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

Percentage Accurate: 97.5% → 98.8%

Time: 9.2s

Alternatives: 10

Speedup: 1.0×

• 33.1% 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: 97.5% 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: 98.8% 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 98.8%

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

   1. +-commutative 98.8%

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

   2. fma-define 99.2%

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

   3. fma-define 99.6%

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

3. Simplified 99.6%

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

5. Final simplification 99.6%

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

Alternative 2: 97.9% 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 98.8%

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

   1. fma-define 99.2%

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

3. Simplified 99.2%

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

5. Final simplification 99.2%

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

Alternative 3: 54.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1.05 \cdot 10^{+62}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;z \cdot t \leq -1.25 \cdot 10^{-86}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;z \cdot t \leq -5.2 \cdot 10^{-247}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;z \cdot t \leq 4.7 \cdot 10^{-290}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;z \cdot t \leq 7.4 \cdot 10^{-140}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;z \cdot t \leq 1.85 \cdot 10^{+49}:\\ \;\;\;\;a \cdot b\\ \mathbf{else}:\\ \;\;\;\;z \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* z t) -1.05e+62)
   (* z t)
   (if (<= (* z t) -1.25e-86)
     (* a b)
     (if (<= (* z t) -5.2e-247)
       (* x y)
       (if (<= (* z t) 4.7e-290)
         (* a b)
         (if (<= (* z t) 7.4e-140)
           (* x y)
           (if (<= (* z t) 1.85e+49) (* a b) (* z t))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z * t) <= -1.05e+62) {
		tmp = z * t;
	} else if ((z * t) <= -1.25e-86) {
		tmp = a * b;
	} else if ((z * t) <= -5.2e-247) {
		tmp = x * y;
	} else if ((z * t) <= 4.7e-290) {
		tmp = a * b;
	} else if ((z * t) <= 7.4e-140) {
		tmp = x * y;
	} else if ((z * t) <= 1.85e+49) {
		tmp = a * b;
	} else {
		tmp = z * t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((z * t) <= (-1.05d+62)) then
        tmp = z * t
    else if ((z * t) <= (-1.25d-86)) then
        tmp = a * b
    else if ((z * t) <= (-5.2d-247)) then
        tmp = x * y
    else if ((z * t) <= 4.7d-290) then
        tmp = a * b
    else if ((z * t) <= 7.4d-140) then
        tmp = x * y
    else if ((z * t) <= 1.85d+49) then
        tmp = a * b
    else
        tmp = z * t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z * t) <= -1.05e+62) {
		tmp = z * t;
	} else if ((z * t) <= -1.25e-86) {
		tmp = a * b;
	} else if ((z * t) <= -5.2e-247) {
		tmp = x * y;
	} else if ((z * t) <= 4.7e-290) {
		tmp = a * b;
	} else if ((z * t) <= 7.4e-140) {
		tmp = x * y;
	} else if ((z * t) <= 1.85e+49) {
		tmp = a * b;
	} else {
		tmp = z * t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (z * t) <= -1.05e+62:
		tmp = z * t
	elif (z * t) <= -1.25e-86:
		tmp = a * b
	elif (z * t) <= -5.2e-247:
		tmp = x * y
	elif (z * t) <= 4.7e-290:
		tmp = a * b
	elif (z * t) <= 7.4e-140:
		tmp = x * y
	elif (z * t) <= 1.85e+49:
		tmp = a * b
	else:
		tmp = z * t
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(z * t) <= -1.05e+62)
		tmp = Float64(z * t);
	elseif (Float64(z * t) <= -1.25e-86)
		tmp = Float64(a * b);
	elseif (Float64(z * t) <= -5.2e-247)
		tmp = Float64(x * y);
	elseif (Float64(z * t) <= 4.7e-290)
		tmp = Float64(a * b);
	elseif (Float64(z * t) <= 7.4e-140)
		tmp = Float64(x * y);
	elseif (Float64(z * t) <= 1.85e+49)
		tmp = Float64(a * b);
	else
		tmp = Float64(z * t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((z * t) <= -1.05e+62)
		tmp = z * t;
	elseif ((z * t) <= -1.25e-86)
		tmp = a * b;
	elseif ((z * t) <= -5.2e-247)
		tmp = x * y;
	elseif ((z * t) <= 4.7e-290)
		tmp = a * b;
	elseif ((z * t) <= 7.4e-140)
		tmp = x * y;
	elseif ((z * t) <= 1.85e+49)
		tmp = a * b;
	else
		tmp = z * t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(z * t), $MachinePrecision], -1.05e+62], N[(z * t), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], -1.25e-86], N[(a * b), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], -5.2e-247], N[(x * y), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 4.7e-290], N[(a * b), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 7.4e-140], N[(x * y), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 1.85e+49], N[(a * b), $MachinePrecision], N[(z * t), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -1.05e62 or 1.85000000000000009e49 < (*.f64 z t)

