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

Percentage Accurate: 98.0% → 98.7%

Time: 8.1s

Alternatives: 9

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: 98.0% 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: 98.7% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.4%

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

   1. +-commutative N/A

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

   2. associate-+l+ N/A

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

   3. fma-define N/A

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

   4. fma-lowering-fma.f64 N/A

      \[\leadsto \mathsf{fma.f64}\left(z, \color{blue}{t}, \left(x \cdot y + a \cdot b\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{fma.f64}\left(z, t, \mathsf{+.f64}\left(\left(x \cdot y\right), \left(a \cdot b\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{fma.f64}\left(z, t, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(a \cdot b\right)\right)\right) \]

   7. *-lowering-*.f64 99.2%

      \[\leadsto \mathsf{fma.f64}\left(z, t, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(a, b\right)\right)\right) \]

4. Applied egg-rr 99.2%

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

Alternative 2: 98.6% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.4%

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

   1. associate-+l+ N/A

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

   2. *-commutative N/A

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

   3. fma-define N/A

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

   4. fma-lowering-fma.f64 N/A

      \[\leadsto \mathsf{fma.f64}\left(y, \color{blue}{x}, \left(z \cdot t + a \cdot b\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{fma.f64}\left(y, x, \mathsf{+.f64}\left(\left(z \cdot t\right), \left(a \cdot b\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{fma.f64}\left(y, x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, t\right), \left(a \cdot b\right)\right)\right) \]

   7. *-lowering-*.f64 99.2%

      \[\leadsto \mathsf{fma.f64}\left(y, x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(z, t\right), \mathsf{*.f64}\left(a, b\right)\right)\right) \]

4. Applied egg-rr 99.2%

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

5. Final simplification 99.2%

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

Alternative 3: 48.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.15 \cdot 10^{-43}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;t \leq 3.7 \cdot 10^{-297}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{-83}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{+28}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;t \leq 1.5 \cdot 10^{+63}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t -1.15e-43)
   (* z t)
   (if (<= t 3.7e-297)
     (* a b)
     (if (<= t 2.1e-83)
       (* x y)
       (if (<= t 5.6e+28) (* a b) (if (<= t 1.5e+63) (* x y) (* z t)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= -1.15e-43) {
		tmp = z * t;
	} else if (t <= 3.7e-297) {
		tmp = a * b;
	} else if (t <= 2.1e-83) {
		tmp = x * y;
	} else if (t <= 5.6e+28) {
		tmp = a * b;
	} else if (t <= 1.5e+63) {
		tmp = x * y;
	} 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 (t <= (-1.15d-43)) then
        tmp = z * t
    else if (t <= 3.7d-297) then
        tmp = a * b
    else if (t <= 2.1d-83) then
        tmp = x * y
    else if (t <= 5.6d+28) then
        tmp = a * b
    else if (t <= 1.5d+63) then
        tmp = x * y
    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 (t <= -1.15e-43) {
		tmp = z * t;
	} else if (t <= 3.7e-297) {
		tmp = a * b;
	} else if (t <= 2.1e-83) {
		tmp = x * y;
	} else if (t <= 5.6e+28) {
		tmp = a * b;
	} else if (t <= 1.5e+63) {
		tmp = x * y;
	} else {
		tmp = z * t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if t <= -1.15e-43:
		tmp = z * t
	elif t <= 3.7e-297:
		tmp = a * b
	elif t <= 2.1e-83:
		tmp = x * y
	elif t <= 5.6e+28:
		tmp = a * b
	elif t <= 1.5e+63:
		tmp = x * y
	else:
		tmp = z * t
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= -1.15e-43)
		tmp = Float64(z * t);
	elseif (t <= 3.7e-297)
		tmp = Float64(a * b);
	elseif (t <= 2.1e-83)
		tmp = Float64(x * y);
	elseif (t <= 5.6e+28)
		tmp = Float64(a * b);
	elseif (t <= 1.5e+63)
		tmp = Float64(x * y);
	else
		tmp = Float64(z * t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (t <= -1.15e-43)
		tmp = z * t;
	elseif (t <= 3.7e-297)
		tmp = a * b;
	elseif (t <= 2.1e-83)
		tmp = x * y;
	elseif (t <= 5.6e+28)
		tmp = a * b;
	elseif (t <= 1.5e+63)
		tmp = x * y;
	else
		tmp = z * t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, -1.15e-43], N[(z * t), $MachinePrecision], If[LessEqual[t, 3.7e-297], N[(a * b), $MachinePrecision], If[LessEqual[t, 2.1e-83], N[(x * y), $MachinePrecision], If[LessEqual[t, 5.6e+28], N[(a * b), $MachinePrecision], If[LessEqual[t, 1.5e+63], N[(x * y), $MachinePrecision], N[(z * t), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.1499999999999999e-43 or 1.5e63 < t

