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

Percentage Accurate: 97.8% → 98.9%

Time: 5.1s

Alternatives: 9

Speedup: 1.0×

• 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: 97.8% 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: 98.9% 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-def 99.6%

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

   3. fma-def 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 98.1% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.4%

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

   1. fma-def 98.8%

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

3. Simplified 98.8%

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

4. Final simplification 98.8%

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

Alternative 3: 52.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -52000:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \cdot y \leq -4 \cdot 10^{-70}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;x \cdot y \leq -6.8 \cdot 10^{-304}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;x \cdot y \leq 1.2 \cdot 10^{-262}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;x \cdot y \leq 1.85 \cdot 10^{-172}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;x \cdot y \leq 1.9 \cdot 10^{-114}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;x \cdot y \leq 4.4 \cdot 10^{+133}:\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* x y) -52000.0)
   (* x y)
   (if (<= (* x y) -4e-70)
     (* a b)
     (if (<= (* x y) -6.8e-304)
       (* z t)
       (if (<= (* x y) 1.2e-262)
         (* a b)
         (if (<= (* x y) 1.85e-172)
           (* z t)
           (if (<= (* x y) 1.9e-114)
             (* a b)
             (if (<= (* x y) 4.4e+133) (* z t) (* x y)))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((x * y) <= -52000.0) {
		tmp = x * y;
	} else if ((x * y) <= -4e-70) {
		tmp = a * b;
	} else if ((x * y) <= -6.8e-304) {
		tmp = z * t;
	} else if ((x * y) <= 1.2e-262) {
		tmp = a * b;
	} else if ((x * y) <= 1.85e-172) {
		tmp = z * t;
	} else if ((x * y) <= 1.9e-114) {
		tmp = a * b;
	} else if ((x * y) <= 4.4e+133) {
		tmp = 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) <= (-52000.0d0)) then
        tmp = x * y
    else if ((x * y) <= (-4d-70)) then
        tmp = a * b
    else if ((x * y) <= (-6.8d-304)) then
        tmp = z * t
    else if ((x * y) <= 1.2d-262) then
        tmp = a * b
    else if ((x * y) <= 1.85d-172) then
        tmp = z * t
    else if ((x * y) <= 1.9d-114) then
        tmp = a * b
    else if ((x * y) <= 4.4d+133) then
        tmp = 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) <= -52000.0) {
		tmp = x * y;
	} else if ((x * y) <= -4e-70) {
		tmp = a * b;
	} else if ((x * y) <= -6.8e-304) {
		tmp = z * t;
	} else if ((x * y) <= 1.2e-262) {
		tmp = a * b;
	} else if ((x * y) <= 1.85e-172) {
		tmp = z * t;
	} else if ((x * y) <= 1.9e-114) {
		tmp = a * b;
	} else if ((x * y) <= 4.4e+133) {
		tmp = z * t;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (x * y) <= -52000.0:
		tmp = x * y
	elif (x * y) <= -4e-70:
		tmp = a * b
	elif (x * y) <= -6.8e-304:
		tmp = z * t
	elif (x * y) <= 1.2e-262:
		tmp = a * b
	elif (x * y) <= 1.85e-172:
		tmp = z * t
	elif (x * y) <= 1.9e-114:
		tmp = a * b
	elif (x * y) <= 4.4e+133:
		tmp = z * t
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(x * y) <= -52000.0)
		tmp = Float64(x * y);
	elseif (Float64(x * y) <= -4e-70)
		tmp = Float64(a * b);
	elseif (Float64(x * y) <= -6.8e-304)
		tmp = Float64(z * t);
	elseif (Float64(x * y) <= 1.2e-262)
		tmp = Float64(a * b);
	elseif (Float64(x * y) <= 1.85e-172)
		tmp = Float64(z * t);
	elseif (Float64(x * y) <= 1.9e-114)
		tmp = Float64(a * b);
	elseif (Float64(x * y) <= 4.4e+133)
		tmp = 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) <= -52000.0)
		tmp = x * y;
	elseif ((x * y) <= -4e-70)
		tmp = a * b;
	elseif ((x * y) <= -6.8e-304)
		tmp = z * t;
	elseif ((x * y) <= 1.2e-262)
		tmp = a * b;
	elseif ((x * y) <= 1.85e-172)
		tmp = z * t;
	elseif ((x * y) <= 1.9e-114)
		tmp = a * b;
	elseif ((x * y) <= 4.4e+133)
		tmp = 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], -52000.0], N[(x * y), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], -4e-70], N[(a * b), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], -6.8e-304], N[(z * t), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 1.2e-262], N[(a * b), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 1.85e-172], N[(z * t), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 1.9e-114], N[(a * b), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 4.4e+133], N[(z * t), $MachinePrecision], N[(x * y), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -52000 or 4.4e133 < (*.f64 x y)

