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

Percentage Accurate: 97.7% → 98.8%

Time: 4.9s

Alternatives: 7

Speedup: 1.2×

• 32.6% 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, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 96.5%

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

   1. lift-+.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. associate-+l+ N/A

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

   5. lift-*.f64 N/A

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

   6. lower-fma.f64 N/A

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

   7. +-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 98.0

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 98.0

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

4. Applied rewrites 98.0%

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

Alternative 2: 52.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{+208}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \cdot y \leq -4 \cdot 10^{+68}:\\ \;\;\;\;t \cdot z\\ \mathbf{elif}\;x \cdot y \leq -2.5 \cdot 10^{-208}:\\ \;\;\;\;b \cdot a\\ \mathbf{elif}\;x \cdot y \leq 10^{-158}:\\ \;\;\;\;t \cdot z\\ \mathbf{elif}\;x \cdot y \leq 10^{+119}:\\ \;\;\;\;b \cdot a\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* x y) -1e+208)
   (* y x)
   (if (<= (* x y) -4e+68)
     (* t z)
     (if (<= (* x y) -2.5e-208)
       (* b a)
       (if (<= (* x y) 1e-158)
         (* t z)
         (if (<= (* x y) 1e+119) (* b a) (* y x)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((x * y) <= -1e+208) {
		tmp = y * x;
	} else if ((x * y) <= -4e+68) {
		tmp = t * z;
	} else if ((x * y) <= -2.5e-208) {
		tmp = b * a;
	} else if ((x * y) <= 1e-158) {
		tmp = t * z;
	} else if ((x * y) <= 1e+119) {
		tmp = b * a;
	} else {
		tmp = y * x;
	}
	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) <= (-1d+208)) then
        tmp = y * x
    else if ((x * y) <= (-4d+68)) then
        tmp = t * z
    else if ((x * y) <= (-2.5d-208)) then
        tmp = b * a
    else if ((x * y) <= 1d-158) then
        tmp = t * z
    else if ((x * y) <= 1d+119) then
        tmp = b * a
    else
        tmp = y * x
    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) <= -1e+208) {
		tmp = y * x;
	} else if ((x * y) <= -4e+68) {
		tmp = t * z;
	} else if ((x * y) <= -2.5e-208) {
		tmp = b * a;
	} else if ((x * y) <= 1e-158) {
		tmp = t * z;
	} else if ((x * y) <= 1e+119) {
		tmp = b * a;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (x * y) <= -1e+208:
		tmp = y * x
	elif (x * y) <= -4e+68:
		tmp = t * z
	elif (x * y) <= -2.5e-208:
		tmp = b * a
	elif (x * y) <= 1e-158:
		tmp = t * z
	elif (x * y) <= 1e+119:
		tmp = b * a
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(x * y) <= -1e+208)
		tmp = Float64(y * x);
	elseif (Float64(x * y) <= -4e+68)
		tmp = Float64(t * z);
	elseif (Float64(x * y) <= -2.5e-208)
		tmp = Float64(b * a);
	elseif (Float64(x * y) <= 1e-158)
		tmp = Float64(t * z);
	elseif (Float64(x * y) <= 1e+119)
		tmp = Float64(b * a);
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((x * y) <= -1e+208)
		tmp = y * x;
	elseif ((x * y) <= -4e+68)
		tmp = t * z;
	elseif ((x * y) <= -2.5e-208)
		tmp = b * a;
	elseif ((x * y) <= 1e-158)
		tmp = t * z;
	elseif ((x * y) <= 1e+119)
		tmp = b * a;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(x * y), $MachinePrecision], -1e+208], N[(y * x), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], -4e+68], N[(t * z), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], -2.5e-208], N[(b * a), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 1e-158], N[(t * z), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 1e+119], N[(b * a), $MachinePrecision], N[(y * x), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -9.9999999999999998e207 or 9.99999999999999944e118 < (*.f64 x y)

