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

Percentage Accurate: 97.8% → 98.9%

Time: 6.4s

Alternatives: 8

Speedup: 0.4×

• 32.5% 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, 0.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

      1. +-commutative 100.0%

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

      2. fma-define 100.0%

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

      3. fma-define 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. Add Preprocessing

   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. Add Preprocessing

   3. Taylor expanded in x around inf 85.7%

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

   4. Taylor expanded in a around inf 100.0%

      \[\leadsto \color{blue}{a \cdot \left(b + \frac{x \cdot y}{a}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 53.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -5 \cdot 10^{+132}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq -3.2 \cdot 10^{-308}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;a \cdot b \leq 1.8 \cdot 10^{-209}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;a \cdot b \leq 2.4 \cdot 10^{+77}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;a \cdot b \leq 1.8 \cdot 10^{+96} \lor \neg \left(a \cdot b \leq 1.35 \cdot 10^{+153}\right):\\ \;\;\;\;a \cdot b\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* a b) -5e+132)
   (* a b)
   (if (<= (* a b) -3.2e-308)
     (* z t)
     (if (<= (* a b) 1.8e-209)
       (* x y)
       (if (<= (* a b) 2.4e+77)
         (* z t)
         (if (or (<= (* a b) 1.8e+96) (not (<= (* a b) 1.35e+153)))
           (* a b)
           (* x y)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a * b) <= -5e+132) {
		tmp = a * b;
	} else if ((a * b) <= -3.2e-308) {
		tmp = z * t;
	} else if ((a * b) <= 1.8e-209) {
		tmp = x * y;
	} else if ((a * b) <= 2.4e+77) {
		tmp = z * t;
	} else if (((a * b) <= 1.8e+96) || !((a * b) <= 1.35e+153)) {
		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) <= (-5d+132)) then
        tmp = a * b
    else if ((a * b) <= (-3.2d-308)) then
        tmp = z * t
    else if ((a * b) <= 1.8d-209) then
        tmp = x * y
    else if ((a * b) <= 2.4d+77) then
        tmp = z * t
    else if (((a * b) <= 1.8d+96) .or. (.not. ((a * b) <= 1.35d+153))) 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) <= -5e+132) {
		tmp = a * b;
	} else if ((a * b) <= -3.2e-308) {
		tmp = z * t;
	} else if ((a * b) <= 1.8e-209) {
		tmp = x * y;
	} else if ((a * b) <= 2.4e+77) {
		tmp = z * t;
	} else if (((a * b) <= 1.8e+96) || !((a * b) <= 1.35e+153)) {
		tmp = a * b;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (a * b) <= -5e+132:
		tmp = a * b
	elif (a * b) <= -3.2e-308:
		tmp = z * t
	elif (a * b) <= 1.8e-209:
		tmp = x * y
	elif (a * b) <= 2.4e+77:
		tmp = z * t
	elif ((a * b) <= 1.8e+96) or not ((a * b) <= 1.35e+153):
		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) <= -5e+132)
		tmp = Float64(a * b);
	elseif (Float64(a * b) <= -3.2e-308)
		tmp = Float64(z * t);
	elseif (Float64(a * b) <= 1.8e-209)
		tmp = Float64(x * y);
	elseif (Float64(a * b) <= 2.4e+77)
		tmp = Float64(z * t);
	elseif ((Float64(a * b) <= 1.8e+96) || !(Float64(a * b) <= 1.35e+153))
		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) <= -5e+132)
		tmp = a * b;
	elseif ((a * b) <= -3.2e-308)
		tmp = z * t;
	elseif ((a * b) <= 1.8e-209)
		tmp = x * y;
	elseif ((a * b) <= 2.4e+77)
		tmp = z * t;
	elseif (((a * b) <= 1.8e+96) || ~(((a * b) <= 1.35e+153)))
		tmp = a * b;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(a * b), $MachinePrecision], -5e+132], N[(a * b), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], -3.2e-308], N[(z * t), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 1.8e-209], N[(x * y), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 2.4e+77], N[(z * t), $MachinePrecision], If[Or[LessEqual[N[(a * b), $MachinePrecision], 1.8e+96], N[Not[LessEqual[N[(a * b), $MachinePrecision], 1.35e+153]], $MachinePrecision]], N[(a * b), $MachinePrecision], N[(x * y), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 a b) < -5.0000000000000001e132 or 2.3999999999999999e77 < (*.f64 a b) < 1.80000000000000007e96 or 1.35e153 < (*.f64 a b)

