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

Percentage Accurate: 97.7% → 99.0%

Time: 3.6s

Alternatives: 8

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: 99.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 96.9%

   \[\left(x \cdot y + z \cdot t\right) + a \cdot b \]
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. associate-+l+ N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. +-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 98.0

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 98.0

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

4. Applied rewrites 98.0%

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

5. Final simplification 98.0%

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

Alternative 2: 54.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+78}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \cdot y \leq 10^{-188}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;x \cdot y \leq 4 \cdot 10^{+44}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+75}:\\ \;\;\;\;a \cdot b\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* x y) -5e+78)
   (* x y)
   (if (<= (* x y) 1e-188)
     (* a b)
     (if (<= (* x y) 4e+44) (* z t) (if (<= (* x y) 5e+75) (* a b) (* x y))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((x * y) <= -5e+78) {
		tmp = x * y;
	} else if ((x * y) <= 1e-188) {
		tmp = a * b;
	} else if ((x * y) <= 4e+44) {
		tmp = z * t;
	} else if ((x * y) <= 5e+75) {
		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 ((x * y) <= (-5d+78)) then
        tmp = x * y
    else if ((x * y) <= 1d-188) then
        tmp = a * b
    else if ((x * y) <= 4d+44) then
        tmp = z * t
    else if ((x * y) <= 5d+75) 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 ((x * y) <= -5e+78) {
		tmp = x * y;
	} else if ((x * y) <= 1e-188) {
		tmp = a * b;
	} else if ((x * y) <= 4e+44) {
		tmp = z * t;
	} else if ((x * y) <= 5e+75) {
		tmp = a * b;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (x * y) <= -5e+78:
		tmp = x * y
	elif (x * y) <= 1e-188:
		tmp = a * b
	elif (x * y) <= 4e+44:
		tmp = z * t
	elif (x * y) <= 5e+75:
		tmp = a * b
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(x * y) <= -5e+78)
		tmp = Float64(x * y);
	elseif (Float64(x * y) <= 1e-188)
		tmp = Float64(a * b);
	elseif (Float64(x * y) <= 4e+44)
		tmp = Float64(z * t);
	elseif (Float64(x * y) <= 5e+75)
		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 ((x * y) <= -5e+78)
		tmp = x * y;
	elseif ((x * y) <= 1e-188)
		tmp = a * b;
	elseif ((x * y) <= 4e+44)
		tmp = z * t;
	elseif ((x * y) <= 5e+75)
		tmp = a * b;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(x * y), $MachinePrecision], -5e+78], N[(x * y), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 1e-188], N[(a * b), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 4e+44], N[(z * t), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 5e+75], N[(a * b), $MachinePrecision], N[(x * y), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -4.99999999999999984e78 or 5.0000000000000002e75 < (*.f64 x y)

   1. Initial program 94.1%

      \[\left(x \cdot y + z \cdot t\right) + a \cdot b \]
   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 29.2

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

   5. Applied rewrites 29.2%

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

   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 88.0

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

   8. Applied rewrites 88.0%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 17.7%

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

   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 73.7

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

   14. Applied rewrites 73.7%

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

   if -4.99999999999999984e78 < (*.f64 x y) < 9.9999999999999995e-189 or 4.0000000000000004e44 < (*.f64 x y) < 5.0000000000000002e75

   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 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 63.1

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

   5. Applied rewrites 63.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 53.9%

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

   if 9.9999999999999995e-189 < (*.f64 x y) < 4.0000000000000004e44

   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

      \[\leadsto \color{blue}{a \cdot b + t \cdot z} \]
   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 85.6

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

   5. Applied rewrites 85.6%

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

   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 73.8

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

   8. Applied rewrites 73.8%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 59.2%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+78}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \cdot y \leq 10^{-188}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;x \cdot y \leq 4 \cdot 10^{+44}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+75}:\\ \;\;\;\;a \cdot b\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{-35}:\\ \;\;\;\;\mathsf{fma}\left(y, x, a \cdot b\right)\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+75}:\\ \;\;\;\;\mathsf{fma}\left(b, a, z \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, z, x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* x y) -2e-35)
   (fma y x (* a b))
   (if (<= (* x y) 5e+75) (fma b a (* z t)) (fma t z (* x y)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((x * y) <= -2e-35) {
		tmp = fma(y, x, (a * b));
	} else if ((x * y) <= 5e+75) {
		tmp = fma(b, a, (z * t));
	} else {
		tmp = fma(t, z, (x * y));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(x * y) <= -2e-35)
		tmp = fma(y, x, Float64(a * b));
	elseif (Float64(x * y) <= 5e+75)
		tmp = fma(b, a, Float64(z * t));
	else
		tmp = fma(t, z, Float64(x * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(x * y), $MachinePrecision], -2e-35], N[(y * x + N[(a * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 5e+75], N[(b * a + N[(z * t), $MachinePrecision]), $MachinePrecision], N[(t * z + N[(x * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -2.00000000000000002e-35

