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

Percentage Accurate: 97.9% → 98.1%

Time: 4.2s

Alternatives: 8

Speedup: 1.0×

• 32.7% 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: 97.9% 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: 98.1% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.8%

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

   1. associate-+l+ 98.8%

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

   2. fma-def 98.8%

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

   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. Step-by-step derivation

   1. fma-udef 99.2%

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

   2. +-commutative 99.2%

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

5. Applied egg-rr 99.2%

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

6. Final simplification 99.2%

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

Alternative 2: 69.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{+262}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq -4.6 \cdot 10^{+258} \lor \neg \left(x \leq -1.56 \cdot 10^{+205}\right) \land \left(x \leq -1.6 \cdot 10^{+132} \lor \neg \left(x \leq -3.6 \cdot 10^{+77}\right) \land x \leq 4150000000000\right):\\ \;\;\;\;a \cdot b + z \cdot t\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= x -6.5e+262)
   (* x y)
   (if (or (<= x -4.6e+258)
           (and (not (<= x -1.56e+205))
                (or (<= x -1.6e+132)
                    (and (not (<= x -3.6e+77)) (<= x 4150000000000.0)))))
     (+ (* 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 <= -6.5e+262) {
		tmp = x * y;
	} else if ((x <= -4.6e+258) || (!(x <= -1.56e+205) && ((x <= -1.6e+132) || (!(x <= -3.6e+77) && (x <= 4150000000000.0))))) {
		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 <= (-6.5d+262)) then
        tmp = x * y
    else if ((x <= (-4.6d+258)) .or. (.not. (x <= (-1.56d+205))) .and. (x <= (-1.6d+132)) .or. (.not. (x <= (-3.6d+77))) .and. (x <= 4150000000000.0d0)) 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 <= -6.5e+262) {
		tmp = x * y;
	} else if ((x <= -4.6e+258) || (!(x <= -1.56e+205) && ((x <= -1.6e+132) || (!(x <= -3.6e+77) && (x <= 4150000000000.0))))) {
		tmp = (a * b) + (z * t);
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if x <= -6.5e+262:
		tmp = x * y
	elif (x <= -4.6e+258) or (not (x <= -1.56e+205) and ((x <= -1.6e+132) or (not (x <= -3.6e+77) and (x <= 4150000000000.0)))):
		tmp = (a * b) + (z * t)
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (x <= -6.5e+262)
		tmp = Float64(x * y);
	elseif ((x <= -4.6e+258) || (!(x <= -1.56e+205) && ((x <= -1.6e+132) || (!(x <= -3.6e+77) && (x <= 4150000000000.0)))))
		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 <= -6.5e+262)
		tmp = x * y;
	elseif ((x <= -4.6e+258) || (~((x <= -1.56e+205)) && ((x <= -1.6e+132) || (~((x <= -3.6e+77)) && (x <= 4150000000000.0)))))
		tmp = (a * b) + (z * t);
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[x, -6.5e+262], N[(x * y), $MachinePrecision], If[Or[LessEqual[x, -4.6e+258], And[N[Not[LessEqual[x, -1.56e+205]], $MachinePrecision], Or[LessEqual[x, -1.6e+132], And[N[Not[LessEqual[x, -3.6e+77]], $MachinePrecision], LessEqual[x, 4150000000000.0]]]]], N[(N[(a * b), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -6.5000000000000005e262 or -4.6000000000000002e258 < x < -1.56000000000000007e205 or -1.5999999999999999e132 < x < -3.5999999999999998e77 or 4.15e12 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 63.5%

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

   if -6.5000000000000005e262 < x < -4.6000000000000002e258 or -1.56000000000000007e205 < x < -1.5999999999999999e132 or -3.5999999999999998e77 < x < 4.15e12

