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

Percentage Accurate: 99.3% → 99.6%

Time: 3.8s

Alternatives: 4

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 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

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. lower-fma.f64 100.0

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

4. Applied rewrites 100.0%

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

Alternative 2: 75.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+63}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;z \cdot t \leq 10000000:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z t) -2e+63) (* z t) (if (<= (* z t) 10000000.0) (* x y) (* z t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * t) <= -2e+63) {
		tmp = z * t;
	} else if ((z * t) <= 10000000.0) {
		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 ((z * t) <= (-2d+63)) then
        tmp = z * t
    else if ((z * t) <= 10000000.0d0) 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 ((z * t) <= -2e+63) {
		tmp = z * t;
	} else if ((z * t) <= 10000000.0) {
		tmp = x * y;
	} else {
		tmp = z * t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z * t) <= -2e+63:
		tmp = z * t
	elif (z * t) <= 10000000.0:
		tmp = x * y
	else:
		tmp = z * t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * t) <= -2e+63)
		tmp = Float64(z * t);
	elseif (Float64(z * t) <= 10000000.0)
		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 ((z * t) <= -2e+63)
		tmp = z * t;
	elseif ((z * t) <= 10000000.0)
		tmp = x * y;
	else
		tmp = z * t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * t), $MachinePrecision], -2e+63], N[(z * t), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 10000000.0], N[(x * y), $MachinePrecision], N[(z * t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -2.00000000000000012e63 or 1e7 < (*.f64 z t)

   1. Initial program 99.1%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 79.1

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

   5. Applied rewrites 79.1%

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

   if -2.00000000000000012e63 < (*.f64 z t) < 1e7

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 80.2

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

   5. Applied rewrites 80.2%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+63}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;z \cdot t \leq 10000000:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. *-commutative N/A

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

   3. lower-fma.f64 99.6

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

4. Applied rewrites 99.6%

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

Alternative 4: 52.6% 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. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. lower-*.f64 49.4

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

5. Applied rewrites 49.4%

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

6. Final simplification 49.4%

   \[\leadsto z \cdot t \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024216 
(FPCore (x y z t)
  :name "Linear.V2:$cdot from linear-1.19.1.3, A"
  :precision binary64
  (+ (* x y) (* z t)))