Linear.V3:cross from linear-1.19.1.3

Percentage Accurate: 99.2% → 99.6%

Time: 3.4s

Alternatives: 4

Speedup: 1.0×

• 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: 99.2% 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, 1.0× speedup

Math

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

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 * Float64(-z)))
end

Wolfram

code[x_, y_, z_, t_] := N[(y * x + N[(t * (-z)), $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 99.6

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

4. Applied rewrites 99.6%

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

5. Final simplification 99.6%

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

Alternative 2: 76.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot x \leq -4.8 \cdot 10^{-48}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \cdot x \leq 3.6 \cdot 10^{-12}:\\ \;\;\;\;t \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* y x) -4.8e-48)
   (* y x)
   (if (<= (* y x) 3.6e-12) (* t (- z)) (* y x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y * x) <= -4.8e-48) {
		tmp = y * x;
	} else if ((y * x) <= 3.6e-12) {
		tmp = t * -z;
	} 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 ((y * x) <= (-4.8d-48)) then
        tmp = y * x
    else if ((y * x) <= 3.6d-12) then
        tmp = t * -z
    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 ((y * x) <= -4.8e-48) {
		tmp = y * x;
	} else if ((y * x) <= 3.6e-12) {
		tmp = t * -z;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (y * x) <= -4.8e-48:
		tmp = y * x
	elif (y * x) <= 3.6e-12:
		tmp = t * -z
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(y * x) <= -4.8e-48)
		tmp = Float64(y * x);
	elseif (Float64(y * x) <= 3.6e-12)
		tmp = Float64(t * Float64(-z));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((y * x) <= -4.8e-48)
		tmp = y * x;
	elseif ((y * x) <= 3.6e-12)
		tmp = t * -z;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(y * x), $MachinePrecision], -4.8e-48], N[(y * x), $MachinePrecision], If[LessEqual[N[(y * x), $MachinePrecision], 3.6e-12], N[(t * (-z)), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -4.8e-48 or 3.6e-12 < (*.f64 x y)

   1. Initial program 98.6%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 73.5

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

   5. Applied rewrites 73.5%

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

   if -4.8e-48 < (*.f64 x y) < 3.6e-12

   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 80.2

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

   5. Applied rewrites 80.2%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot x \leq -4.8 \cdot 10^{-48}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \cdot x \leq 3.6 \cdot 10^{-12}:\\ \;\;\;\;t \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024230 
(FPCore (x y z t)
  :name "Linear.V3:cross from linear-1.19.1.3"
  :precision binary64
  (- (* x y) (* z t)))