   1. Initial program 97.9%

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

   3. Taylor expanded in z around inf 97.1%

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

   4. Taylor expanded in a around 0 85.6%

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

   5. Taylor expanded in t around inf 74.0%

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

   if -1.05e62 < (*.f64 z t) < -1.25e-86 or -5.2e-247 < (*.f64 z t) < 4.7000000000000001e-290 or 7.39999999999999955e-140 < (*.f64 z t) < 1.85000000000000009e49

   1. Initial program 100.0%

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

   3. Taylor expanded in a around inf 58.8%

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

   if -1.25e-86 < (*.f64 z t) < -5.2e-247 or 4.7000000000000001e-290 < (*.f64 z t) < 7.39999999999999955e-140

   1. Initial program 97.7%

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

   3. Taylor expanded in x around inf 94.0%

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

   4. Taylor expanded in b around inf 85.6%

      \[\leadsto \color{blue}{b \cdot \left(a + \frac{x \cdot y}{b}\right)} \]
   5. Step-by-step derivation

      1. associate-/l* 83.4%

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

   6. Simplified 83.4%

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

   7. Taylor expanded in b around 0 62.0%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1.05 \cdot 10^{+62}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;z \cdot t \leq -1.25 \cdot 10^{-86}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;z \cdot t \leq -5.2 \cdot 10^{-247}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;z \cdot t \leq 4.7 \cdot 10^{-290}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;z \cdot t \leq 7.4 \cdot 10^{-140}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;z \cdot t \leq 1.85 \cdot 10^{+49}:\\ \;\;\;\;a \cdot b\\ \mathbf{else}:\\ \;\;\;\;z \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -3.4 \cdot 10^{+220}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \cdot y \leq -2.9 \cdot 10^{+185} \lor \neg \left(x \cdot y \leq -1.35 \cdot 10^{+24}\right) \land x \cdot y \leq 4 \cdot 10^{+53}:\\ \;\;\;\;z \cdot t + a \cdot b\\ \mathbf{else}:\\ \;\;\;\;z \cdot t + x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* x y) -3.4e+220)
   (* x y)
   (if (or (<= (* x y) -2.9e+185)
           (and (not (<= (* x y) -1.35e+24)) (<= (* x y) 4e+53)))
     (+ (* z t) (* 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) <= -3.4e+220) {
		tmp = x * y;
	} else if (((x * y) <= -2.9e+185) || (!((x * y) <= -1.35e+24) && ((x * y) <= 4e+53))) {
		tmp = (z * t) + (a * b);
	} 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 ((x * y) <= (-3.4d+220)) then
        tmp = x * y
    else if (((x * y) <= (-2.9d+185)) .or. (.not. ((x * y) <= (-1.35d+24))) .and. ((x * y) <= 4d+53)) then
        tmp = (z * t) + (a * b)
    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 ((x * y) <= -3.4e+220) {
		tmp = x * y;
	} else if (((x * y) <= -2.9e+185) || (!((x * y) <= -1.35e+24) && ((x * y) <= 4e+53))) {
		tmp = (z * t) + (a * b);
	} else {
		tmp = (z * t) + (x * y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (x * y) <= -3.4e+220:
		tmp = x * y
	elif ((x * y) <= -2.9e+185) or (not ((x * y) <= -1.35e+24) and ((x * y) <= 4e+53)):
		tmp = (z * t) + (a * b)
	else:
		tmp = (z * t) + (x * y)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(x * y) <= -3.4e+220)
		tmp = Float64(x * y);
	elseif ((Float64(x * y) <= -2.9e+185) || (!(Float64(x * y) <= -1.35e+24) && (Float64(x * y) <= 4e+53)))
		tmp = Float64(Float64(z * t) + Float64(a * b));
	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 ((x * y) <= -3.4e+220)
		tmp = x * y;
	elseif (((x * y) <= -2.9e+185) || (~(((x * y) <= -1.35e+24)) && ((x * y) <= 4e+53)))
		tmp = (z * t) + (a * b);
	else
		tmp = (z * t) + (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(x * y), $MachinePrecision], -3.4e+220], N[(x * y), $MachinePrecision], If[Or[LessEqual[N[(x * y), $MachinePrecision], -2.9e+185], And[N[Not[LessEqual[N[(x * y), $MachinePrecision], -1.35e+24]], $MachinePrecision], LessEqual[N[(x * y), $MachinePrecision], 4e+53]]], N[(N[(z * t), $MachinePrecision] + N[(a * b), $MachinePrecision]), $MachinePrecision], N[(N[(z * t), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -3.4e220

