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

Percentage Accurate: 97.7% → 98.8%

Time: 7.5s

Alternatives: 9

Speedup: 1.0×

• 33.4% 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: 97.7% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 97.6%

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

   1. associate-+l+ 97.6%

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

   2. fma-define 98.8%

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

   3. fma-define 99.6%

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

3. Simplified 99.6%

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

5. Final simplification 99.6%

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

Alternative 2: 98.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 97.6%

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

   1. +-commutative 97.6%

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

   2. *-commutative 97.6%

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

   3. fma-define 98.0%

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

   4. fma-define 99.2%

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

4. Applied egg-rr 99.2%

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

5. Final simplification 99.2%

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

Alternative 3: 98.1% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 97.6%

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

   1. fma-define 98.8%

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

3. Simplified 98.8%

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

5. Final simplification 98.8%

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

Alternative 4: 52.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -3600000000000:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq -2.6 \cdot 10^{-56}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;a \cdot b \leq -7.8 \cdot 10^{-60}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq 0:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;a \cdot b \leq 1.8 \cdot 10^{-110}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;a \cdot b \leq 2.1 \cdot 10^{-52}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;a \cdot b \leq 2.3 \cdot 10^{+177}:\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* a b) -3600000000000.0)
   (* a b)
   (if (<= (* a b) -2.6e-56)
     (* z t)
     (if (<= (* a b) -7.8e-60)
       (* a b)
       (if (<= (* a b) 0.0)
         (* x y)
         (if (<= (* a b) 1.8e-110)
           (* z t)
           (if (<= (* a b) 2.1e-52)
             (* x y)
             (if (<= (* a b) 2.3e+177) (* z t) (* a b)))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a * b) <= -3600000000000.0) {
		tmp = a * b;
	} else if ((a * b) <= -2.6e-56) {
		tmp = z * t;
	} else if ((a * b) <= -7.8e-60) {
		tmp = a * b;
	} else if ((a * b) <= 0.0) {
		tmp = x * y;
	} else if ((a * b) <= 1.8e-110) {
		tmp = z * t;
	} else if ((a * b) <= 2.1e-52) {
		tmp = x * y;
	} else if ((a * b) <= 2.3e+177) {
		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) <= (-3600000000000.0d0)) then
        tmp = a * b
    else if ((a * b) <= (-2.6d-56)) then
        tmp = z * t
    else if ((a * b) <= (-7.8d-60)) then
        tmp = a * b
    else if ((a * b) <= 0.0d0) then
        tmp = x * y
    else if ((a * b) <= 1.8d-110) then
        tmp = z * t
    else if ((a * b) <= 2.1d-52) then
        tmp = x * y
    else if ((a * b) <= 2.3d+177) 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) <= -3600000000000.0) {
		tmp = a * b;
	} else if ((a * b) <= -2.6e-56) {
		tmp = z * t;
	} else if ((a * b) <= -7.8e-60) {
		tmp = a * b;
	} else if ((a * b) <= 0.0) {
		tmp = x * y;
	} else if ((a * b) <= 1.8e-110) {
		tmp = z * t;
	} else if ((a * b) <= 2.1e-52) {
		tmp = x * y;
	} else if ((a * b) <= 2.3e+177) {
		tmp = z * t;
	} else {
		tmp = a * b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (a * b) <= -3600000000000.0:
		tmp = a * b
	elif (a * b) <= -2.6e-56:
		tmp = z * t
	elif (a * b) <= -7.8e-60:
		tmp = a * b
	elif (a * b) <= 0.0:
		tmp = x * y
	elif (a * b) <= 1.8e-110:
		tmp = z * t
	elif (a * b) <= 2.1e-52:
		tmp = x * y
	elif (a * b) <= 2.3e+177:
		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) <= -3600000000000.0)
		tmp = Float64(a * b);
	elseif (Float64(a * b) <= -2.6e-56)
		tmp = Float64(z * t);
	elseif (Float64(a * b) <= -7.8e-60)
		tmp = Float64(a * b);
	elseif (Float64(a * b) <= 0.0)
		tmp = Float64(x * y);
	elseif (Float64(a * b) <= 1.8e-110)
		tmp = Float64(z * t);
	elseif (Float64(a * b) <= 2.1e-52)
		tmp = Float64(x * y);
	elseif (Float64(a * b) <= 2.3e+177)
		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) <= -3600000000000.0)
		tmp = a * b;
	elseif ((a * b) <= -2.6e-56)
		tmp = z * t;
	elseif ((a * b) <= -7.8e-60)
		tmp = a * b;
	elseif ((a * b) <= 0.0)
		tmp = x * y;
	elseif ((a * b) <= 1.8e-110)
		tmp = z * t;
	elseif ((a * b) <= 2.1e-52)
		tmp = x * y;
	elseif ((a * b) <= 2.3e+177)
		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], -3600000000000.0], N[(a * b), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], -2.6e-56], N[(z * t), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], -7.8e-60], N[(a * b), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 0.0], N[(x * y), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 1.8e-110], N[(z * t), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 2.1e-52], N[(x * y), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 2.3e+177], N[(z * t), $MachinePrecision], N[(a * b), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 a b) < -3.6e12 or -2.59999999999999997e-56 < (*.f64 a b) < -7.8000000000000004e-60 or 2.2999999999999999e177 < (*.f64 a b)

