Linear.V3:cross from linear-1.19.1.3

Percentage Accurate: 99.1% → 99.6%

Time: 3.6s

Alternatives: 4

Speedup: 1.0×

• 23.6% 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

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

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. lift-*.f64 N/A

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

   3. fp-cancel-sub-sign-inv N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-neg.f64 100.0

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

4. Applied rewrites 100.0%

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

Alternative 2: 76.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+51} \lor \neg \left(x \cdot y \leq 5 \cdot 10^{+30}\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (* x y) -2e+51) (not (<= (* x y) 5e+30))) (* y x) (* (- z) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((x * y) <= -2e+51) || !((x * y) <= 5e+30)) {
		tmp = y * x;
	} 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 (((x * y) <= (-2d+51)) .or. (.not. ((x * y) <= 5d+30))) then
        tmp = y * x
    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 (((x * y) <= -2e+51) || !((x * y) <= 5e+30)) {
		tmp = y * x;
	} else {
		tmp = -z * t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((x * y) <= -2e+51) or not ((x * y) <= 5e+30):
		tmp = y * x
	else:
		tmp = -z * t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(x * y) <= -2e+51) || !(Float64(x * y) <= 5e+30))
		tmp = Float64(y * x);
	else
		tmp = Float64(Float64(-z) * t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((x * y) <= -2e+51) || ~(((x * y) <= 5e+30)))
		tmp = y * x;
	else
		tmp = -z * t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[N[(x * y), $MachinePrecision], -2e+51], N[Not[LessEqual[N[(x * y), $MachinePrecision], 5e+30]], $MachinePrecision]], N[(y * x), $MachinePrecision], N[((-z) * t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -2e51 or 4.9999999999999998e30 < (*.f64 x y)

   1. Initial program 98.2%

      \[x \cdot y - z \cdot t \]
   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. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 23.7

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

   5. Applied rewrites 23.7%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 80.3

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

   8. Applied rewrites 80.3%

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

   if -2e51 < (*.f64 x y) < 4.9999999999999998e30

   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. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 80.1

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

   5. Applied rewrites 80.1%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+51} \lor \neg \left(x \cdot y \leq 5 \cdot 10^{+30}\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 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 4: 52.0% 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 99.2%

   \[x \cdot y - z \cdot t \]
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. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f64 N/A

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

   4. mul-1-neg N/A

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

   5. lower-neg.f64 55.9

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

5. Applied rewrites 55.9%

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

6. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 49.9

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

8. Applied rewrites 49.9%

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

Reproduce

herbie shell --seed 2024320 
(FPCore (x y z t)
  :name "Linear.V3:cross from linear-1.19.1.3"
  :precision binary64
  (- (* x y) (* z t)))