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

Percentage Accurate: 97.6% → 98.8%

Time: 5.5s

Alternatives: 7

Speedup: 1.2×

• 33.2% 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: 97.6% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.0%

   \[\left(x \cdot y + z \cdot t\right) + a \cdot b \]
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. lift-*.f64 N/A

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

   8. lower-fma.f64 98.8

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

4. Applied rewrites 98.8%

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

Alternative 2: 52.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{-107}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;z \cdot t \leq -5 \cdot 10^{-324}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{-39}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;z \cdot t \leq 50000000:\\ \;\;\;\;a \cdot b\\ \mathbf{else}:\\ \;\;\;\;z \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* z t) -2e-107)
   (* z t)
   (if (<= (* z t) -5e-324)
     (* a b)
     (if (<= (* z t) 5e-39)
       (* x y)
       (if (<= (* z t) 50000000.0) (* a b) (* z t))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z * t) <= -2e-107) {
		tmp = z * t;
	} else if ((z * t) <= -5e-324) {
		tmp = a * b;
	} else if ((z * t) <= 5e-39) {
		tmp = x * y;
	} else if ((z * t) <= 50000000.0) {
		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) <= (-2d-107)) then
        tmp = z * t
    else if ((z * t) <= (-5d-324)) then
        tmp = a * b
    else if ((z * t) <= 5d-39) then
        tmp = x * y
    else if ((z * t) <= 50000000.0d0) 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) <= -2e-107) {
		tmp = z * t;
	} else if ((z * t) <= -5e-324) {
		tmp = a * b;
	} else if ((z * t) <= 5e-39) {
		tmp = x * y;
	} else if ((z * t) <= 50000000.0) {
		tmp = a * b;
	} else {
		tmp = z * t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (z * t) <= -2e-107:
		tmp = z * t
	elif (z * t) <= -5e-324:
		tmp = a * b
	elif (z * t) <= 5e-39:
		tmp = x * y
	elif (z * t) <= 50000000.0:
		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) <= -2e-107)
		tmp = Float64(z * t);
	elseif (Float64(z * t) <= -5e-324)
		tmp = Float64(a * b);
	elseif (Float64(z * t) <= 5e-39)
		tmp = Float64(x * y);
	elseif (Float64(z * t) <= 50000000.0)
		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) <= -2e-107)
		tmp = z * t;
	elseif ((z * t) <= -5e-324)
		tmp = a * b;
	elseif ((z * t) <= 5e-39)
		tmp = x * y;
	elseif ((z * t) <= 50000000.0)
		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], -2e-107], N[(z * t), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], -5e-324], N[(a * b), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 5e-39], N[(x * y), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 50000000.0], N[(a * b), $MachinePrecision], N[(z * t), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -2e-107 or 5e7 < (*.f64 z t)

   1. Initial program 96.4%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 55.1

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

   5. Applied rewrites 55.1%

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

   if -2e-107 < (*.f64 z t) < -4.94066e-324 or 4.9999999999999998e-39 < (*.f64 z t) < 5e7

   1. Initial program 99.0%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 46.7

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

   5. Applied rewrites 46.7%

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

   if -4.94066e-324 < (*.f64 z t) < 4.9999999999999998e-39

   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 inf

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

      1. lower-*.f64 49.2

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

   5. Applied rewrites 49.2%

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

4. Final simplification 52.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{-107}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;z \cdot t \leq -5 \cdot 10^{-324}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{-39}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;z \cdot t \leq 50000000:\\ \;\;\;\;a \cdot b\\ \mathbf{else}:\\ \;\;\;\;z \cdot t\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024223 
(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)))