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

Percentage Accurate: 99.2% → 99.6%

Time: 4.3s

Alternatives: 4

Speedup: 1.2×

• 23.5% 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, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

Alternative 2: 75.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{-9} \lor \neg \left(x \cdot y \leq 4 \cdot 10^{-79}\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;t \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (* x y) -1e-9) (not (<= (* x y) 4e-79))) (* y x) (* t z)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((x * y) <= -1e-9) || !((x * y) <= 4e-79)) {
		tmp = y * x;
	} else {
		tmp = t * 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) :: tmp
    if (((x * y) <= (-1d-9)) .or. (.not. ((x * y) <= 4d-79))) then
        tmp = y * x
    else
        tmp = t * z
    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-9) || !((x * y) <= 4e-79)) {
		tmp = y * x;
	} else {
		tmp = t * z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((x * y) <= -1e-9) or not ((x * y) <= 4e-79):
		tmp = y * x
	else:
		tmp = t * z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(x * y) <= -1e-9) || !(Float64(x * y) <= 4e-79))
		tmp = Float64(y * x);
	else
		tmp = Float64(t * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((x * y) <= -1e-9) || ~(((x * y) <= 4e-79)))
		tmp = y * x;
	else
		tmp = t * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[N[(x * y), $MachinePrecision], -1e-9], N[Not[LessEqual[N[(x * y), $MachinePrecision], 4e-79]], $MachinePrecision]], N[(y * x), $MachinePrecision], N[(t * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -1.00000000000000006e-9 or 4e-79 < (*.f64 x y)

   1. Initial program 99.2%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 22.6

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

   5. Applied rewrites 22.6%

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

   7. Applied rewrites 11.5%

      \[\leadsto \left|t \cdot z\right| \]

   8. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 77.8

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

   10. Applied rewrites 77.8%

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

   if -1.00000000000000006e-9 < (*.f64 x y) < 4e-79

   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. lower-*.f64 84.1

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

   5. Applied rewrites 84.1%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{-9} \lor \neg \left(x \cdot y \leq 4 \cdot 10^{-79}\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;t \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.6% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 99.6

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 99.6

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

4. Applied rewrites 99.6%

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

Alternative 4: 52.3% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

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 = t * z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in x around 0

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

   1. lower-*.f64 51.4

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

5. Applied rewrites 51.4%

   \[\leadsto \color{blue}{t \cdot z} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024338 
(FPCore (x y z t)
  :name "Linear.V2:$cdot from linear-1.19.1.3, A"
  :precision binary64
  (+ (* x y) (* z t)))