Linear.V2:$cdot from linear-1.19.1.3, A

Percentage Accurate: 99.2% → 99.6%

Time: 3.0s

Alternatives: 4

Speedup: 1.0×

• 23.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ x \cdot y + z \cdot t \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot y + z \cdot t \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

   \[x \cdot y + z \cdot t \]
2. Step-by-step derivation

   1. fma-define 99.6%

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

3. Simplified 99.6%

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

5. Final simplification 99.6%

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

Alternative 2: 75.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5.2 \cdot 10^{-63} \lor \neg \left(x \cdot y \leq 6 \cdot 10^{-33}\right) \land \left(x \cdot y \leq 0.042 \lor \neg \left(x \cdot y \leq 2.7 \cdot 10^{+95}\right)\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (* x y) -5.2e-63)
         (and (not (<= (* x y) 6e-33))
              (or (<= (* x y) 0.042) (not (<= (* x y) 2.7e+95)))))
   (* x y)
   (* z t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((x * y) <= -5.2e-63) || (!((x * y) <= 6e-33) && (((x * y) <= 0.042) || !((x * y) <= 2.7e+95)))) {
		tmp = x * y;
	} else {
		tmp = z * t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (((x * y) <= (-5.2d-63)) .or. (.not. ((x * y) <= 6d-33)) .and. ((x * y) <= 0.042d0) .or. (.not. ((x * y) <= 2.7d+95))) 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 tmp;
	if (((x * y) <= -5.2e-63) || (!((x * y) <= 6e-33) && (((x * y) <= 0.042) || !((x * y) <= 2.7e+95)))) {
		tmp = x * y;
	} else {
		tmp = z * t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((x * y) <= -5.2e-63) or (not ((x * y) <= 6e-33) and (((x * y) <= 0.042) or not ((x * y) <= 2.7e+95))):
		tmp = x * y
	else:
		tmp = z * t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(x * y) <= -5.2e-63) || (!(Float64(x * y) <= 6e-33) && ((Float64(x * y) <= 0.042) || !(Float64(x * y) <= 2.7e+95))))
		tmp = Float64(x * y);
	else
		tmp = Float64(z * t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((x * y) <= -5.2e-63) || (~(((x * y) <= 6e-33)) && (((x * y) <= 0.042) || ~(((x * y) <= 2.7e+95)))))
		tmp = x * y;
	else
		tmp = z * t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[N[(x * y), $MachinePrecision], -5.2e-63], And[N[Not[LessEqual[N[(x * y), $MachinePrecision], 6e-33]], $MachinePrecision], Or[LessEqual[N[(x * y), $MachinePrecision], 0.042], N[Not[LessEqual[N[(x * y), $MachinePrecision], 2.7e+95]], $MachinePrecision]]]], N[(x * y), $MachinePrecision], N[(z * t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -5.2000000000000003e-63 or 6.0000000000000003e-33 < (*.f64 x y) < 0.0420000000000000026 or 2.7e95 < (*.f64 x y)

   1. Initial program 99.2%

      \[x \cdot y + z \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 80.4%

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

   if -5.2000000000000003e-63 < (*.f64 x y) < 6.0000000000000003e-33 or 0.0420000000000000026 < (*.f64 x y) < 2.7e95

   1. Initial program 100.0%

      \[x \cdot y + z \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 80.1%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5.2 \cdot 10^{-63} \lor \neg \left(x \cdot y \leq 6 \cdot 10^{-33}\right) \land \left(x \cdot y \leq 0.042 \lor \neg \left(x \cdot y \leq 2.7 \cdot 10^{+95}\right)\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ z \cdot t + x \cdot y \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

   \[x \cdot y + z \cdot t \]
2. Add Preprocessing

3. Final simplification 99.6%

   \[\leadsto z \cdot t + x \cdot y \]
4. Add Preprocessing

Alternative 4: 51.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ z \cdot t \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[(z * t), $MachinePrecision]

Derivation

1. Initial program 99.6%

   \[x \cdot y + z \cdot t \]
2. Add Preprocessing

3. Taylor expanded in x around 0 51.9%

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

4. Final simplification 51.9%

   \[\leadsto z \cdot t \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024095 
(FPCore (x y z t)
  :name "Linear.V2:$cdot from linear-1.19.1.3, A"
  :precision binary64
  (+ (* x y) (* z t)))