   1. Initial program 98.1%

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

   2. Taylor expanded in x around inf 68.9%

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

   if -52000 < (*.f64 x y) < -3.99999999999999998e-70 or -6.7999999999999997e-304 < (*.f64 x y) < 1.2e-262 or 1.85e-172 < (*.f64 x y) < 1.8999999999999999e-114

   1. Initial program 98.5%

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

   2. Taylor expanded in a around inf 68.0%

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

   if -3.99999999999999998e-70 < (*.f64 x y) < -6.7999999999999997e-304 or 1.2e-262 < (*.f64 x y) < 1.85e-172 or 1.8999999999999999e-114 < (*.f64 x y) < 4.4e133

   1. Initial program 98.8%

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

   2. Taylor expanded in z around inf 63.9%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -52000:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \cdot y \leq -4 \cdot 10^{-70}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;x \cdot y \leq -6.8 \cdot 10^{-304}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;x \cdot y \leq 1.2 \cdot 10^{-262}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;x \cdot y \leq 1.85 \cdot 10^{-172}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;x \cdot y \leq 1.9 \cdot 10^{-114}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;x \cdot y \leq 4.4 \cdot 10^{+133}:\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 4: 85.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot b + x \cdot y\\ \mathbf{if}\;a \cdot b \leq -3.3 \cdot 10^{+177}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \cdot b \leq -1.86 \cdot 10^{-59}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \mathbf{elif}\;a \cdot b \leq 2.7 \cdot 10^{+53}:\\ \;\;\;\;x \cdot y + z \cdot t\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (* a b) (* x y))))
   (if (<= (* a b) -3.3e+177)
     t_1
     (if (<= (* a b) -1.86e-59)
       (+ (* a b) (* z t))
       (if (<= (* a b) 2.7e+53) (+ (* x y) (* z t)) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a * b) + (x * y);
	double tmp;
	if ((a * b) <= -3.3e+177) {
		tmp = t_1;
	} else if ((a * b) <= -1.86e-59) {
		tmp = (a * b) + (z * t);
	} else if ((a * b) <= 2.7e+53) {
		tmp = (x * y) + (z * t);
	} 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 = (a * b) + (x * y)
    if ((a * b) <= (-3.3d+177)) then
        tmp = t_1
    else if ((a * b) <= (-1.86d-59)) then
        tmp = (a * b) + (z * t)
    else if ((a * b) <= 2.7d+53) then
        tmp = (x * y) + (z * t)
    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 = (a * b) + (x * y);
	double tmp;
	if ((a * b) <= -3.3e+177) {
		tmp = t_1;
	} else if ((a * b) <= -1.86e-59) {
		tmp = (a * b) + (z * t);
	} else if ((a * b) <= 2.7e+53) {
		tmp = (x * y) + (z * t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (a * b) + (x * y)
	tmp = 0
	if (a * b) <= -3.3e+177:
		tmp = t_1
	elif (a * b) <= -1.86e-59:
		tmp = (a * b) + (z * t)
	elif (a * b) <= 2.7e+53:
		tmp = (x * y) + (z * t)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a * b) + Float64(x * y))
	tmp = 0.0
	if (Float64(a * b) <= -3.3e+177)
		tmp = t_1;
	elseif (Float64(a * b) <= -1.86e-59)
		tmp = Float64(Float64(a * b) + Float64(z * t));
	elseif (Float64(a * b) <= 2.7e+53)
		tmp = Float64(Float64(x * y) + Float64(z * t));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (a * b) + (x * y);
	tmp = 0.0;
	if ((a * b) <= -3.3e+177)
		tmp = t_1;
	elseif ((a * b) <= -1.86e-59)
		tmp = (a * b) + (z * t);
	elseif ((a * b) <= 2.7e+53)
		tmp = (x * y) + (z * t);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a * b), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(a * b), $MachinePrecision], -3.3e+177], t$95$1, If[LessEqual[N[(a * b), $MachinePrecision], -1.86e-59], N[(N[(a * b), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 2.7e+53], N[(N[(x * y), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 a b) < -3.3000000000000001e177 or 2.70000000000000019e53 < (*.f64 a b)

   1. Initial program 96.4%

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

   2. Taylor expanded in z around 0 87.9%

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

   if -3.3000000000000001e177 < (*.f64 a b) < -1.86000000000000004e-59

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 89.1%

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

   if -1.86000000000000004e-59 < (*.f64 a b) < 2.70000000000000019e53

   1. Initial program 99.2%

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

   2. Taylor expanded in a around 0 93.1%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -3.3 \cdot 10^{+177}:\\ \;\;\;\;a \cdot b + x \cdot y\\ \mathbf{elif}\;a \cdot b \leq -1.86 \cdot 10^{-59}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \mathbf{elif}\;a \cdot b \leq 2.7 \cdot 10^{+53}:\\ \;\;\;\;x \cdot y + z \cdot t\\ \mathbf{else}:\\ \;\;\;\;a \cdot b + x \cdot y\\ \end{array} \]