   1. Initial program 98.2%

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

   2. Taylor expanded in x around 0 82.6%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{+262}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq -4.6 \cdot 10^{+258} \lor \neg \left(x \leq -1.56 \cdot 10^{+205}\right) \land \left(x \leq -1.6 \cdot 10^{+132} \lor \neg \left(x \leq -3.6 \cdot 10^{+77}\right) \land x \leq 4150000000000\right):\\ \;\;\;\;a \cdot b + z \cdot t\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 3: 47.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.5 \cdot 10^{-78}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;b \leq 1.8 \cdot 10^{-191}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;b \leq 5.2 \cdot 10^{-136}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;b \leq 1.15 \cdot 10^{-82}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;b \leq 1.52 \cdot 10^{+26}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;b \leq 2.85 \cdot 10^{+78}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;b \leq 3.3 \cdot 10^{+118}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b -1.5e-78)
   (* a b)
   (if (<= b 1.8e-191)
     (* z t)
     (if (<= b 5.2e-136)
       (* x y)
       (if (<= b 1.15e-82)
         (* z t)
         (if (<= b 1.52e+26)
           (* x y)
           (if (<= b 2.85e+78)
             (* z t)
             (if (<= b 3.3e+118) (* x y) (* a b)))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -1.5e-78) {
		tmp = a * b;
	} else if (b <= 1.8e-191) {
		tmp = z * t;
	} else if (b <= 5.2e-136) {
		tmp = x * y;
	} else if (b <= 1.15e-82) {
		tmp = z * t;
	} else if (b <= 1.52e+26) {
		tmp = x * y;
	} else if (b <= 2.85e+78) {
		tmp = z * t;
	} else if (b <= 3.3e+118) {
		tmp = x * y;
	} 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 (b <= (-1.5d-78)) then
        tmp = a * b
    else if (b <= 1.8d-191) then
        tmp = z * t
    else if (b <= 5.2d-136) then
        tmp = x * y
    else if (b <= 1.15d-82) then
        tmp = z * t
    else if (b <= 1.52d+26) then
        tmp = x * y
    else if (b <= 2.85d+78) then
        tmp = z * t
    else if (b <= 3.3d+118) then
        tmp = x * y
    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 (b <= -1.5e-78) {
		tmp = a * b;
	} else if (b <= 1.8e-191) {
		tmp = z * t;
	} else if (b <= 5.2e-136) {
		tmp = x * y;
	} else if (b <= 1.15e-82) {
		tmp = z * t;
	} else if (b <= 1.52e+26) {
		tmp = x * y;
	} else if (b <= 2.85e+78) {
		tmp = z * t;
	} else if (b <= 3.3e+118) {
		tmp = x * y;
	} else {
		tmp = a * b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if b <= -1.5e-78:
		tmp = a * b
	elif b <= 1.8e-191:
		tmp = z * t
	elif b <= 5.2e-136:
		tmp = x * y
	elif b <= 1.15e-82:
		tmp = z * t
	elif b <= 1.52e+26:
		tmp = x * y
	elif b <= 2.85e+78:
		tmp = z * t
	elif b <= 3.3e+118:
		tmp = x * y
	else:
		tmp = a * b
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= -1.5e-78)
		tmp = Float64(a * b);
	elseif (b <= 1.8e-191)
		tmp = Float64(z * t);
	elseif (b <= 5.2e-136)
		tmp = Float64(x * y);
	elseif (b <= 1.15e-82)
		tmp = Float64(z * t);
	elseif (b <= 1.52e+26)
		tmp = Float64(x * y);
	elseif (b <= 2.85e+78)
		tmp = Float64(z * t);
	elseif (b <= 3.3e+118)
		tmp = Float64(x * y);
	else
		tmp = Float64(a * b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (b <= -1.5e-78)
		tmp = a * b;
	elseif (b <= 1.8e-191)
		tmp = z * t;
	elseif (b <= 5.2e-136)
		tmp = x * y;
	elseif (b <= 1.15e-82)
		tmp = z * t;
	elseif (b <= 1.52e+26)
		tmp = x * y;
	elseif (b <= 2.85e+78)
		tmp = z * t;
	elseif (b <= 3.3e+118)
		tmp = x * y;
	else
		tmp = a * b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, -1.5e-78], N[(a * b), $MachinePrecision], If[LessEqual[b, 1.8e-191], N[(z * t), $MachinePrecision], If[LessEqual[b, 5.2e-136], N[(x * y), $MachinePrecision], If[LessEqual[b, 1.15e-82], N[(z * t), $MachinePrecision], If[LessEqual[b, 1.52e+26], N[(x * y), $MachinePrecision], If[LessEqual[b, 2.85e+78], N[(z * t), $MachinePrecision], If[LessEqual[b, 3.3e+118], N[(x * y), $MachinePrecision], N[(a * b), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.49999999999999994e-78 or 3.3e118 < b

   1. Initial program 97.1%

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

   2. Taylor expanded in a around inf 53.7%

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

   if -1.49999999999999994e-78 < b < 1.8000000000000001e-191 or 5.19999999999999993e-136 < b < 1.14999999999999998e-82 or 1.52e26 < b < 2.84999999999999993e78

