Linear.V3:cross from linear-1.19.1.3

Percentage Accurate: 99.1% → 99.6%

Time: 2.1s

Alternatives: 4

Speedup: 0.5×

• 24.4% 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

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

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 97.2%

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

   1. *-commutative 97.2%

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

   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 2: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot x - z \cdot t\\ \mathbf{if}\;t_1 \leq \infty:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* y x) (* z t)))) (if (<= t_1 INFINITY) t_1 (* z (- t)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (y * x) - (z * t);
	double tmp;
	if (t_1 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = z * -t;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (y * x) - (z * t);
	double tmp;
	if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else {
		tmp = z * -t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (y * x) - (z * t)
	tmp = 0
	if t_1 <= math.inf:
		tmp = t_1
	else:
		tmp = z * -t
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(y * x) - Float64(z * t))
	tmp = 0.0
	if (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(z * Float64(-t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (y * x) - (z * t);
	tmp = 0.0;
	if (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = z * -t;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x y) (*.f64 z t)) < +inf.0

   1. Initial program 100.0%

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

   if +inf.0 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 0.0%

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

   2. Taylor expanded in x around 0 71.4%

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

      1. associate-*r* 71.4%

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

      2. neg-mul-1 71.4%

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

      3. *-commutative 71.4%

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

   4. Simplified 71.4%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot x - z \cdot t \leq \infty:\\ \;\;\;\;y \cdot x - z \cdot t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-t\right)\\ \end{array} \]

Alternative 3: 75.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot x \leq -8 \cdot 10^{-59}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \cdot x \leq 1.25 \cdot 10^{+34}:\\ \;\;\;\;z \cdot \left(-t\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* y x) -8e-59)
   (* y x)
   (if (<= (* y x) 1.25e+34) (* z (- t)) (* y x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y * x) <= -8e-59) {
		tmp = y * x;
	} else if ((y * x) <= 1.25e+34) {
		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 ((y * x) <= (-8d-59)) then
        tmp = y * x
    else if ((y * x) <= 1.25d+34) 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 ((y * x) <= -8e-59) {
		tmp = y * x;
	} else if ((y * x) <= 1.25e+34) {
		tmp = z * -t;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (y * x) <= -8e-59:
		tmp = y * x
	elif (y * x) <= 1.25e+34:
		tmp = z * -t
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(y * x) <= -8e-59)
		tmp = Float64(y * x);
	elseif (Float64(y * x) <= 1.25e+34)
		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 ((y * x) <= -8e-59)
		tmp = y * x;
	elseif ((y * x) <= 1.25e+34)
		tmp = z * -t;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(y * x), $MachinePrecision], -8e-59], N[(y * x), $MachinePrecision], If[LessEqual[N[(y * x), $MachinePrecision], 1.25e+34], N[(z * (-t)), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -8.0000000000000002e-59 or 1.25e34 < (*.f64 x y)

   1. Initial program 95.2%

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

   2. Taylor expanded in x around inf 76.0%

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

   if -8.0000000000000002e-59 < (*.f64 x y) < 1.25e34

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 81.3%

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

      1. associate-*r* 81.3%

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

      2. neg-mul-1 81.3%

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

      3. *-commutative 81.3%

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

   4. Simplified 81.3%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot x \leq -8 \cdot 10^{-59}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \cdot x \leq 1.25 \cdot 10^{+34}:\\ \;\;\;\;z \cdot \left(-t\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 4: 52.5% 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 97.2%

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

2. Taylor expanded in x around inf 53.9%

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

3. Final simplification 53.9%

   \[\leadsto y \cdot x \]

Reproduce

herbie shell --seed 2023275 
(FPCore (x y z t)
  :name "Linear.V3:cross from linear-1.19.1.3"
  :precision binary64
  (- (* x y) (* z t)))