   1. Initial program 98.2%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{t \cdot z} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 55.7%

         \[\leadsto \mathsf{*.f64}\left(t, \color{blue}{z}\right) \]

   5. Simplified 55.7%

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

   if -1.1499999999999999e-43 < t < 3.7e-297 or 2.0999999999999999e-83 < t < 5.6000000000000003e28

   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

      \[\leadsto \color{blue}{a \cdot b} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 54.3%

         \[\leadsto \mathsf{*.f64}\left(a, \color{blue}{b}\right) \]

   5. Simplified 54.3%

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

   if 3.7e-297 < t < 2.0999999999999999e-83 or 5.6000000000000003e28 < t < 1.5e63

   1. Initial program 98.1%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot y} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 60.5%

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{y}\right) \]

   5. Simplified 60.5%

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

4. Final simplification 56.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.15 \cdot 10^{-43}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;t \leq 3.7 \cdot 10^{-297}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{-83}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{+28}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;t \leq 1.5 \cdot 10^{+63}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -5 \cdot 10^{-63}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \mathbf{elif}\;a \cdot b \leq 2 \cdot 10^{+31}:\\ \;\;\;\;x \cdot y + z \cdot t\\ \mathbf{else}:\\ \;\;\;\;x \cdot y + a \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* a b) -5e-63)
   (+ (* a b) (* z t))
   (if (<= (* a b) 2e+31) (+ (* x y) (* z t)) (+ (* x y) (* a b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a * b) <= -5e-63) {
		tmp = (a * b) + (z * t);
	} else if ((a * b) <= 2e+31) {
		tmp = (x * y) + (z * t);
	} else {
		tmp = (x * y) + (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) <= (-5d-63)) then
        tmp = (a * b) + (z * t)
    else if ((a * b) <= 2d+31) then
        tmp = (x * y) + (z * t)
    else
        tmp = (x * y) + (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) <= -5e-63) {
		tmp = (a * b) + (z * t);
	} else if ((a * b) <= 2e+31) {
		tmp = (x * y) + (z * t);
	} else {
		tmp = (x * y) + (a * b);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (a * b) <= -5e-63:
		tmp = (a * b) + (z * t)
	elif (a * b) <= 2e+31:
		tmp = (x * y) + (z * t)
	else:
		tmp = (x * y) + (a * b)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(a * b) <= -5e-63)
		tmp = Float64(Float64(a * b) + Float64(z * t));
	elseif (Float64(a * b) <= 2e+31)
		tmp = Float64(Float64(x * y) + Float64(z * t));
	else
		tmp = Float64(Float64(x * y) + Float64(a * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((a * b) <= -5e-63)
		tmp = (a * b) + (z * t);
	elseif ((a * b) <= 2e+31)
		tmp = (x * y) + (z * t);
	else
		tmp = (x * y) + (a * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(a * b), $MachinePrecision], -5e-63], N[(N[(a * b), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 2e+31], N[(N[(x * y), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $MachinePrecision] + N[(a * b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 a b) < -5.0000000000000002e-63

   1. Initial program 97.4%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(t \cdot z\right)}, \mathsf{*.f64}\left(a, b\right)\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 81.4%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(t, z\right), \mathsf{*.f64}\left(\color{blue}{a}, b\right)\right) \]

   5. Simplified 81.4%

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

   if -5.0000000000000002e-63 < (*.f64 a b) < 1.9999999999999999e31

   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 0

      \[\leadsto \color{blue}{t \cdot z + x \cdot y} \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(t \cdot z\right), \color{blue}{\left(x \cdot y\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(t, z\right), \left(\color{blue}{x} \cdot y\right)\right) \]