   1. Initial program 93.6%

      \[\left(x \cdot y + z \cdot t\right) + a \cdot b \]
   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 83.9

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

   5. Applied rewrites 83.9%

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

   7. Applied rewrites 86.5%

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

   if -2.00000000000000002e-35 < (*.f64 x y) < 5.0000000000000002e75

   1. Initial program 99.2%

      \[\left(x \cdot y + z \cdot t\right) + a \cdot b \]
   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 92.2

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

   5. Applied rewrites 92.2%

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

   if 5.0000000000000002e75 < (*.f64 x y)

   1. Initial program 95.5%

      \[\left(x \cdot y + z \cdot t\right) + a \cdot b \]
   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 31.5

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

   5. Applied rewrites 31.5%

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

   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 87.6

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

   8. Applied rewrites 87.6%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{-35}:\\ \;\;\;\;\mathsf{fma}\left(y, x, a \cdot b\right)\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+75}:\\ \;\;\;\;\mathsf{fma}\left(b, a, z \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, z, x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{-35}:\\ \;\;\;\;\mathsf{fma}\left(b, a, x \cdot y\right)\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+75}:\\ \;\;\;\;\mathsf{fma}\left(b, a, z \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, z, x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* x y) -2e-35)
   (fma b a (* x y))
   (if (<= (* x y) 5e+75) (fma b a (* z t)) (fma t z (* x y)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((x * y) <= -2e-35) {
		tmp = fma(b, a, (x * y));
	} else if ((x * y) <= 5e+75) {
		tmp = fma(b, a, (z * t));
	} else {
		tmp = fma(t, z, (x * y));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(x * y) <= -2e-35)
		tmp = fma(b, a, Float64(x * y));
	elseif (Float64(x * y) <= 5e+75)
		tmp = fma(b, a, Float64(z * t));
	else
		tmp = fma(t, z, Float64(x * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(x * y), $MachinePrecision], -2e-35], N[(b * a + N[(x * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 5e+75], N[(b * a + N[(z * t), $MachinePrecision]), $MachinePrecision], N[(t * z + N[(x * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -2.00000000000000002e-35

   1. Initial program 93.6%

      \[\left(x \cdot y + z \cdot t\right) + a \cdot b \]
   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 83.9

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

   5. Applied rewrites 83.9%

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

   if -2.00000000000000002e-35 < (*.f64 x y) < 5.0000000000000002e75

   1. Initial program 99.2%

      \[\left(x \cdot y + z \cdot t\right) + a \cdot b \]
   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 92.2

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

   5. Applied rewrites 92.2%

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

   if 5.0000000000000002e75 < (*.f64 x y)

   1. Initial program 95.5%

      \[\left(x \cdot y + z \cdot t\right) + a \cdot b \]
   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 31.5

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

   5. Applied rewrites 31.5%

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

   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 87.6

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

   8. Applied rewrites 87.6%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{-35}:\\ \;\;\;\;\mathsf{fma}\left(b, a, x \cdot y\right)\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+75}:\\ \;\;\;\;\mathsf{fma}\left(b, a, z \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, z, x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b, a, x \cdot y\right)\\ \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{-35}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \cdot y \leq 4 \cdot 10^{+44}:\\ \;\;\;\;\mathsf{fma}\left(b, a, z \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma b a (* x y))))
   (if (<= (* x y) -2e-35) t_1 (if (<= (* x y) 4e+44) (fma b a (* z t)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(b, a, (x * y));
	double tmp;
	if ((x * y) <= -2e-35) {
		tmp = t_1;
	} else if ((x * y) <= 4e+44) {
		tmp = fma(b, a, (z * t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(b, a, Float64(x * y))
	tmp = 0.0
	if (Float64(x * y) <= -2e-35)
		tmp = t_1;
	elseif (Float64(x * y) <= 4e+44)
		tmp = fma(b, a, Float64(z * t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(b * a + N[(x * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * y), $MachinePrecision], -2e-35], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], 4e+44], N[(b * a + N[(z * t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -2.00000000000000002e-35 or 4.0000000000000004e44 < (*.f64 x y)

   1. Initial program 94.7%

      \[\left(x \cdot y + z \cdot t\right) + a \cdot b \]
   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 83.5