   1. Initial program 100.0%

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

   2. Taylor expanded in z around inf 48.8%

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

   if 1.8000000000000001e-191 < b < 5.19999999999999993e-136 or 1.14999999999999998e-82 < b < 1.52e26 or 2.84999999999999993e78 < b < 3.3e118

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 57.1%

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

4. Final simplification 52.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.5 \cdot 10^{-78}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;b \leq 1.8 \cdot 10^{-191}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;b \leq 5.2 \cdot 10^{-136}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;b \leq 1.15 \cdot 10^{-82}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;b \leq 1.52 \cdot 10^{+26}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;b \leq 2.85 \cdot 10^{+78}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;b \leq 3.3 \cdot 10^{+118}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \]

Alternative 4: 77.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot b + z \cdot t\\ \mathbf{if}\;z \leq -1.1 \cdot 10^{+87}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -8.5 \cdot 10^{+80}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;z \leq -1.55 \cdot 10^{+19} \lor \neg \left(z \leq 1.3 \cdot 10^{-121}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;a \cdot b + x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (* a b) (* z t))))
   (if (<= z -1.1e+87)
     t_1
     (if (<= z -8.5e+80)
       (* x y)
       (if (or (<= z -1.55e+19) (not (<= z 1.3e-121)))
         t_1
         (+ (* a b) (* x y)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a * b) + (z * t);
	double tmp;
	if (z <= -1.1e+87) {
		tmp = t_1;
	} else if (z <= -8.5e+80) {
		tmp = x * y;
	} else if ((z <= -1.55e+19) || !(z <= 1.3e-121)) {
		tmp = t_1;
	} else {
		tmp = (a * b) + (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) :: t_1
    real(8) :: tmp
    t_1 = (a * b) + (z * t)
    if (z <= (-1.1d+87)) then
        tmp = t_1
    else if (z <= (-8.5d+80)) then
        tmp = x * y
    else if ((z <= (-1.55d+19)) .or. (.not. (z <= 1.3d-121))) then
        tmp = t_1
    else
        tmp = (a * b) + (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 t_1 = (a * b) + (z * t);
	double tmp;
	if (z <= -1.1e+87) {
		tmp = t_1;
	} else if (z <= -8.5e+80) {
		tmp = x * y;
	} else if ((z <= -1.55e+19) || !(z <= 1.3e-121)) {
		tmp = t_1;
	} else {
		tmp = (a * b) + (x * y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (a * b) + (z * t)
	tmp = 0
	if z <= -1.1e+87:
		tmp = t_1
	elif z <= -8.5e+80:
		tmp = x * y
	elif (z <= -1.55e+19) or not (z <= 1.3e-121):
		tmp = t_1
	else:
		tmp = (a * b) + (x * y)
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a * b) + Float64(z * t))
	tmp = 0.0
	if (z <= -1.1e+87)
		tmp = t_1;
	elseif (z <= -8.5e+80)
		tmp = Float64(x * y);
	elseif ((z <= -1.55e+19) || !(z <= 1.3e-121))
		tmp = t_1;
	else
		tmp = Float64(Float64(a * b) + Float64(x * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (a * b) + (z * t);
	tmp = 0.0;
	if (z <= -1.1e+87)
		tmp = t_1;
	elseif (z <= -8.5e+80)
		tmp = x * y;
	elseif ((z <= -1.55e+19) || ~((z <= 1.3e-121)))
		tmp = t_1;
	else
		tmp = (a * b) + (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a * b), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.1e+87], t$95$1, If[LessEqual[z, -8.5e+80], N[(x * y), $MachinePrecision], If[Or[LessEqual[z, -1.55e+19], N[Not[LessEqual[z, 1.3e-121]], $MachinePrecision]], t$95$1, N[(N[(a * b), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.1e87 or -8.50000000000000007e80 < z < -1.55e19 or 1.29999999999999993e-121 < z

   1. Initial program 97.9%

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

   2. Taylor expanded in x around 0 78.7%

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

   if -1.1e87 < z < -8.50000000000000007e80

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 0.3%

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

   if -1.55e19 < z < 1.29999999999999993e-121

   1. Initial program 100.0%

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

   2. Taylor expanded in z around 0 83.8%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{+87}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \mathbf{elif}\;z \leq -8.5 \cdot 10^{+80}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;z \leq -1.55 \cdot 10^{+19} \lor \neg \left(z \leq 1.3 \cdot 10^{-121}\right):\\ \;\;\;\;a \cdot b + z \cdot t\\ \mathbf{else}:\\ \;\;\;\;a \cdot b + x \cdot y\\ \end{array} \]