   1. Initial program 96.0%

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

   3. Taylor expanded in a around inf 71.2%

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

   if -3.6e12 < (*.f64 a b) < -2.59999999999999997e-56 or 0.0 < (*.f64 a b) < 1.79999999999999997e-110 or 2.0999999999999999e-52 < (*.f64 a b) < 2.2999999999999999e177

   1. Initial program 97.5%

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

      1. +-commutative 97.5%

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

      2. *-commutative 97.5%

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

      3. fma-define 97.5%

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

      4. fma-define 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in z around inf 61.0%

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

   if -7.8000000000000004e-60 < (*.f64 a b) < 0.0 or 1.79999999999999997e-110 < (*.f64 a b) < 2.0999999999999999e-52

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. *-commutative 100.0%

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

      3. fma-define 100.0%

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

      4. fma-define 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf 63.4%

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

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -3600000000000:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq -2.6 \cdot 10^{-56}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;a \cdot b \leq -7.8 \cdot 10^{-60}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq 0:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;a \cdot b \leq 1.8 \cdot 10^{-110}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;a \cdot b \leq 2.1 \cdot 10^{-52}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;a \cdot b \leq 2.3 \cdot 10^{+177}:\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1.06 \cdot 10^{+114} \lor \neg \left(x \cdot y \leq 6.8 \cdot 10^{+209}\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= (* x y) -1.06e+114) (not (<= (* x y) 6.8e+209)))
   (* x y)
   (+ (* a b) (* z t))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((x * y) <= -1.06e+114) || !((x * y) <= 6.8e+209)) {
		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 * y) <= (-1.06d+114)) .or. (.not. ((x * y) <= 6.8d+209))) 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 * y) <= -1.06e+114) || !((x * y) <= 6.8e+209)) {
		tmp = x * y;
	} else {
		tmp = (a * b) + (z * t);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if ((x * y) <= -1.06e+114) or not ((x * y) <= 6.8e+209):
		tmp = x * y
	else:
		tmp = (a * b) + (z * t)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((Float64(x * y) <= -1.06e+114) || !(Float64(x * y) <= 6.8e+209))
		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 * y) <= -1.06e+114) || ~(((x * y) <= 6.8e+209)))
		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[N[(x * y), $MachinePrecision], -1.06e+114], N[Not[LessEqual[N[(x * y), $MachinePrecision], 6.8e+209]], $MachinePrecision]], N[(x * y), $MachinePrecision], N[(N[(a * b), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -1.05999999999999993e114 or 6.7999999999999993e209 < (*.f64 x y)

