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

Percentage Accurate: 99.1% → 99.5%

Time: 2.0s

Alternatives: 4

Speedup: 1.0×

• 23.3% 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: 99.5% 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-def 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 67.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.16 \cdot 10^{-82} \lor \neg \left(t \leq -8.2 \cdot 10^{-131}\right) \land \left(t \leq -3.5 \cdot 10^{-147} \lor \neg \left(t \leq 0.028\right)\right):\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -1.16e-82)
         (and (not (<= t -8.2e-131)) (or (<= t -3.5e-147) (not (<= t 0.028)))))
   (* z t)
   (* x y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -1.16e-82) || (!(t <= -8.2e-131) && ((t <= -3.5e-147) || !(t <= 0.028)))) {
		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 ((t <= (-1.16d-82)) .or. (.not. (t <= (-8.2d-131))) .and. (t <= (-3.5d-147)) .or. (.not. (t <= 0.028d0))) 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 ((t <= -1.16e-82) || (!(t <= -8.2e-131) && ((t <= -3.5e-147) || !(t <= 0.028)))) {
		tmp = z * t;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -1.16e-82) or (not (t <= -8.2e-131) and ((t <= -3.5e-147) or not (t <= 0.028))):
		tmp = z * t
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -1.16e-82) || (!(t <= -8.2e-131) && ((t <= -3.5e-147) || !(t <= 0.028))))
		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 ((t <= -1.16e-82) || (~((t <= -8.2e-131)) && ((t <= -3.5e-147) || ~((t <= 0.028)))))
		tmp = z * t;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -1.16e-82], And[N[Not[LessEqual[t, -8.2e-131]], $MachinePrecision], Or[LessEqual[t, -3.5e-147], N[Not[LessEqual[t, 0.028]], $MachinePrecision]]]], N[(z * t), $MachinePrecision], N[(x * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.16e-82 or -8.2000000000000004e-131 < t < -3.50000000000000004e-147 or 0.0280000000000000006 < t

   1. Initial program 99.3%

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

   2. Taylor expanded in x around 0 70.0%

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

   if -1.16e-82 < t < -8.2000000000000004e-131 or -3.50000000000000004e-147 < t < 0.0280000000000000006

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 71.5%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.16 \cdot 10^{-82} \lor \neg \left(t \leq -8.2 \cdot 10^{-131}\right) \land \left(t \leq -3.5 \cdot 10^{-147} \lor \neg \left(t \leq 0.028\right)\right):\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 3: 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 99.6%

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

2. Final simplification 99.6%

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

Alternative 4: 52.0% 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. Taylor expanded in x around 0 55.0%

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

3. Final simplification 55.0%

   \[\leadsto z \cdot t \]

Reproduce

herbie shell --seed 2023242 
(FPCore (x y z t)
  :name "Linear.V2:$cdot from linear-1.19.1.3, A"
  :precision binary64
  (+ (* x y) (* z t)))