   1. Initial program 95.2%

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

   3. Taylor expanded in a around inf 77.0%

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

   if -5.0000000000000001e132 < (*.f64 a b) < -3.2000000000000001e-308 or 1.80000000000000008e-209 < (*.f64 a b) < 2.3999999999999999e77

   1. Initial program 98.4%

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

   3. Taylor expanded in y around inf 89.0%

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

      1. associate-+r+ 89.0%

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

      2. associate-/l* 82.6%

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

      3. associate-/l* 79.4%

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

   5. Simplified 79.4%

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

   6. Taylor expanded in t around inf 55.8%

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

   if -3.2000000000000001e-308 < (*.f64 a b) < 1.80000000000000008e-209 or 1.80000000000000007e96 < (*.f64 a b) < 1.35e153

   1. Initial program 98.0%

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

   3. Taylor expanded in y around inf 88.8%

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

      1. associate-+r+ 88.8%

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

      2. associate-/l* 88.8%

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

      3. associate-/l* 88.7%

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

   5. Simplified 88.7%

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

   6. Taylor expanded in x around inf 67.5%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -5 \cdot 10^{+132}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq -3.2 \cdot 10^{-308}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;a \cdot b \leq 1.8 \cdot 10^{-209}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;a \cdot b \leq 2.4 \cdot 10^{+77}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;a \cdot b \leq 1.8 \cdot 10^{+96} \lor \neg \left(a \cdot b \leq 1.35 \cdot 10^{+153}\right):\\ \;\;\;\;a \cdot b\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x \cdot y + z \cdot t\right) + a \cdot b\\ \mathbf{if}\;t\_1 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(b + \frac{x \cdot y}{a}\right)\\ \end{array} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   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. Add Preprocessing

   3. Taylor expanded in x around inf 85.7%

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

   4. Taylor expanded in a around inf 100.0%

      \[\leadsto \color{blue}{a \cdot \left(b + \frac{x \cdot y}{a}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 83.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+134}:\\ \;\;\;\;x \cdot y + a \cdot b\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+87}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \mathbf{else}:\\ \;\;\;\;x \cdot y + z \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* x y) -5e+134)
   (+ (* x y) (* a b))
   (if (<= (* x y) 5e+87) (+ (* a b) (* z t)) (+ (* x y) (* z t)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((x * y) <= -5e+134) {
		tmp = (x * y) + (a * b);
	} else if ((x * y) <= 5e+87) {
		tmp = (a * b) + (z * t);
	} else {
		tmp = (x * y) + (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) <= (-5d+134)) then
        tmp = (x * y) + (a * b)
    else if ((x * y) <= 5d+87) then
        tmp = (a * b) + (z * t)
    else
        tmp = (x * y) + (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) <= -5e+134) {
		tmp = (x * y) + (a * b);
	} else if ((x * y) <= 5e+87) {
		tmp = (a * b) + (z * t);
	} else {
		tmp = (x * y) + (z * t);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (x * y) <= -5e+134:
		tmp = (x * y) + (a * b)
	elif (x * y) <= 5e+87:
		tmp = (a * b) + (z * t)
	else:
		tmp = (x * y) + (z * t)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(x * y) <= -5e+134)
		tmp = Float64(Float64(x * y) + Float64(a * b));
	elseif (Float64(x * y) <= 5e+87)
		tmp = Float64(Float64(a * b) + Float64(z * t));
	else
		tmp = Float64(Float64(x * y) + Float64(z * t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((x * y) <= -5e+134)
		tmp = (x * y) + (a * b);
	elseif ((x * y) <= 5e+87)
		tmp = (a * b) + (z * t);
	else
		tmp = (x * y) + (z * t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(x * y), $MachinePrecision], -5e+134], N[(N[(x * y), $MachinePrecision] + N[(a * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 5e+87], N[(N[(a * b), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -4.99999999999999981e134

