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

Percentage Accurate: 99.3% → 99.6%

Time: 2.7s

Alternatives: 3

Speedup: 1.2×

• 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: 99.3% 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, 1.2× 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 100.0%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 100.0

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 100.0

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

4. Applied rewrites 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 76.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{-53}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{-42}:\\ \;\;\;\;t \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* x y) -1e-53) (* x y) (if (<= (* x y) 2e-42) (* t z) (* x y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * y) <= -1e-53) {
		tmp = x * y;
	} else if ((x * y) <= 2e-42) {
		tmp = t * z;
	} 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) <= (-1d-53)) then
        tmp = x * y
    else if ((x * y) <= 2d-42) then
        tmp = t * z
    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) <= -1e-53) {
		tmp = x * y;
	} else if ((x * y) <= 2e-42) {
		tmp = t * z;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x * y) <= -1e-53:
		tmp = x * y
	elif (x * y) <= 2e-42:
		tmp = t * z
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x * y) <= -1e-53)
		tmp = Float64(x * y);
	elseif (Float64(x * y) <= 2e-42)
		tmp = Float64(t * z);
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x * y) <= -1e-53)
		tmp = x * y;
	elseif ((x * y) <= 2e-42)
		tmp = t * z;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x * y), $MachinePrecision], -1e-53], N[(x * y), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 2e-42], N[(t * z), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -1.00000000000000003e-53 or 2.00000000000000008e-42 < (*.f64 x y)

   1. Initial program 98.7%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 72.2

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

   5. Applied rewrites 72.2%

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

   if -1.00000000000000003e-53 < (*.f64 x y) < 2.00000000000000008e-42

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 81.7

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

   5. Applied rewrites 81.7%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{-53}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{-42}:\\ \;\;\;\;t \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024230 
(FPCore (x y z t)
  :name "Linear.V2:$cdot from linear-1.19.1.3, A"
  :precision binary64
  (+ (* x y) (* z t)))