Linear.V3:cross from linear-1.19.1.3

Percentage Accurate: 99.2% → 99.5%

Time: 2.9s

Alternatives: 4

Speedup: 1.0×

• 23.3% 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: 99.5% 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 99.2%

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

   1. *-commutative 99.2%

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

   2. fma-neg 99.6%

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

   3. distribute-rgt-neg-in 99.6%

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

3. Applied egg-rr 99.6%

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

4. Final simplification 99.6%

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

Alternative 2: 74.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot x \leq -6.6 \cdot 10^{+28} \lor \neg \left(y \cdot x \leq -1.15 \cdot 10^{-14}\right) \land \left(y \cdot x \leq -1.1 \cdot 10^{-71} \lor \neg \left(y \cdot x \leq 3.4 \cdot 10^{-63} \lor \neg \left(y \cdot x \leq 1.1 \cdot 10^{-54}\right) \land y \cdot x \leq 44000000000000\right)\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (* y x) -6.6e+28)
         (and (not (<= (* y x) -1.15e-14))
              (or (<= (* y x) -1.1e-71)
                  (not
                   (or (<= (* y x) 3.4e-63)
                       (and (not (<= (* y x) 1.1e-54))
                            (<= (* y x) 44000000000000.0)))))))
   (* y x)
   (* z (- t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((y * x) <= -6.6e+28) || (!((y * x) <= -1.15e-14) && (((y * x) <= -1.1e-71) || !(((y * x) <= 3.4e-63) || (!((y * x) <= 1.1e-54) && ((y * x) <= 44000000000000.0)))))) {
		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 (((y * x) <= (-6.6d+28)) .or. (.not. ((y * x) <= (-1.15d-14))) .and. ((y * x) <= (-1.1d-71)) .or. (.not. ((y * x) <= 3.4d-63) .or. (.not. ((y * x) <= 1.1d-54)) .and. ((y * x) <= 44000000000000.0d0))) 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 (((y * x) <= -6.6e+28) || (!((y * x) <= -1.15e-14) && (((y * x) <= -1.1e-71) || !(((y * x) <= 3.4e-63) || (!((y * x) <= 1.1e-54) && ((y * x) <= 44000000000000.0)))))) {
		tmp = y * x;
	} else {
		tmp = z * -t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((y * x) <= -6.6e+28) or (not ((y * x) <= -1.15e-14) and (((y * x) <= -1.1e-71) or not (((y * x) <= 3.4e-63) or (not ((y * x) <= 1.1e-54) and ((y * x) <= 44000000000000.0))))):
		tmp = y * x
	else:
		tmp = z * -t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(y * x) <= -6.6e+28) || (!(Float64(y * x) <= -1.15e-14) && ((Float64(y * x) <= -1.1e-71) || !((Float64(y * x) <= 3.4e-63) || (!(Float64(y * x) <= 1.1e-54) && (Float64(y * x) <= 44000000000000.0))))))
		tmp = Float64(y * x);
	else
		tmp = Float64(z * Float64(-t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((y * x) <= -6.6e+28) || (~(((y * x) <= -1.15e-14)) && (((y * x) <= -1.1e-71) || ~((((y * x) <= 3.4e-63) || (~(((y * x) <= 1.1e-54)) && ((y * x) <= 44000000000000.0)))))))
		tmp = y * x;
	else
		tmp = z * -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[N[(y * x), $MachinePrecision], -6.6e+28], And[N[Not[LessEqual[N[(y * x), $MachinePrecision], -1.15e-14]], $MachinePrecision], Or[LessEqual[N[(y * x), $MachinePrecision], -1.1e-71], N[Not[Or[LessEqual[N[(y * x), $MachinePrecision], 3.4e-63], And[N[Not[LessEqual[N[(y * x), $MachinePrecision], 1.1e-54]], $MachinePrecision], LessEqual[N[(y * x), $MachinePrecision], 44000000000000.0]]]], $MachinePrecision]]]], N[(y * x), $MachinePrecision], N[(z * (-t)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -6.6e28 or -1.14999999999999999e-14 < (*.f64 x y) < -1.09999999999999999e-71 or 3.39999999999999998e-63 < (*.f64 x y) < 1.1e-54 or 4.4e13 < (*.f64 x y)

   1. Initial program 98.5%

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

   2. Taylor expanded in x around inf 81.0%

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

   if -6.6e28 < (*.f64 x y) < -1.14999999999999999e-14 or -1.09999999999999999e-71 < (*.f64 x y) < 3.39999999999999998e-63 or 1.1e-54 < (*.f64 x y) < 4.4e13

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 83.0%

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

      1. associate-*r* 83.0%

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

      2. neg-mul-1 83.0%

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

      3. *-commutative 83.0%

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

   4. Simplified 83.0%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot x \leq -6.6 \cdot 10^{+28} \lor \neg \left(y \cdot x \leq -1.15 \cdot 10^{-14}\right) \land \left(y \cdot x \leq -1.1 \cdot 10^{-71} \lor \neg \left(y \cdot x \leq 3.4 \cdot 10^{-63} \lor \neg \left(y \cdot x \leq 1.1 \cdot 10^{-54}\right) \land y \cdot x \leq 44000000000000\right)\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-t\right)\\ \end{array} \]

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 99.2%

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

2. Final simplification 99.2%

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

Alternative 4: 52.2% 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. Taylor expanded in x around inf 53.2%

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

3. Final simplification 53.2%

   \[\leadsto y \cdot x \]

Reproduce

herbie shell --seed 2023301 
(FPCore (x y z t)
  :name "Linear.V3:cross from linear-1.19.1.3"
  :precision binary64
  (- (* x y) (* z t)))