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

Percentage Accurate: 99.2% → 99.6%

Time: 3.9s

Alternatives: 5

Speedup: 1.0×

• 23.9% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.8%

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

   1. +-commutative 98.8%

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

   2. fma-def 99.6%

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

3. Applied egg-rr 99.6%

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

4. Final simplification 99.6%

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

Alternative 2: 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 98.8%

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

   1. fma-def 99.2%

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

3. Simplified 99.2%

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

4. Final simplification 99.2%

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

Alternative 3: 75.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1.12 \cdot 10^{+113}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \cdot y \leq -6.8 \cdot 10^{+97} \lor \neg \left(x \cdot y \leq -4.2 \cdot 10^{+46}\right) \land \left(x \cdot y \leq -3.8 \cdot 10^{-26} \lor \neg \left(x \cdot y \leq -1.95 \cdot 10^{-61}\right) \land x \cdot y \leq 5.2 \cdot 10^{+51}\right):\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* x y) -1.12e+113)
   (* x y)
   (if (or (<= (* x y) -6.8e+97)
           (and (not (<= (* x y) -4.2e+46))
                (or (<= (* x y) -3.8e-26)
                    (and (not (<= (* x y) -1.95e-61)) (<= (* x y) 5.2e+51)))))
     (* z t)
     (* x y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * y) <= -1.12e+113) {
		tmp = x * y;
	} else if (((x * y) <= -6.8e+97) || (!((x * y) <= -4.2e+46) && (((x * y) <= -3.8e-26) || (!((x * y) <= -1.95e-61) && ((x * y) <= 5.2e+51))))) {
		tmp = z * t;
	} else {
		tmp = x * y;
	}
	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) <= (-1.12d+113)) then
        tmp = x * y
    else if (((x * y) <= (-6.8d+97)) .or. (.not. ((x * y) <= (-4.2d+46))) .and. ((x * y) <= (-3.8d-26)) .or. (.not. ((x * y) <= (-1.95d-61))) .and. ((x * y) <= 5.2d+51)) then
        tmp = 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 tmp;
	if ((x * y) <= -1.12e+113) {
		tmp = x * y;
	} else if (((x * y) <= -6.8e+97) || (!((x * y) <= -4.2e+46) && (((x * y) <= -3.8e-26) || (!((x * y) <= -1.95e-61) && ((x * y) <= 5.2e+51))))) {
		tmp = z * t;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x * y) <= -1.12e+113:
		tmp = x * y
	elif ((x * y) <= -6.8e+97) or (not ((x * y) <= -4.2e+46) and (((x * y) <= -3.8e-26) or (not ((x * y) <= -1.95e-61) and ((x * y) <= 5.2e+51)))):
		tmp = z * t
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x * y) <= -1.12e+113)
		tmp = Float64(x * y);
	elseif ((Float64(x * y) <= -6.8e+97) || (!(Float64(x * y) <= -4.2e+46) && ((Float64(x * y) <= -3.8e-26) || (!(Float64(x * y) <= -1.95e-61) && (Float64(x * y) <= 5.2e+51)))))
		tmp = Float64(z * t);
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x * y) <= -1.12e+113)
		tmp = x * y;
	elseif (((x * y) <= -6.8e+97) || (~(((x * y) <= -4.2e+46)) && (((x * y) <= -3.8e-26) || (~(((x * y) <= -1.95e-61)) && ((x * y) <= 5.2e+51)))))
		tmp = z * t;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x * y), $MachinePrecision], -1.12e+113], N[(x * y), $MachinePrecision], If[Or[LessEqual[N[(x * y), $MachinePrecision], -6.8e+97], And[N[Not[LessEqual[N[(x * y), $MachinePrecision], -4.2e+46]], $MachinePrecision], Or[LessEqual[N[(x * y), $MachinePrecision], -3.8e-26], And[N[Not[LessEqual[N[(x * y), $MachinePrecision], -1.95e-61]], $MachinePrecision], LessEqual[N[(x * y), $MachinePrecision], 5.2e+51]]]]], N[(z * t), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -1.1200000000000001e113 or -6.8000000000000002e97 < (*.f64 x y) < -4.2e46 or -3.80000000000000015e-26 < (*.f64 x y) < -1.95000000000000016e-61 or 5.2000000000000002e51 < (*.f64 x y)

   1. Initial program 97.3%

      \[x \cdot y + z \cdot t \]

   2. Taylor expanded in x around inf 84.2%

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

   if -1.1200000000000001e113 < (*.f64 x y) < -6.8000000000000002e97 or -4.2e46 < (*.f64 x y) < -3.80000000000000015e-26 or -1.95000000000000016e-61 < (*.f64 x y) < 5.2000000000000002e51

   1. Initial program 100.0%

      \[x \cdot y + z \cdot t \]

   2. Taylor expanded in x around 0 83.6%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1.12 \cdot 10^{+113}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \cdot y \leq -6.8 \cdot 10^{+97} \lor \neg \left(x \cdot y \leq -4.2 \cdot 10^{+46}\right) \land \left(x \cdot y \leq -3.8 \cdot 10^{-26} \lor \neg \left(x \cdot y \leq -1.95 \cdot 10^{-61}\right) \land x \cdot y \leq 5.2 \cdot 10^{+51}\right):\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 4: 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]

Derivation

1. Initial program 98.8%

   \[x \cdot y + z \cdot t \]

2. Final simplification 98.8%

   \[\leadsto x \cdot y + z \cdot t \]

Alternative 5: 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 98.8%

   \[x \cdot y + z \cdot t \]

2. Taylor expanded in x around 0 55.7%

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

3. Final simplification 55.7%

   \[\leadsto z \cdot t \]

Reproduce

herbie shell --seed 2023287 
(FPCore (x y z t)
  :name "Linear.V2:$cdot from linear-1.19.1.3, A"
  :precision binary64
  (+ (* x y) (* z t)))