b) <= 7.2e+23)
		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], -3.1e+73], N[(a * b), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 7.2e+23], N[(z * t), $MachinePrecision], N[(a * b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a b) < -3.1e73 or 7.1999999999999997e23 < (*.f64 a b)

   1. Initial program 94.8%

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

   2. Taylor expanded in a around inf 69.0%

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

   if -3.1e73 < (*.f64 a b) < 7.1999999999999997e23

   1. Initial program 98.7%

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

   2. Taylor expanded in z around inf 48.5%

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

4. Final simplification 56.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -3.1 \cdot 10^{+73}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq 7.2 \cdot 10^{+23}:\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \]

Alternative 10: 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 97.3%

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

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 2023275 
(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)))