   1. Initial program 94.9%

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

   3. Taylor expanded in x around inf 99.1%

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

   4. Taylor expanded in b around inf 86.3%

      \[\leadsto \color{blue}{b \cdot \left(a + \frac{x \cdot y}{b}\right)} \]
   5. Step-by-step derivation

      1. associate-/l* 86.3%

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

   6. Simplified 86.3%

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

   7. Taylor expanded in b around 0 94.1%

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

   if -3.4e220 < (*.f64 x y) < -2.89999999999999988e185 or -1.35e24 < (*.f64 x y) < 4e53

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 91.6%

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

   if -2.89999999999999988e185 < (*.f64 x y) < -1.35e24 or 4e53 < (*.f64 x y)

   1. Initial program 97.6%

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

   3. Taylor expanded in b around inf 74.5%

      \[\leadsto \color{blue}{b \cdot \left(a + \left(\frac{t \cdot z}{b} + \frac{x \cdot y}{b}\right)\right)} \]
   4. Step-by-step derivation

      1. associate-+r+ 74.5%

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

      2. associate-/l* 73.3%

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

      3. associate-/l* 69.9%

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

   5. Simplified 69.9%

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

   6. Taylor expanded in b around 0 82.2%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -3.4 \cdot 10^{+220}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \cdot y \leq -2.9 \cdot 10^{+185} \lor \neg \left(x \cdot y \leq -1.35 \cdot 10^{+24}\right) \land x \cdot y \leq 4 \cdot 10^{+53}:\\ \;\;\;\;z \cdot t + a \cdot b\\ \mathbf{else}:\\ \;\;\;\;z \cdot t + x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+38}:\\ \;\;\;\;z \cdot t + a \cdot b\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+96}:\\ \;\;\;\;a \cdot b + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(t + \frac{x \cdot y}{z}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* z t) -1e+38)
   (+ (* z t) (* a b))
   (if (<= (* z t) 2e+96) (+ (* a b) (* x y)) (* z (+ t (/ (* x y) z))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z * t) <= -1e+38) {
		tmp = (z * t) + (a * b);
	} else if ((z * t) <= 2e+96) {
		tmp = (a * b) + (x * y);
	} else {
		tmp = z * (t + ((x * y) / z));
	}
	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 ((z * t) <= (-1d+38)) then
        tmp = (z * t) + (a * b)
    else if ((z * t) <= 2d+96) then
        tmp = (a * b) + (x * y)
    else
        tmp = z * (t + ((x * y) / z))
    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 ((z * t) <= -1e+38) {
		tmp = (z * t) + (a * b);
	} else if ((z * t) <= 2e+96) {
		tmp = (a * b) + (x * y);
	} else {
		tmp = z * (t + ((x * y) / z));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (z * t) <= -1e+38:
		tmp = (z * t) + (a * b)
	elif (z * t) <= 2e+96:
		tmp = (a * b) + (x * y)
	else:
		tmp = z * (t + ((x * y) / z))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(z * t) <= -1e+38)
		tmp = Float64(Float64(z * t) + Float64(a * b));
	elseif (Float64(z * t) <= 2e+96)
		tmp = Float64(Float64(a * b) + Float64(x * y));
	else
		tmp = Float64(z * Float64(t + Float64(Float64(x * y) / z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((z * t) <= -1e+38)
		tmp = (z * t) + (a * b);
	elseif ((z * t) <= 2e+96)
		tmp = (a * b) + (x * y);
	else
		tmp = z * (t + ((x * y) / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(z * t), $MachinePrecision], -1e+38], N[(N[(z * t), $MachinePrecision] + N[(a * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 2e+96], N[(N[(a * b), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision], N[(z * N[(t + N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -9.99999999999999977e37