   1. Initial program 88.6%

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

   3. Taylor expanded in x around inf 91.6%

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

   if -4.99999999999999981e134 < (*.f64 x y) < 4.9999999999999998e87

   1. Initial program 98.8%

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

   3. Taylor expanded in x around 0 89.8%

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

   if 4.9999999999999998e87 < (*.f64 x y)

   1. Initial program 98.0%

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

   3. Taylor expanded in y around inf 98.0%

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

      1. associate-+r+ 98.0%

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

      2. associate-/l* 100.0%

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

      3. associate-/l* 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in a around 0 93.6%

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

   7. Taylor expanded in y around 0 93.6%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+134}:\\ \;\;\;\;x \cdot y + a \cdot b\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+87}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \mathbf{else}:\\ \;\;\;\;x \cdot y + z \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+134}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+87}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \mathbf{else}:\\ \;\;\;\;x \cdot y + z \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* x y) -5e+134)
   (* x y)
   (if (<= (* x y) 5e+87) (+ (* a b) (* z t)) (+ (* x y) (* z t)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((x * y) <= -5e+134) {
		tmp = x * y;
	} else if ((x * y) <= 5e+87) {
		tmp = (a * b) + (z * t);
	} else {
		tmp = (x * y) + (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) <= (-5d+134)) then
        tmp = x * y
    else if ((x * y) <= 5d+87) then
        tmp = (a * b) + (z * t)
    else
        tmp = (x * y) + (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) <= -5e+134) {
		tmp = x * y;
	} else if ((x * y) <= 5e+87) {
		tmp = (a * b) + (z * t);
	} else {
		tmp = (x * y) + (z * t);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (x * y) <= -5e+134:
		tmp = x * y
	elif (x * y) <= 5e+87:
		tmp = (a * b) + (z * t)
	else:
		tmp = (x * y) + (z * t)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(x * y) <= -5e+134)
		tmp = Float64(x * y);
	elseif (Float64(x * y) <= 5e+87)
		tmp = Float64(Float64(a * b) + Float64(z * t));
	else
		tmp = Float64(Float64(x * y) + Float64(z * t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((x * y) <= -5e+134)
		tmp = x * y;
	elseif ((x * y) <= 5e+87)
		tmp = (a * b) + (z * t);
	else
		tmp = (x * y) + (z * t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(x * y), $MachinePrecision], -5e+134], N[(x * y), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 5e+87], N[(N[(a * b), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -4.99999999999999981e134

   1. Initial program 88.6%

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

   3. Taylor expanded in y around inf 88.5%

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

      1. associate-+r+ 88.5%

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

      2. associate-/l* 88.5%

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

      3. associate-/l* 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf 81.0%

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

   if -4.99999999999999981e134 < (*.f64 x y) < 4.9999999999999998e87

   1. Initial program 98.8%

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

   3. Taylor expanded in x around 0 89.8%

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

   if 4.9999999999999998e87 < (*.f64 x y)

   1. Initial program 98.0%

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

   3. Taylor expanded in y around inf 98.0%

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

      1. associate-+r+ 98.0%

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

      2. associate-/l* 100.0%

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

      3. associate-/l* 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in a around 0 93.6%