      3. *-lowering-*.f64 92.9%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(t, z\right), \mathsf{*.f64}\left(x, \color{blue}{y}\right)\right) \]

   5. Simplified 92.9%

      \[\leadsto \color{blue}{t \cdot z + x \cdot y} \]

   if 1.9999999999999999e31 < (*.f64 a b)

   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

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(x \cdot y\right)}, \mathsf{*.f64}\left(a, b\right)\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 89.2%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\color{blue}{a}, b\right)\right) \]

   5. Simplified 89.2%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -5 \cdot 10^{-63}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \mathbf{elif}\;a \cdot b \leq 2 \cdot 10^{+31}:\\ \;\;\;\;x \cdot y + z \cdot t\\ \mathbf{else}:\\ \;\;\;\;x \cdot y + a \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot b + z \cdot t\\ \mathbf{if}\;a \cdot b \leq -6 \cdot 10^{-63}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \cdot b \leq 1.25 \cdot 10^{+36}:\\ \;\;\;\;x \cdot y + z \cdot t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (* a b) (* z t))))
   (if (<= (* a b) -6e-63)
     t_1
     (if (<= (* a b) 1.25e+36) (+ (* x y) (* z t)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a * b) + (z * t);
	double tmp;
	if ((a * b) <= -6e-63) {
		tmp = t_1;
	} else if ((a * b) <= 1.25e+36) {
		tmp = (x * y) + (z * t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (a * b) + (z * t)
    if ((a * b) <= (-6d-63)) then
        tmp = t_1
    else if ((a * b) <= 1.25d+36) then
        tmp = (x * y) + (z * t)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a * b) + (z * t);
	double tmp;
	if ((a * b) <= -6e-63) {
		tmp = t_1;
	} else if ((a * b) <= 1.25e+36) {
		tmp = (x * y) + (z * t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (a * b) + (z * t)
	tmp = 0
	if (a * b) <= -6e-63:
		tmp = t_1
	elif (a * b) <= 1.25e+36:
		tmp = (x * y) + (z * t)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a * b) + Float64(z * t))
	tmp = 0.0
	if (Float64(a * b) <= -6e-63)
		tmp = t_1;
	elseif (Float64(a * b) <= 1.25e+36)
		tmp = Float64(Float64(x * y) + Float64(z * t));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (a * b) + (z * t);
	tmp = 0.0;
	if ((a * b) <= -6e-63)
		tmp = t_1;
	elseif ((a * b) <= 1.25e+36)
		tmp = (x * y) + (z * t);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a * b), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(a * b), $MachinePrecision], -6e-63], t$95$1, If[LessEqual[N[(a * b), $MachinePrecision], 1.25e+36], N[(N[(x * y), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a b) < -5.99999999999999959e-63 or 1.24999999999999994e36 < (*.f64 a b)

   1. Initial program 97.5%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(t \cdot z\right)}, \mathsf{*.f64}\left(a, b\right)\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 83.2%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(t, z\right), \mathsf{*.f64}\left(\color{blue}{a}, b\right)\right) \]

   5. Simplified 83.2%

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

   if -5.99999999999999959e-63 < (*.f64 a b) < 1.24999999999999994e36

   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 0

      \[\leadsto \color{blue}{t \cdot z + x \cdot y} \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(t \cdot z\right), \color{blue}{\left(x \cdot y\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(t, z\right), \left(\color{blue}{x} \cdot y\right)\right) \]

      3. *-lowering-*.f64 92.9%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(t, z\right), \mathsf{*.f64}\left(x, \color{blue}{y}\right)\right) \]