   1. Initial program 98.0%

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

   3. Taylor expanded in x around 0 86.5%

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

   if -9.99999999999999977e37 < (*.f64 z t) < 2.0000000000000001e96

   1. Initial program 99.3%

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

   3. Taylor expanded in x around inf 90.9%

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

   if 2.0000000000000001e96 < (*.f64 z t)

   1. Initial program 97.5%

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

   3. Taylor expanded in z around inf 99.9%

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

   4. Taylor expanded in a around 0 95.2%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+38}:\\ \;\;\;\;z \cdot t + a \cdot b\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+96}:\\ \;\;\;\;a \cdot b + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(t + \frac{x \cdot y}{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -4 \cdot 10^{+220} \lor \neg \left(x \cdot y \leq 7.6 \cdot 10^{+78}\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot t + a \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= (* x y) -4e+220) (not (<= (* x y) 7.6e+78)))
   (* x y)
   (+ (* z t) (* a b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((x * y) <= -4e+220) || !((x * y) <= 7.6e+78)) {
		tmp = x * y;
	} else {
		tmp = (z * t) + (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 (((x * y) <= (-4d+220)) .or. (.not. ((x * y) <= 7.6d+78))) then
        tmp = x * y
    else
        tmp = (z * t) + (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 (((x * y) <= -4e+220) || !((x * y) <= 7.6e+78)) {
		tmp = x * y;
	} else {
		tmp = (z * t) + (a * b);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if ((x * y) <= -4e+220) or not ((x * y) <= 7.6e+78):
		tmp = x * y
	else:
		tmp = (z * t) + (a * b)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((Float64(x * y) <= -4e+220) || !(Float64(x * y) <= 7.6e+78))
		tmp = Float64(x * y);
	else
		tmp = Float64(Float64(z * t) + Float64(a * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (((x * y) <= -4e+220) || ~(((x * y) <= 7.6e+78)))
		tmp = x * y;
	else
		tmp = (z * t) + (a * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[N[(x * y), $MachinePrecision], -4e+220], N[Not[LessEqual[N[(x * y), $MachinePrecision], 7.6e+78]], $MachinePrecision]], N[(x * y), $MachinePrecision], N[(N[(z * t), $MachinePrecision] + N[(a * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -4e220 or 7.5999999999999998e78 < (*.f64 x y)

   1. Initial program 95.0%

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

   3. Taylor expanded in x around inf 90.0%

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

   4. Taylor expanded in b around inf 78.3%

      \[\leadsto \color{blue}{b \cdot \left(a + \frac{x \cdot y}{b}\right)} \]
   5. Step-by-step derivation

