Linear.V3:cross from linear-1.19.1.3

Percentage Accurate: 99.1% → 99.6%

Time: 3.2s

Alternatives: 5

Speedup: 1.0×

• 22.6% 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, 0.1× 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 98.8%

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

   1. sub-neg 98.8%

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

   2. +-commutative 98.8%

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

   3. distribute-lft-neg-in 98.8%

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

   4. fma-def 99.6%

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

3. Applied egg-rr 99.6%

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

4. Final simplification 99.6%

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

Alternative 2: 99.5% 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 98.8%

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

   1. *-commutative 98.8%

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

   2. fma-neg 99.2%

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

   3. distribute-rgt-neg-in 99.2%

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

3. Applied egg-rr 99.2%

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

4. Final simplification 99.2%

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

Alternative 3: 75.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -58 \lor \neg \left(x \cdot y \leq 1.35 \cdot 10^{+83}\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (* x y) -58.0) (not (<= (* x y) 1.35e+83))) (* x y) (* z (- t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((x * y) <= -58.0) || !((x * y) <= 1.35e+83)) {
		tmp = x * y;
	} 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) <= (-58.0d0)) .or. (.not. ((x * y) <= 1.35d+83))) then
        tmp = x * y
    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) <= -58.0) || !((x * y) <= 1.35e+83)) {
		tmp = x * y;
	} else {
		tmp = z * -t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((x * y) <= -58.0) or not ((x * y) <= 1.35e+83):
		tmp = x * y
	else:
		tmp = z * -t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(x * y) <= -58.0) || !(Float64(x * y) <= 1.35e+83))
		tmp = Float64(x * y);
	else
		tmp = Float64(z * Float64(-t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((x * y) <= -58.0) || ~(((x * y) <= 1.35e+83)))
		tmp = x * y;
	else
		tmp = z * -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[N[(x * y), $MachinePrecision], -58.0], N[Not[LessEqual[N[(x * y), $MachinePrecision], 1.35e+83]], $MachinePrecision]], N[(x * y), $MachinePrecision], N[(z * (-t)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -58 or 1.35000000000000003e83 < (*.f64 x y)

   1. Initial program 97.3%

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

   2. Taylor expanded in x around inf 85.6%

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

   if -58 < (*.f64 x y) < 1.35000000000000003e83

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 81.9%

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

      1. associate-*r* 81.9%

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

      2. neg-mul-1 81.9%

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

      3. *-commutative 81.9%

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

   4. Simplified 81.9%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -58 \lor \neg \left(x \cdot y \leq 1.35 \cdot 10^{+83}\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-t\right)\\ \end{array} \]

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 98.8%

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

2. Final simplification 98.8%

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

Alternative 5: 51.7% 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 98.8%

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

2. Taylor expanded in x around inf 48.6%

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

3. Final simplification 48.6%

   \[\leadsto x \cdot y \]

Reproduce

herbie shell --seed 2023308 
(FPCore (x y z t)
  :name "Linear.V3:cross from linear-1.19.1.3"
  :precision binary64
  (- (* x y) (* z t)))