   1. Initial program 91.3%

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

      1. +-commutative 91.3%

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

      2. *-commutative 91.3%

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

      3. fma-define 92.8%

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

      4. fma-define 97.1%

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

   4. Applied egg-rr 97.1%

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

   5. Taylor expanded in x around inf 81.6%

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

   if -1.05999999999999993e114 < (*.f64 x y) < 6.7999999999999993e209

   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 83.7%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1.06 \cdot 10^{+114} \lor \neg \left(x \cdot y \leq 6.8 \cdot 10^{+209}\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -1.95 \cdot 10^{+42} \lor \neg \left(a \cdot b \leq 1.2 \cdot 10^{+31}\right):\\ \;\;\;\;a \cdot b + z \cdot t\\ \mathbf{else}:\\ \;\;\;\;x \cdot y + z \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= (* a b) -1.95e+42) (not (<= (* a b) 1.2e+31)))
   (+ (* 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 (((a * b) <= -1.95e+42) || !((a * b) <= 1.2e+31)) {
		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 (((a * b) <= (-1.95d+42)) .or. (.not. ((a * b) <= 1.2d+31))) 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 (((a * b) <= -1.95e+42) || !((a * b) <= 1.2e+31)) {
		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 ((a * b) <= -1.95e+42) or not ((a * b) <= 1.2e+31):
		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(a * b) <= -1.95e+42) || !(Float64(a * b) <= 1.2e+31))
		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 (((a * b) <= -1.95e+42) || ~(((a * b) <= 1.2e+31)))
		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[Or[LessEqual[N[(a * b), $MachinePrecision], -1.95e+42], N[Not[LessEqual[N[(a * b), $MachinePrecision], 1.2e+31]], $MachinePrecision]], 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 2 regimes

2. if (*.f64 a b) < -1.94999999999999985e42 or 1.19999999999999991e31 < (*.f64 a b)

   1. Initial program 96.5%

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

   3. Taylor expanded in x around 0 85.3%

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

   if -1.94999999999999985e42 < (*.f64 a b) < 1.19999999999999991e31

   1. Initial program 98.6%

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

      1. +-commutative 98.6%

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

      2. *-commutative 98.6%

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

      3. fma-define 98.6%

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

      4. fma-define 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in b around 0 88.2%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -1.95 \cdot 10^{+42} \lor \neg \left(a \cdot b \leq 1.2 \cdot 10^{+31}\right):\\ \;\;\;\;a \cdot b + z \cdot t\\ \mathbf{else}:\\ \;\;\;\;x \cdot y + z \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 53.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -8.5 \cdot 10^{+15} \lor \neg \left(a \cdot b \leq 2.3 \cdot 10^{+177}\right):\\ \;\;\;\;a \cdot b\\ \mathbf{else}:\\ \;\;\;\;z \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= (* a b) -8.5e+15) (not (<= (* a b) 2.3e+177))) (* a b) (* z t)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((a * b) <= -8.5e+15) || !((a * b) <= 2.3e+177)) {
		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) <= (-8.5d+15)) .or. (.not. ((a * b) <= 2.3d+177))) 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) <= -8.5e+15) || !((a * b) <= 2.3e+177)) {
		tmp = a * b;
	} else {
		tmp = z * t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if ((a * b) <= -8.5e+15) or not ((a * b) <= 2.3e+177):
		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) <= -8.5e+15) || !(Float64(a * b) <= 2.3e+177))
		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) <= -8.5e+15) || ~(((a * b) <= 2.3e+177)))
		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], -8.5e+15], N[Not[LessEqual[N[(a * b), $MachinePrecision], 2.3e+177]], $MachinePrecision]], N[(a * b), $MachinePrecision], N[(z * t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a b) < -8.5e15 or 2.2999999999999999e177 < (*.f64 a b)

   1. Initial program 95.9%

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

   3. Taylor expanded in a around inf 71.0%

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

   if -8.5e15 < (*.f64 a b) < 2.2999999999999999e177

   1. Initial program 98.7%

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

      1. +-commutative 98.7%

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

      2. *-commutative 98.7%

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

      3. fma-define 98.7%

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

      4. fma-define 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in z around inf 48.1%

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

4. Final simplification 56.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -8.5 \cdot 10^{+15} \lor \neg \left(a \cdot b \leq 2.3 \cdot 10^{+177}\right):\\ \;\;\;\;a \cdot b\\ \mathbf{else}:\\ \;\;\;\;z \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (a * b) + ((x * y) + (z * t))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.6%

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

3. Final simplification 97.6%

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

Alternative 9: 35.3% 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 97.6%

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

3. Taylor expanded in a around inf 34.6%

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

4. Final simplification 34.6%

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

Reproduce

herbie shell --seed 2024046 
(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)))