Linear.V3:cross from linear-1.19.1.3

Percentage Accurate: 99.1% → 99.6%

Time: 3.5s

Alternatives: 5

Speedup: 1.0×

• 24.2% 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: 99.6% accurate, 1.0× 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(Float64(-z), t, Float64(x * y))
end

Wolfram

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

Derivation

1. Initial program 99.2%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. distribute-lft-neg-in N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-neg.f64 100.0

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

4. Applied rewrites 100.0%

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

Alternative 2: 76.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -3.6 \cdot 10^{+59}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \cdot y \leq 7.1 \cdot 10^{+56}:\\ \;\;\;\;-z \cdot t\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* x y) -3.6e+59)
   (* x y)
   (if (<= (* x y) 7.1e+56) (- (* z t)) (* x y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * y) <= -3.6e+59) {
		tmp = x * y;
	} else if ((x * y) <= 7.1e+56) {
		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 ((x * y) <= (-3.6d+59)) then
        tmp = x * y
    else if ((x * y) <= 7.1d+56) 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 ((x * y) <= -3.6e+59) {
		tmp = x * y;
	} else if ((x * y) <= 7.1e+56) {
		tmp = -(z * t);
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x * y) <= -3.6e+59:
		tmp = x * y
	elif (x * y) <= 7.1e+56:
		tmp = -(z * t)
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x * y) <= -3.6e+59)
		tmp = Float64(x * y);
	elseif (Float64(x * y) <= 7.1e+56)
		tmp = Float64(-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 ((x * y) <= -3.6e+59)
		tmp = x * y;
	elseif ((x * y) <= 7.1e+56)
		tmp = -(z * t);
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x * y), $MachinePrecision], -3.6e+59], N[(x * y), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 7.1e+56], (-N[(z * t), $MachinePrecision]), N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -3.5999999999999999e59 or 7.1e56 < (*.f64 x y)

   1. Initial program 98.1%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 84.3

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

   5. Applied rewrites 84.3%

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

   if -3.5999999999999999e59 < (*.f64 x y) < 7.1e56

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-1 \cdot \left(t \cdot z\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(t \cdot z\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(t \cdot z\right)} \]

      3. lower-*.f64 75.2

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

   5. Applied rewrites 75.2%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -3.6 \cdot 10^{+59}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \cdot y \leq 7.1 \cdot 10^{+56}:\\ \;\;\;\;-z \cdot t\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 1.0× 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(-Float64(z * t)))
end

Wolfram

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

Derivation

1. Initial program 99.2%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-neg.f64 99.2

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

4. Applied rewrites 99.2%

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

Alternative 4: 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]

Derivation

1. Initial program 99.2%

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

Alternative 5: 51.5% 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 99.2%

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

3. Taylor expanded in x around inf

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

   1. lower-*.f64 52.6

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

5. Applied rewrites 52.6%

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

Reproduce

herbie shell --seed 2024238 
(FPCore (x y z t)
  :name "Linear.V3:cross from linear-1.19.1.3"
  :precision binary64
  (- (* x y) (* z t)))