Alternative 5: 83.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5.5 \lor \neg \left(x \cdot y \leq 7 \cdot 10^{+133}\right):\\ \;\;\;\;a \cdot b + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= (* x y) -5.5) (not (<= (* x y) 7e+133)))
   (+ (* 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) <= -5.5) || !((x * y) <= 7e+133)) {
		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) <= (-5.5d0)) .or. (.not. ((x * y) <= 7d+133))) 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) <= -5.5) || !((x * y) <= 7e+133)) {
		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) <= -5.5) or not ((x * y) <= 7e+133):
		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) <= -5.5) || !(Float64(x * y) <= 7e+133))
		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) <= -5.5) || ~(((x * y) <= 7e+133)))
		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], -5.5], N[Not[LessEqual[N[(x * y), $MachinePrecision], 7e+133]], $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) < -5.5 or 6.9999999999999997e133 < (*.f64 x y)

   1. Initial program 98.1%

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

   2. Taylor expanded in z around 0 85.4%

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

   if -5.5 < (*.f64 x y) < 6.9999999999999997e133

   1. Initial program 98.6%

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

   2. Taylor expanded in x around 0 88.4%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5.5 \lor \neg \left(x \cdot y \leq 7 \cdot 10^{+133}\right):\\ \;\;\;\;a \cdot b + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \end{array} \]

Alternative 6: 80.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2.9 \cdot 10^{+99}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \cdot y \leq 6.1 \cdot 10^{+140}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* x y) -2.9e+99)
   (* x y)
   (if (<= (* x y) 6.1e+140) (+ (* 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) <= -2.9e+99) {
		tmp = x * y;
	} else if ((x * y) <= 6.1e+140) {
		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) <= (-2.9d+99)) then
        tmp = x * y
    else if ((x * y) <= 6.1d+140) 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) <= -2.9e+99) {
		tmp = x * y;
	} else if ((x * y) <= 6.1e+140) {
		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) <= -2.9e+99:
		tmp = x * y
	elif (x * y) <= 6.1e+140:
		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) <= -2.9e+99)
		tmp = Float64(x * y);
	elseif (Float64(x * y) <= 6.1e+140)
		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) <= -2.9e+99)
		tmp = x * y;
	elseif ((x * y) <= 6.1e+140)
		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], -2.9e+99], N[(x * y), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 6.1e+140], 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) < -2.9000000000000002e99 or 6.0999999999999996e140 < (*.f64 x y)

   1. Initial program 97.9%

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

   2. Taylor expanded in x around inf 73.6%

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

   if -2.9000000000000002e99 < (*.f64 x y) < 6.0999999999999996e140

   1. Initial program 98.7%

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

   2. Taylor expanded in x around 0 86.4%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2.9 \cdot 10^{+99}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \cdot y \leq 6.1 \cdot 10^{+140}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 7: 54.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -740000:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq 7.4 \cdot 10^{+102}:\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* a b) -740000.0)
   (* a b)
   (if (<= (* a b) 7.4e+102) (* z t) (* a b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a * b) <= -740000.0) {
		tmp = a * b;
	} else if ((a * b) <= 7.4e+102) {
		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) <= (-740000.0d0)) then
        tmp = a * b
    else if ((a * b) <= 7.4d+102) 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) <= -740000.0) {
		tmp = a * b;
	} else if ((a * b) <= 7.4e+102) {
		tmp = z * t;
	} else {
		tmp = a * b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (a * b) <= -740000.0:
		tmp = a * b
	elif (a * b) <= 7.4e+102:
		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) <= -740000.0)
		tmp = Float64(a * b);
	elseif (Float64(a * b) <= 7.4e+102)
		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) <= -740000.0)
		tmp = a * b;
	elseif ((a * b) <= 7.4e+102)
		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], -740000.0], N[(a * b), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 7.4e+102], N[(z * t), $MachinePrecision], N[(a * b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a b) < -7.4e5 or 7.40000000000000045e102 < (*.f64 a b)

   1. Initial program 97.0%

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

   2. Taylor expanded in a around inf 66.6%

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

   if -7.4e5 < (*.f64 a b) < 7.40000000000000045e102

   1. Initial program 99.3%

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

   2. Taylor expanded in z around inf 51.2%

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

4. Final simplification 57.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -740000:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq 7.4 \cdot 10^{+102}:\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \]

Alternative 8: 97.8% 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. Final simplification 98.4%

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

Alternative 9: 35.6% 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. Taylor expanded in a around inf 33.1%

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

3. Final simplification 33.1%

   \[\leadsto a \cdot b \]

Reproduce

herbie shell --seed 2023290 
(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)))