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

   5. Applied rewrites 83.5%

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

   if -2.00000000000000002e-35 < (*.f64 x y) < 4.0000000000000004e44

   1. Initial program 99.2%

      \[\left(x \cdot y + z \cdot t\right) + a \cdot b \]
   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 93.6

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

   5. Applied rewrites 93.6%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{-35}:\\ \;\;\;\;\mathsf{fma}\left(b, a, x \cdot y\right)\\ \mathbf{elif}\;x \cdot y \leq 4 \cdot 10^{+44}:\\ \;\;\;\;\mathsf{fma}\left(b, a, z \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+78}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+140}:\\ \;\;\;\;\mathsf{fma}\left(b, a, z \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* x y) -5e+78)
   (* x y)
   (if (<= (* x y) 5e+140) (fma b a (* z t)) (* x y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((x * y) <= -5e+78) {
		tmp = x * y;
	} else if ((x * y) <= 5e+140) {
		tmp = fma(b, a, (z * t));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(x * y) <= -5e+78)
		tmp = Float64(x * y);
	elseif (Float64(x * y) <= 5e+140)
		tmp = fma(b, a, Float64(z * t));
	else
		tmp = Float64(x * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(x * y), $MachinePrecision], -5e+78], N[(x * y), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 5e+140], N[(b * a + N[(z * t), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -4.99999999999999984e78 or 5.00000000000000008e140 < (*.f64 x y)

   1. Initial program 93.4%

      \[\left(x \cdot y + z \cdot t\right) + a \cdot b \]
   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 25.2

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

   5. Applied rewrites 25.2%

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

   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 90.2

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

   8. Applied rewrites 90.2%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 15.5%

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

   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 78.3

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

   14. Applied rewrites 78.3%

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

   if -4.99999999999999984e78 < (*.f64 x y) < 5.00000000000000008e140

   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 86.0

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

   5. Applied rewrites 86.0%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+78}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+140}:\\ \;\;\;\;\mathsf{fma}\left(b, a, z \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 54.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{-5}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+79}:\\ \;\;\;\;a \cdot b\\ \mathbf{else}:\\ \;\;\;\;z \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* z t) -1e-5) (* z t) (if (<= (* z t) 2e+79) (* a b) (* z t))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z * t) <= -1e-5) {
		tmp = z * t;
	} else if ((z * t) <= 2e+79) {
		tmp = a * b;
	} else {
		tmp = z * t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((z * t) <= (-1d-5)) then
        tmp = z * t
    else if ((z * t) <= 2d+79) then
        tmp = a * b
    else
        tmp = z * t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z * t) <= -1e-5) {
		tmp = z * t;
	} else if ((z * t) <= 2e+79) {
		tmp = a * b;
	} else {
		tmp = z * t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (z * t) <= -1e-5:
		tmp = z * t
	elif (z * t) <= 2e+79:
		tmp = a * b
	else:
		tmp = z * t
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(z * t) <= -1e-5)
		tmp = Float64(z * t);
	elseif (Float64(z * t) <= 2e+79)
		tmp = Float64(a * b);
	else
		tmp = Float64(z * t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((z * t) <= -1e-5)
		tmp = z * t;
	elseif ((z * t) <= 2e+79)
		tmp = a * b;
	else
		tmp = z * t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(z * t), $MachinePrecision], -1e-5], N[(z * t), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 2e+79], N[(a * b), $MachinePrecision], N[(z * t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -1.00000000000000008e-5 or 1.99999999999999993e79 < (*.f64 z t)

   1. Initial program 94.0%

      \[\left(x \cdot y + z \cdot t\right) + a \cdot b \]
   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 74.3

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

   5. Applied rewrites 74.3%

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

   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 85.1

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

   8. Applied rewrites 85.1%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 61.3%

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

   if -1.00000000000000008e-5 < (*.f64 z t) < 1.99999999999999993e79

   1. Initial program 99.3%

      \[\left(x \cdot y + z \cdot t\right) + a \cdot b \]
   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 91.8

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

   5. Applied rewrites 91.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 48.6%

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

4. Final simplification 54.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{-5}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+79}:\\ \;\;\;\;a \cdot b\\ \mathbf{else}:\\ \;\;\;\;z \cdot t\\ \end{array} \]
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 96.9%

   \[\left(x \cdot y + z \cdot t\right) + a \cdot b \]
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 67.8

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

5. Applied rewrites 67.8%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 34.1%

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

9. Final simplification 34.1%

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

Reproduce

herbie shell --seed 2024332 
(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)))