Linear.V3:cross from linear-1.19.1.3

Percentage Accurate: 99.3% → 99.6%

Time: 2.4s

Alternatives: 4

Speedup: 1.0×

• 23.7% 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, 0.1× speedup

Math

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

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 * Float64(-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. Step-by-step derivation

   1. *-commutative 99.6%

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

   2. fma-neg 100.0%

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

   3. distribute-rgt-neg-in 100.0%

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

3. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 69.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.8 \cdot 10^{-28} \lor \neg \left(z \leq 6.5 \cdot 10^{+17}\right):\\ \;\;\;\;z \cdot \left(-t\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -6.8e-28) (not (<= z 6.5e+17))) (* z (- t)) (* y x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -6.8e-28) || !(z <= 6.5e+17)) {
		tmp = z * -t;
	} else {
		tmp = y * x;
	}
	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 <= (-6.8d-28)) .or. (.not. (z <= 6.5d+17))) then
        tmp = z * -t
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -6.8e-28) || !(z <= 6.5e+17)) {
		tmp = z * -t;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -6.8e-28) or not (z <= 6.5e+17):
		tmp = z * -t
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -6.8e-28) || !(z <= 6.5e+17))
		tmp = Float64(z * Float64(-t));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -6.8e-28) || ~((z <= 6.5e+17)))
		tmp = z * -t;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -6.8e-28], N[Not[LessEqual[z, 6.5e+17]], $MachinePrecision]], N[(z * (-t)), $MachinePrecision], N[(y * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -6.8000000000000001e-28 or 6.5e17 < z

   1. Initial program 99.3%

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

   2. Taylor expanded in x around 0 69.4%

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

      1. associate-*r* 69.4%

         \[\leadsto \color{blue}{\left(-1 \cdot t\right) \cdot z} \]

      2. neg-mul-1 69.4%

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

      3. *-commutative 69.4%

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

   4. Simplified 69.4%

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

   if -6.8000000000000001e-28 < z < 6.5e17

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 71.8%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.8 \cdot 10^{-28} \lor \neg \left(z \leq 6.5 \cdot 10^{+17}\right):\\ \;\;\;\;z \cdot \left(-t\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 3: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return (y * x) - (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 = (y * x) - (z * t)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

2. Final simplification 99.6%

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

Alternative 4: 51.8% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ y \cdot x \end{array} \]

FPCore

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

C

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

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 = y * x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in x around inf 50.6%

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

3. Final simplification 50.6%

   \[\leadsto y \cdot x \]

Reproduce

herbie shell --seed 2023268 
(FPCore (x y z t)
  :name "Linear.V3:cross from linear-1.19.1.3"
  :precision binary64
  (- (* x y) (* z t)))