Linear.V3:cross from linear-1.19.1.3

Percentage Accurate: 99.3% → 99.7%

Time: 3.8s

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: 99.3% 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: 99.7% 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. lift-*.f64 N/A

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

   3. sub-neg N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-neg.f64 100.0

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

4. Applied rewrites 100.0%

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

Alternative 2: 75.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -z \cdot t\\ \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+63}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \cdot t \leq 4 \cdot 10^{-41}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* z t))))
   (if (<= (* z t) -2e+63) t_1 (if (<= (* z t) 4e-41) (* y x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = -(z * t);
	double tmp;
	if ((z * t) <= -2e+63) {
		tmp = t_1;
	} else if ((z * t) <= 4e-41) {
		tmp = y * x;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = -(z * t)
    if ((z * t) <= (-2d+63)) then
        tmp = t_1
    else if ((z * t) <= 4d-41) then
        tmp = y * x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = -(z * t);
	double tmp;
	if ((z * t) <= -2e+63) {
		tmp = t_1;
	} else if ((z * t) <= 4e-41) {
		tmp = y * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = -(z * t)
	tmp = 0
	if (z * t) <= -2e+63:
		tmp = t_1
	elif (z * t) <= 4e-41:
		tmp = y * x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(-Float64(z * t))
	tmp = 0.0
	if (Float64(z * t) <= -2e+63)
		tmp = t_1;
	elseif (Float64(z * t) <= 4e-41)
		tmp = Float64(y * x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = -(z * t);
	tmp = 0.0;
	if ((z * t) <= -2e+63)
		tmp = t_1;
	elseif ((z * t) <= 4e-41)
		tmp = y * x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = (-N[(z * t), $MachinePrecision])}, If[LessEqual[N[(z * t), $MachinePrecision], -2e+63], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], 4e-41], N[(y * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -2.00000000000000012e63 or 4.00000000000000002e-41 < (*.f64 z t)

   1. Initial program 98.3%

      \[x \cdot y - z \cdot t \]
   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 78.4

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

   5. Applied rewrites 78.4%

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

   if -2.00000000000000012e63 < (*.f64 z t) < 4.00000000000000002e-41

   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 81.5

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

   5. Applied rewrites 81.5%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+63}:\\ \;\;\;\;-z \cdot t\\ \mathbf{elif}\;z \cdot t \leq 4 \cdot 10^{-41}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;-z \cdot t\\ \end{array} \]
5. Add Preprocessing

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 99.2%

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

3. Final simplification 99.2%

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

Alternative 4: 51.7% 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 inf

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

   1. lower-*.f64 55.5

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

5. Applied rewrites 55.5%

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

6. Final simplification 55.5%

   \[\leadsto y \cdot x \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024216 
(FPCore (x y z t)
  :name "Linear.V3:cross from linear-1.19.1.3"
  :precision binary64
  (- (* x y) (* z t)))