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

Percentage Accurate: 99.1% → 99.4%

Time: 8.7s

Alternatives: 5

Speedup: 1.0×

• 24.6% 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: 99.1% 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: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot t + x \cdot y\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{+280}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(t + x \cdot \frac{y}{z}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (* z t) (* x y))))
   (if (<= t_1 5e+280) t_1 (* z (+ t (* x (/ y z)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * t) + (x * y);
	double tmp;
	if (t_1 <= 5e+280) {
		tmp = t_1;
	} else {
		tmp = z * (t + (x * (y / z)));
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = (z * t) + (x * y)
    if (t_1 <= 5d+280) then
        tmp = t_1
    else
        tmp = z * (t + (x * (y / z)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (z * t) + (x * y);
	double tmp;
	if (t_1 <= 5e+280) {
		tmp = t_1;
	} else {
		tmp = z * (t + (x * (y / z)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (z * t) + (x * y)
	tmp = 0
	if t_1 <= 5e+280:
		tmp = t_1
	else:
		tmp = z * (t + (x * (y / z)))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * t) + Float64(x * y))
	tmp = 0.0
	if (t_1 <= 5e+280)
		tmp = t_1;
	else
		tmp = Float64(z * Float64(t + Float64(x * Float64(y / z))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (z * t) + (x * y);
	tmp = 0.0;
	if (t_1 <= 5e+280)
		tmp = t_1;
	else
		tmp = z * (t + (x * (y / z)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * t), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 5e+280], t$95$1, N[(z * N[(t + N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 x y) (*.f64 z t)) < 5.0000000000000002e280

   1. Initial program 100.0%

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

   if 5.0000000000000002e280 < (+.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 84.4%

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

   3. Taylor expanded in z around inf 93.8%

      \[\leadsto \color{blue}{z \cdot \left(t + \frac{x \cdot y}{z}\right)} \]
   4. Step-by-step derivation

      1. associate-/l* 100.0%

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

   5. Simplified 100.0%

      \[\leadsto \color{blue}{z \cdot \left(t + x \cdot \frac{y}{z}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t + x \cdot y \leq 5 \cdot 10^{+280}:\\ \;\;\;\;z \cdot t + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(t + x \cdot \frac{y}{z}\right)\\ \end{array} \]
5. Add Preprocessing

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 98.0%

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

   1. fma-define 98.8%

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

3. Simplified 98.8%

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

Alternative 3: 75.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2.85 \cdot 10^{+49} \lor \neg \left(x \cdot y \leq 340\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (* x y) -2.85e+49) (not (<= (* x y) 340.0))) (* x y) (* z t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((x * y) <= -2.85e+49) || !((x * y) <= 340.0)) {
		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) <= (-2.85d+49)) .or. (.not. ((x * y) <= 340.0d0))) 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) <= -2.85e+49) || !((x * y) <= 340.0)) {
		tmp = x * y;
	} else {
		tmp = z * t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((x * y) <= -2.85e+49) or not ((x * y) <= 340.0):
		tmp = x * y
	else:
		tmp = z * t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(x * y) <= -2.85e+49) || !(Float64(x * y) <= 340.0))
		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) <= -2.85e+49) || ~(((x * y) <= 340.0)))
		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], -2.85e+49], N[Not[LessEqual[N[(x * y), $MachinePrecision], 340.0]], $MachinePrecision]], N[(x * y), $MachinePrecision], N[(z * t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -2.8499999999999999e49 or 340 < (*.f64 x y)

   1. Initial program 95.2%

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

   3. Taylor expanded in x around inf 81.0%

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

   if -2.8499999999999999e49 < (*.f64 x y) < 340

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 100.0%

      \[\leadsto \color{blue}{z \cdot \left(t + \frac{x \cdot y}{z}\right)} \]
   4. Step-by-step derivation

      1. associate-/l* 97.3%

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

   5. Simplified 97.3%

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

   6. Taylor expanded in z around inf 81.7%

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

      1. *-commutative 81.7%

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

   8. Simplified 81.7%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2.85 \cdot 10^{+49} \lor \neg \left(x \cdot y \leq 340\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.1% 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 98.0%

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

3. Final simplification 98.0%

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

Alternative 5: 53.2% accurate, 2.3× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return 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 = x * y
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.0%

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

3. Taylor expanded in x around inf 45.7%

   \[\leadsto \color{blue}{x \cdot y} \]
4. Add Preprocessing

Reproduce

herbie shell --seed 2024143 
(FPCore (x y z t)
  :name "Linear.V2:$cdot from linear-1.19.1.3, A"
  :precision binary64
  (+ (* x y) (* z t)))