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

   7. Taylor expanded in y around 0 93.6%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+134}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+87}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \mathbf{else}:\\ \;\;\;\;x \cdot y + z \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 70.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{+132} \lor \neg \left(x \leq 9 \cdot 10^{+15}\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= x -2.5e+132) (not (<= x 9e+15))) (* 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 <= -2.5e+132) || !(x <= 9e+15)) {
		tmp = 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 <= (-2.5d+132)) .or. (.not. (x <= 9d+15))) then
        tmp = 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 <= -2.5e+132) || !(x <= 9e+15)) {
		tmp = x * y;
	} else {
		tmp = (a * b) + (z * t);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (x <= -2.5e+132) or not (x <= 9e+15):
		tmp = x * y
	else:
		tmp = (a * b) + (z * t)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((x <= -2.5e+132) || !(x <= 9e+15))
		tmp = 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 <= -2.5e+132) || ~((x <= 9e+15)))
		tmp = x * y;
	else
		tmp = (a * b) + (z * t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[x, -2.5e+132], N[Not[LessEqual[x, 9e+15]], $MachinePrecision]], N[(x * y), $MachinePrecision], N[(N[(a * b), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.5000000000000001e132 or 9e15 < x

   1. Initial program 94.2%

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

   3. Taylor expanded in y around inf 82.2%

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

      1. associate-+r+ 82.2%

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

      2. associate-/l* 81.3%

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

      3. associate-/l* 84.2%

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

   5. Simplified 84.2%

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

   6. Taylor expanded in x around inf 60.4%

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

   if -2.5000000000000001e132 < x < 9e15

   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 0 85.7%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{+132} \lor \neg \left(x \leq 9 \cdot 10^{+15}\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 54.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -4.6 \cdot 10^{+132} \lor \neg \left(a \cdot b \leq 3.4 \cdot 10^{+77}\right):\\ \;\;\;\;a \cdot b\\ \mathbf{else}:\\ \;\;\;\;z \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= (* a b) -4.6e+132) (not (<= (* a b) 3.4e+77))) (* a b) (* z t)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((a * b) <= -4.6e+132) || !((a * b) <= 3.4e+77)) {
		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 (((a * b) <= (-4.6d+132)) .or. (.not. ((a * b) <= 3.4d+77))) 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 (((a * b) <= -4.6e+132) || !((a * b) <= 3.4e+77)) {
		tmp = a * b;
	} else {
		tmp = z * t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if ((a * b) <= -4.6e+132) or not ((a * b) <= 3.4e+77):
		tmp = a * b
	else:
		tmp = z * t
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((Float64(a * b) <= -4.6e+132) || !(Float64(a * b) <= 3.4e+77))
		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 (((a * b) <= -4.6e+132) || ~(((a * b) <= 3.4e+77)))
		tmp = a * b;
	else
		tmp = z * t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[N[(a * b), $MachinePrecision], -4.6e+132], N[Not[LessEqual[N[(a * b), $MachinePrecision], 3.4e+77]], $MachinePrecision]], N[(a * b), $MachinePrecision], N[(z * t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a b) < -4.6000000000000003e132 or 3.39999999999999997e77 < (*.f64 a b)

   1. Initial program 95.6%

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

   3. Taylor expanded in a around inf 70.8%

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

   if -4.6000000000000003e132 < (*.f64 a b) < 3.39999999999999997e77

   1. Initial program 98.2%

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

   3. Taylor expanded in y around inf 88.9%

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

      1. associate-+r+ 88.9%

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

      2. associate-/l* 84.1%

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

      3. associate-/l* 81.7%

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

   5. Simplified 81.7%

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

   6. Taylor expanded in t around inf 50.1%

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

4. Final simplification 57.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -4.6 \cdot 10^{+132} \lor \neg \left(a \cdot b \leq 3.4 \cdot 10^{+77}\right):\\ \;\;\;\;a \cdot b\\ \mathbf{else}:\\ \;\;\;\;z \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 35.1% 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. Add Preprocessing

3. Taylor expanded in a around inf 34.1%

   \[\leadsto \color{blue}{a \cdot b} \]
4. Add Preprocessing

Reproduce

herbie shell --seed 2024091 
(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)))