   1. Initial program 89.6%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 26.5

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

   5. Applied rewrites 26.5%

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

   6. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 88.4

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

   8. Applied rewrites 88.4%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 17.1%

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

   12. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 77.8

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

   14. Applied rewrites 77.8%

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

   if -9.9999999999999998e207 < (*.f64 x y) < -3.99999999999999981e68 or -2.49999999999999981e-208 < (*.f64 x y) < 1.00000000000000006e-158

   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

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 89.8

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

   5. Applied rewrites 89.8%

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

   6. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 69.3

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

   8. Applied rewrites 69.3%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 59.0%

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

   if -3.99999999999999981e68 < (*.f64 x y) < -2.49999999999999981e-208 or 1.00000000000000006e-158 < (*.f64 x y) < 9.99999999999999944e118

   1. Initial program 99.0%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 73.6

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

   5. Applied rewrites 73.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 58.5%

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

Alternative 3: 86.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+80} \lor \neg \left(x \cdot y \leq 10^{+102}\right):\\ \;\;\;\;\mathsf{fma}\left(t, z, y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, t \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= (* x y) -2e+80) (not (<= (* x y) 1e+102)))
   (fma t z (* y x))
   (fma b a (* t z))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((x * y) <= -2e+80) || !((x * y) <= 1e+102)) {
		tmp = fma(t, z, (y * x));
	} else {
		tmp = fma(b, a, (t * z));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((Float64(x * y) <= -2e+80) || !(Float64(x * y) <= 1e+102))
		tmp = fma(t, z, Float64(y * x));
	else
		tmp = fma(b, a, Float64(t * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[N[(x * y), $MachinePrecision], -2e+80], N[Not[LessEqual[N[(x * y), $MachinePrecision], 1e+102]], $MachinePrecision]], N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision], N[(b * a + N[(t * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -2e80 or 9.99999999999999977e101 < (*.f64 x y)

   1. Initial program 90.6%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 34.1

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

   5. Applied rewrites 34.1%

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

   6. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 89.7

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

   8. Applied rewrites 89.7%

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

   if -2e80 < (*.f64 x y) < 9.99999999999999977e101

   1. Initial program 99.4%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 89.5

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

   5. Applied rewrites 89.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b, a, t \cdot z\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+80} \lor \neg \left(x \cdot y \leq 10^{+102}\right):\\ \;\;\;\;\mathsf{fma}\left(t, z, y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, t \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{+208} \lor \neg \left(x \cdot y \leq 5 \cdot 10^{+127}\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, t \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= (* x y) -1e+208) (not (<= (* x y) 5e+127)))
   (* y x)
   (fma b a (* t z))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((x * y) <= -1e+208) || !((x * y) <= 5e+127)) {
		tmp = y * x;
	} else {
		tmp = fma(b, a, (t * z));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((Float64(x * y) <= -1e+208) || !(Float64(x * y) <= 5e+127))
		tmp = Float64(y * x);
	else
		tmp = fma(b, a, Float64(t * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[N[(x * y), $MachinePrecision], -1e+208], N[Not[LessEqual[N[(x * y), $MachinePrecision], 5e+127]], $MachinePrecision]], N[(y * x), $MachinePrecision], N[(b * a + N[(t * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -9.9999999999999998e207 or 5.0000000000000004e127 < (*.f64 x y)