   5. Simplified 92.9%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -6 \cdot 10^{-63}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \mathbf{elif}\;a \cdot b \leq 1.25 \cdot 10^{+36}:\\ \;\;\;\;x \cdot y + z \cdot t\\ \mathbf{else}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1.15 \cdot 10^{+157}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \cdot y \leq 3.7 \cdot 10^{+88}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* x y) -1.15e+157)
   (* x y)
   (if (<= (* x y) 3.7e+88) (+ (* 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) <= -1.15e+157) {
		tmp = x * y;
	} else if ((x * y) <= 3.7e+88) {
		tmp = (a * b) + (z * t);
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((x * y) <= (-1.15d+157)) then
        tmp = x * y
    else if ((x * y) <= 3.7d+88) then
        tmp = (a * b) + (z * t)
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((x * y) <= -1.15e+157) {
		tmp = x * y;
	} else if ((x * y) <= 3.7e+88) {
		tmp = (a * b) + (z * t);
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (x * y) <= -1.15e+157:
		tmp = x * y
	elif (x * y) <= 3.7e+88:
		tmp = (a * b) + (z * t)
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(x * y) <= -1.15e+157)
		tmp = Float64(x * y);
	elseif (Float64(x * y) <= 3.7e+88)
		tmp = Float64(Float64(a * b) + Float64(z * t));
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((x * y) <= -1.15e+157)
		tmp = x * y;
	elseif ((x * y) <= 3.7e+88)
		tmp = (a * b) + (z * t);
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(x * y), $MachinePrecision], -1.15e+157], N[(x * y), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 3.7e+88], N[(N[(a * b), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -1.15000000000000002e157 or 3.69999999999999994e88 < (*.f64 x y)

   1. Initial program 97.0%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot y} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 72.0%

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{y}\right) \]

   5. Simplified 72.0%

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

   if -1.15000000000000002e157 < (*.f64 x y) < 3.69999999999999994e88

   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

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(t \cdot z\right)}, \mathsf{*.f64}\left(a, b\right)\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 86.5%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(t, z\right), \mathsf{*.f64}\left(\color{blue}{a}, b\right)\right) \]

   5. Simplified 86.5%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1.15 \cdot 10^{+157}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \cdot y \leq 3.7 \cdot 10^{+88}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 47.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.15 \cdot 10^{-43}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;t \leq 2.9 \cdot 10^{+26}:\\ \;\;\;\;a \cdot b\\ \mathbf{else}:\\ \;\;\;\;z \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t -1.15e-43) (* z t) (if (<= t 2.9e+26) (* a b) (* z t))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= -1.15e-43) {
		tmp = z * t;
	} else if (t <= 2.9e+26) {
		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 (t <= (-1.15d-43)) then
        tmp = z * t
    else if (t <= 2.9d+26) 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 (t <= -1.15e-43) {
		tmp = z * t;
	} else if (t <= 2.9e+26) {
		tmp = a * b;
	} else {
		tmp = z * t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if t <= -1.15e-43:
		tmp = z * t
	elif t <= 2.9e+26:
		tmp = a * b
	else:
		tmp = z * t
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= -1.15e-43)
		tmp = Float64(z * t);
	elseif (t <= 2.9e+26)
		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 (t <= -1.15e-43)
		tmp = z * t;
	elseif (t <= 2.9e+26)
		tmp = a * b;
	else
		tmp = z * t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, -1.15e-43], N[(z * t), $MachinePrecision], If[LessEqual[t, 2.9e+26], N[(a * b), $MachinePrecision], N[(z * t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.1499999999999999e-43 or 2.9e26 < t

   1. Initial program 98.4%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{t \cdot z} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 52.6%

         \[\leadsto \mathsf{*.f64}\left(t, \color{blue}{z}\right) \]

   5. Simplified 52.6%

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

   if -1.1499999999999999e-43 < t < 2.9e26

   1. Initial program 98.4%

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

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a \cdot b} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 45.9%

         \[\leadsto \mathsf{*.f64}\left(a, \color{blue}{b}\right) \]

   5. Simplified 45.9%

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

4. Final simplification 49.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.15 \cdot 10^{-43}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;t \leq 2.9 \cdot 10^{+26}:\\ \;\;\;\;a \cdot b\\ \mathbf{else}:\\ \;\;\;\;z \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.4%

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

3. Final simplification 98.4%

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

Alternative 9: 34.7% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ a \cdot b \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.4%

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

3. Taylor expanded in a around inf

   \[\leadsto \color{blue}{a \cdot b} \]
4. Step-by-step derivation

   1. *-lowering-*.f64 35.3%

      \[\leadsto \mathsf{*.f64}\left(a, \color{blue}{b}\right) \]

5. Simplified 35.3%

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

Reproduce

herbie shell --seed 2024158 
(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)))