Linear.V3:cross from linear-1.19.1.3

Percentage Accurate: 99.2% → 99.7%

Time: 3.3s

Alternatives: 4

Speedup: 1.0×

• 22.9% 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: 99.7% 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 98.4%

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

   1. *-commutative 98.4%

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

   2. fma-neg 98.4%

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

   3. distribute-rgt-neg-in 98.4%

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

3. Applied egg-rr 98.4%

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

4. Final simplification 98.4%

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

Alternative 2: 75.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot x \leq -8 \cdot 10^{-63}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \cdot x \leq 1.48 \cdot 10^{-105} \lor \neg \left(y \cdot x \leq 3.2 \cdot 10^{-60}\right) \land y \cdot x \leq 460:\\ \;\;\;\;z \cdot \left(-t\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* y x) -8e-63)
   (* y x)
   (if (or (<= (* y x) 1.48e-105)
           (and (not (<= (* y x) 3.2e-60)) (<= (* y x) 460.0)))
     (* z (- t))
     (* y x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y * x) <= -8e-63) {
		tmp = y * x;
	} else if (((y * x) <= 1.48e-105) || (!((y * x) <= 3.2e-60) && ((y * x) <= 460.0))) {
		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-63)) then
        tmp = y * x
    else if (((y * x) <= 1.48d-105) .or. (.not. ((y * x) <= 3.2d-60)) .and. ((y * x) <= 460.0d0)) 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-63) {
		tmp = y * x;
	} else if (((y * x) <= 1.48e-105) || (!((y * x) <= 3.2e-60) && ((y * x) <= 460.0))) {
		tmp = z * -t;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (y * x) <= -8e-63:
		tmp = y * x
	elif ((y * x) <= 1.48e-105) or (not ((y * x) <= 3.2e-60) and ((y * x) <= 460.0)):
		tmp = z * -t
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(y * x) <= -8e-63)
		tmp = Float64(y * x);
	elseif ((Float64(y * x) <= 1.48e-105) || (!(Float64(y * x) <= 3.2e-60) && (Float64(y * x) <= 460.0)))
		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-63)
		tmp = y * x;
	elseif (((y * x) <= 1.48e-105) || (~(((y * x) <= 3.2e-60)) && ((y * x) <= 460.0)))
		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-63], N[(y * x), $MachinePrecision], If[Or[LessEqual[N[(y * x), $MachinePrecision], 1.48e-105], And[N[Not[LessEqual[N[(y * x), $MachinePrecision], 3.2e-60]], $MachinePrecision], LessEqual[N[(y * x), $MachinePrecision], 460.0]]], N[(z * (-t)), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -8.00000000000000053e-63 or 1.47999999999999992e-105 < (*.f64 x y) < 3.2000000000000001e-60 or 460 < (*.f64 x y)

   1. Initial program 97.2%

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

   2. Taylor expanded in x around inf 73.8%

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

   if -8.00000000000000053e-63 < (*.f64 x y) < 1.47999999999999992e-105 or 3.2000000000000001e-60 < (*.f64 x y) < 460

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 86.0%

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

      1. associate-*r* 86.0%

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

      2. neg-mul-1 86.0%

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

      3. *-commutative 86.0%

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

   4. Simplified 86.0%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot x \leq -8 \cdot 10^{-63}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \cdot x \leq 1.48 \cdot 10^{-105} \lor \neg \left(y \cdot x \leq 3.2 \cdot 10^{-60}\right) \land y \cdot x \leq 460:\\ \;\;\;\;z \cdot \left(-t\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 3: 99.2% 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 98.4%

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

2. Final simplification 98.4%

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

Alternative 4: 51.9% 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 98.4%

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

2. Taylor expanded in x around inf 50.5%

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

3. Final simplification 50.5%

   \[\leadsto y \cdot x \]

Reproduce

herbie shell --seed 2023279 
(FPCore (x y z t)
  :name "Linear.V3:cross from linear-1.19.1.3"
  :precision binary64
  (- (* x y) (* z t)))