Alternative 5: 84.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -2.6 \cdot 10^{+83} \lor \neg \left(a \cdot b \leq 6.6 \cdot 10^{+114}\right):\\ \;\;\;\;a \cdot b + z \cdot t\\ \mathbf{else}:\\ \;\;\;\;z \cdot t + x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= (* a b) -2.6e+83) (not (<= (* a b) 6.6e+114)))
   (+ (* a b) (* z t))
   (+ (* z t) (* x y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((a * b) <= -2.6e+83) || !((a * b) <= 6.6e+114)) {
		tmp = (a * b) + (z * t);
	} 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) <= (-2.6d+83)) .or. (.not. ((a * b) <= 6.6d+114))) then
        tmp = (a * b) + (z * t)
    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) <= -2.6e+83) || !((a * b) <= 6.6e+114)) {
		tmp = (a * b) + (z * t);
	} else {
		tmp = (z * t) + (x * y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if ((a * b) <= -2.6e+83) or not ((a * b) <= 6.6e+114):
		tmp = (a * b) + (z * t)
	else:
		tmp = (z * t) + (x * y)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((Float64(a * b) <= -2.6e+83) || !(Float64(a * b) <= 6.6e+114))
		tmp = Float64(Float64(a * b) + Float64(z * t));
	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) <= -2.6e+83) || ~(((a * b) <= 6.6e+114)))
		tmp = (a * b) + (z * t);
	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], -2.6e+83], N[Not[LessEqual[N[(a * b), $MachinePrecision], 6.6e+114]], $MachinePrecision]], N[(N[(a * b), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision], N[(N[(z * t), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a b) < -2.6000000000000001e83 or 6.6000000000000001e114 < (*.f64 a b)

   1. Initial program 96.3%

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

   2. Taylor expanded in x around 0 90.3%

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

   if -2.6000000000000001e83 < (*.f64 a b) < 6.6000000000000001e114

   1. Initial program 100.0%

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

   2. Taylor expanded in a around 0 86.1%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -2.6 \cdot 10^{+83} \lor \neg \left(a \cdot b \leq 6.6 \cdot 10^{+114}\right):\\ \;\;\;\;a \cdot b + z \cdot t\\ \mathbf{else}:\\ \;\;\;\;z \cdot t + x \cdot y\\ \end{array} \]

Alternative 6: 53.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -1.45 \cdot 10^{+69}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq 3.9 \cdot 10^{+133}:\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* a b) -1.45e+69)
   (* a b)
   (if (<= (* a b) 3.9e+133) (* z t) (* a b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a * b) <= -1.45e+69) {
		tmp = a * b;
	} else if ((a * b) <= 3.9e+133) {
		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) <= (-1.45d+69)) then
        tmp = a * b
    else if ((a * b) <= 3.9d+133) 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) <= -1.45e+69) {
		tmp = a * b;
	} else if ((a * b) <= 3.9e+133) {
		tmp = z * t;
	} else {
		tmp = a * b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (a * b) <= -1.45e+69:
		tmp = a * b
	elif (a * b) <= 3.9e+133:
		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) <= -1.45e+69)
		tmp = Float64(a * b);
	elseif (Float64(a * b) <= 3.9e+133)
		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) <= -1.45e+69)
		tmp = a * b;
	elseif ((a * b) <= 3.9e+133)
		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], -1.45e+69], N[(a * b), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 3.9e+133], N[(z * t), $MachinePrecision], N[(a * b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a b) < -1.4499999999999999e69 or 3.90000000000000014e133 < (*.f64 a b)

   1. Initial program 96.2%

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

   2. Taylor expanded in a around inf 72.9%

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

   if -1.4499999999999999e69 < (*.f64 a b) < 3.90000000000000014e133

   1. Initial program 100.0%

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

   2. Taylor expanded in z around inf 45.3%

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

4. Final simplification 53.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -1.45 \cdot 10^{+69}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq 3.9 \cdot 10^{+133}:\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \]

Alternative 7: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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) + ((z * t) + (x * y))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.8%

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

2. Final simplification 98.8%

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

Alternative 8: 35.0% 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 98.8%

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

2. Taylor expanded in a around inf 32.5%

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

3. Final simplification 32.5%

   \[\leadsto a \cdot b \]

Reproduce

herbie shell --seed 2023221 
(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)))