      1. associate-/l* 76.7%

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

   6. Simplified 76.7%

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

   7. Taylor expanded in b around 0 79.6%

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

   if -4e220 < (*.f64 x y) < 7.5999999999999998e78

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 84.2%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -4 \cdot 10^{+220} \lor \neg \left(x \cdot y \leq 7.6 \cdot 10^{+78}\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot t + a \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+38}:\\ \;\;\;\;z \cdot t + a \cdot b\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+45}:\\ \;\;\;\;a \cdot b + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot t + x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* z t) -1e+38)
   (+ (* z t) (* a b))
   (if (<= (* z t) 5e+45) (+ (* 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 ((z * t) <= -1e+38) {
		tmp = (z * t) + (a * b);
	} else if ((z * t) <= 5e+45) {
		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 ((z * t) <= (-1d+38)) then
        tmp = (z * t) + (a * b)
    else if ((z * t) <= 5d+45) 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 ((z * t) <= -1e+38) {
		tmp = (z * t) + (a * b);
	} else if ((z * t) <= 5e+45) {
		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 (z * t) <= -1e+38:
		tmp = (z * t) + (a * b)
	elif (z * t) <= 5e+45:
		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(z * t) <= -1e+38)
		tmp = Float64(Float64(z * t) + Float64(a * b));
	elseif (Float64(z * t) <= 5e+45)
		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 ((z * t) <= -1e+38)
		tmp = (z * t) + (a * b);
	elseif ((z * t) <= 5e+45)
		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[LessEqual[N[(z * t), $MachinePrecision], -1e+38], N[(N[(z * t), $MachinePrecision] + N[(a * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 5e+45], 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 3 regimes

2. if (*.f64 z t) < -9.99999999999999977e37

   1. Initial program 98.0%

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

   3. Taylor expanded in x around 0 86.5%

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

   if -9.99999999999999977e37 < (*.f64 z t) < 5e45

   1. Initial program 99.3%

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

   3. Taylor expanded in x around inf 92.8%

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

   if 5e45 < (*.f64 z t)

   1. Initial program 98.1%

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

   3. Taylor expanded in b around inf 80.5%

      \[\leadsto \color{blue}{b \cdot \left(a + \left(\frac{t \cdot z}{b} + \frac{x \cdot y}{b}\right)\right)} \]
   4. Step-by-step derivation

      1. associate-+r+ 80.5%

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

      2. associate-/l* 75.1%

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

      3. associate-/l* 73.2%

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

   5. Simplified 73.2%

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

   6. Taylor expanded in b around 0 87.2%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+38}:\\ \;\;\;\;z \cdot t + a \cdot b\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+45}:\\ \;\;\;\;a \cdot b + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot t + x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 53.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -1.1 \cdot 10^{-22} \lor \neg \left(a \cdot b \leq 7 \cdot 10^{-35}\right):\\ \;\;\;\;a \cdot b\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= (* a b) -1.1e-22) (not (<= (* a b) 7e-35))) (* a b) (* x y)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((a * b) <= -1.1e-22) || !((a * b) <= 7e-35)) {
		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 (((a * b) <= (-1.1d-22)) .or. (.not. ((a * b) <= 7d-35))) 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 (((a * b) <= -1.1e-22) || !((a * b) <= 7e-35)) {
		tmp = a * b;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if ((a * b) <= -1.1e-22) or not ((a * b) <= 7e-35):
		tmp = a * b
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((Float64(a * b) <= -1.1e-22) || !(Float64(a * b) <= 7e-35))
		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 (((a * b) <= -1.1e-22) || ~(((a * b) <= 7e-35)))
		tmp = a * b;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[N[(a * b), $MachinePrecision], -1.1e-22], N[Not[LessEqual[N[(a * b), $MachinePrecision], 7e-35]], $MachinePrecision]], N[(a * b), $MachinePrecision], N[(x * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a b) < -1.1e-22 or 6.99999999999999992e-35 < (*.f64 a b)

   1. Initial program 99.2%

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

   3. Taylor expanded in a around inf 62.3%

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

   if -1.1e-22 < (*.f64 a b) < 6.99999999999999992e-35

   1. Initial program 98.3%

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

   3. Taylor expanded in x around inf 56.4%

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

   4. Taylor expanded in b around inf 43.9%

      \[\leadsto \color{blue}{b \cdot \left(a + \frac{x \cdot y}{b}\right)} \]
   5. Step-by-step derivation

      1. associate-/l* 40.6%

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

   6. Simplified 40.6%

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

   7. Taylor expanded in b around 0 48.3%

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

4. Final simplification 55.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -1.1 \cdot 10^{-22} \lor \neg \left(a \cdot b \leq 7 \cdot 10^{-35}\right):\\ \;\;\;\;a \cdot b\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.5% 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. Add Preprocessing

3. Final simplification 98.8%

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

Alternative 10: 34.9% 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. Add Preprocessing

3. Taylor expanded in a around inf 38.5%

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

4. Final simplification 38.5%

   \[\leadsto a \cdot b \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024082 
(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)))