   1. Initial program 89.1%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 24.6

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

   5. Applied rewrites 24.6%

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

   6. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 87.9

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

   8. Applied rewrites 87.9%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 14.7%

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

   12. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 79.8

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

   14. Applied rewrites 79.8%

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

   if -9.9999999999999998e207 < (*.f64 x y) < 5.0000000000000004e127

   1. Initial program 98.9%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 86.6

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

   5. Applied rewrites 86.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b, a, t \cdot z\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{+208} \lor \neg \left(x \cdot y \leq 5 \cdot 10^{+127}\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, t \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{+208}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+127}:\\ \;\;\;\;\mathsf{fma}\left(b, a, t \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, y \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* x y) -1e+208)
   (* y x)
   (if (<= (* x y) 5e+127) (fma b a (* t z)) (fma b a (* y x)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((x * y) <= -1e+208) {
		tmp = y * x;
	} else if ((x * y) <= 5e+127) {
		tmp = fma(b, a, (t * z));
	} else {
		tmp = fma(b, a, (y * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(x * y) <= -1e+208)
		tmp = Float64(y * x);
	elseif (Float64(x * y) <= 5e+127)
		tmp = fma(b, a, Float64(t * z));
	else
		tmp = fma(b, a, Float64(y * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(x * y), $MachinePrecision], -1e+208], N[(y * x), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 5e+127], N[(b * a + N[(t * z), $MachinePrecision]), $MachinePrecision], N[(b * a + N[(y * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -9.9999999999999998e207

   1. Initial program 84.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 18.8

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

   5. Applied rewrites 18.8%

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

   6. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 92.0

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

   8. Applied rewrites 92.0%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 10.1%

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

   12. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 90.7

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

   14. Applied rewrites 90.7%

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

   if -9.9999999999999998e207 < (*.f64 x y) < 5.0000000000000004e127

   1. Initial program 98.9%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 86.6

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

   5. Applied rewrites 86.6%

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

   if 5.0000000000000004e127 < (*.f64 x y)

   1. Initial program 92.3%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 84.9

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

   5. Applied rewrites 84.9%

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

Alternative 6: 54.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -50000000000000 \lor \neg \left(z \cdot t \leq 10^{+139}\right):\\ \;\;\;\;t \cdot z\\ \mathbf{else}:\\ \;\;\;\;b \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= (* z t) -50000000000000.0) (not (<= (* z t) 1e+139)))
   (* t z)
   (* b a)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((z * t) <= -50000000000000.0) || !((z * t) <= 1e+139)) {
		tmp = t * z;
	} else {
		tmp = b * a;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (((z * t) <= (-50000000000000.0d0)) .or. (.not. ((z * t) <= 1d+139))) then
        tmp = t * z
    else
        tmp = b * a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((z * t) <= -50000000000000.0) || !((z * t) <= 1e+139)) {
		tmp = t * z;
	} else {
		tmp = b * a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if ((z * t) <= -50000000000000.0) or not ((z * t) <= 1e+139):
		tmp = t * z
	else:
		tmp = b * a
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((Float64(z * t) <= -50000000000000.0) || !(Float64(z * t) <= 1e+139))
		tmp = Float64(t * z);
	else
		tmp = Float64(b * a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (((z * t) <= -50000000000000.0) || ~(((z * t) <= 1e+139)))
		tmp = t * z;
	else
		tmp = b * a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[N[(z * t), $MachinePrecision], -50000000000000.0], N[Not[LessEqual[N[(z * t), $MachinePrecision], 1e+139]], $MachinePrecision]], N[(t * z), $MachinePrecision], N[(b * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -5e13 or 1.00000000000000003e139 < (*.f64 z t)

   1. Initial program 93.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 82.7

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

   5. Applied rewrites 82.7%

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

   6. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 84.5

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

   8. Applied rewrites 84.5%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 69.1%

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

   if -5e13 < (*.f64 z t) < 1.00000000000000003e139

   1. Initial program 98.7%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 86.2

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

   5. Applied rewrites 86.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 51.2%

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

4. Final simplification 58.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -50000000000000 \lor \neg \left(z \cdot t \leq 10^{+139}\right):\\ \;\;\;\;t \cdot z\\ \mathbf{else}:\\ \;\;\;\;b \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 35.3% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

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 = b * a
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.5%

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

3. Taylor expanded in z around 0

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

   1. *-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 67.5

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

5. Applied rewrites 67.5%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 39.0%

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

Reproduce

herbie shell --seed 2024313 
(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)))