Linear.V3:cross from linear-1.19.1.3

Percentage Accurate: 99.1% → 99.5%

Time: 3.0s

Alternatives: 4

Speedup: 1.0×

• 23.9% 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: 99.1% 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: 99.5% 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 99.2%

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

   1. sub-neg 99.2%

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

   2. +-commutative 99.2%

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

   3. distribute-lft-neg-in 99.2%

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

   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: 74.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1.26 \cdot 10^{+113}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \cdot y \leq -6.8 \cdot 10^{+97} \lor \neg \left(x \cdot y \leq -1.25 \cdot 10^{+26}\right) \land \left(x \cdot y \leq -9.5 \cdot 10^{-25} \lor \neg \left(x \cdot y \leq -5.7 \cdot 10^{-64}\right) \land x \cdot y \leq 1.95 \cdot 10^{+80}\right):\\ \;\;\;\;z \cdot \left(-t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* x y) -1.26e+113)
   (* x y)
   (if (or (<= (* x y) -6.8e+97)
           (and (not (<= (* x y) -1.25e+26))
                (or (<= (* x y) -9.5e-25)
                    (and (not (<= (* x y) -5.7e-64)) (<= (* x y) 1.95e+80)))))
     (* z (- t))
     (* x y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * y) <= -1.26e+113) {
		tmp = x * y;
	} else if (((x * y) <= -6.8e+97) || (!((x * y) <= -1.25e+26) && (((x * y) <= -9.5e-25) || (!((x * y) <= -5.7e-64) && ((x * y) <= 1.95e+80))))) {
		tmp = z * -t;
	} else {
		tmp = x * y;
	}
	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) <= (-1.26d+113)) then
        tmp = x * y
    else if (((x * y) <= (-6.8d+97)) .or. (.not. ((x * y) <= (-1.25d+26))) .and. ((x * y) <= (-9.5d-25)) .or. (.not. ((x * y) <= (-5.7d-64))) .and. ((x * y) <= 1.95d+80)) then
        tmp = z * -t
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * y) <= -1.26e+113) {
		tmp = x * y;
	} else if (((x * y) <= -6.8e+97) || (!((x * y) <= -1.25e+26) && (((x * y) <= -9.5e-25) || (!((x * y) <= -5.7e-64) && ((x * y) <= 1.95e+80))))) {
		tmp = z * -t;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x * y) <= -1.26e+113:
		tmp = x * y
	elif ((x * y) <= -6.8e+97) or (not ((x * y) <= -1.25e+26) and (((x * y) <= -9.5e-25) or (not ((x * y) <= -5.7e-64) and ((x * y) <= 1.95e+80)))):
		tmp = z * -t
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x * y) <= -1.26e+113)
		tmp = Float64(x * y);
	elseif ((Float64(x * y) <= -6.8e+97) || (!(Float64(x * y) <= -1.25e+26) && ((Float64(x * y) <= -9.5e-25) || (!(Float64(x * y) <= -5.7e-64) && (Float64(x * y) <= 1.95e+80)))))
		tmp = Float64(z * Float64(-t));
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x * y) <= -1.26e+113)
		tmp = x * y;
	elseif (((x * y) <= -6.8e+97) || (~(((x * y) <= -1.25e+26)) && (((x * y) <= -9.5e-25) || (~(((x * y) <= -5.7e-64)) && ((x * y) <= 1.95e+80)))))
		tmp = z * -t;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x * y), $MachinePrecision], -1.26e+113], N[(x * y), $MachinePrecision], If[Or[LessEqual[N[(x * y), $MachinePrecision], -6.8e+97], And[N[Not[LessEqual[N[(x * y), $MachinePrecision], -1.25e+26]], $MachinePrecision], Or[LessEqual[N[(x * y), $MachinePrecision], -9.5e-25], And[N[Not[LessEqual[N[(x * y), $MachinePrecision], -5.7e-64]], $MachinePrecision], LessEqual[N[(x * y), $MachinePrecision], 1.95e+80]]]]], N[(z * (-t)), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -1.2599999999999999e113 or -6.8000000000000002e97 < (*.f64 x y) < -1.25e26 or -9.50000000000000065e-25 < (*.f64 x y) < -5.7000000000000003e-64 or 1.94999999999999999e80 < (*.f64 x y)

   1. Initial program 98.2%

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

   2. Taylor expanded in x around inf 83.7%

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

   if -1.2599999999999999e113 < (*.f64 x y) < -6.8000000000000002e97 or -1.25e26 < (*.f64 x y) < -9.50000000000000065e-25 or -5.7000000000000003e-64 < (*.f64 x y) < 1.94999999999999999e80

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 84.1%

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

      1. associate-*r* 84.1%

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

      2. neg-mul-1 84.1%

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

      3. *-commutative 84.1%

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

   4. Simplified 84.1%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1.26 \cdot 10^{+113}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \cdot y \leq -6.8 \cdot 10^{+97} \lor \neg \left(x \cdot y \leq -1.25 \cdot 10^{+26}\right) \land \left(x \cdot y \leq -9.5 \cdot 10^{-25} \lor \neg \left(x \cdot y \leq -5.7 \cdot 10^{-64}\right) \land x \cdot y \leq 1.95 \cdot 10^{+80}\right):\\ \;\;\;\;z \cdot \left(-t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 3: 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 99.2%

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

2. Final simplification 99.2%

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

Alternative 4: 52.6% 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 99.2%

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

2. Taylor expanded in x around inf 49.0%

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

3. Final simplification 49.0%

   \[\leadsto x \cdot y \]

Reproduce

herbie shell --seed 2023287 
(FPCore (x y z t)
  :name "Linear.V3:cross from linear-1.19.1.3"
  :precision binary64
  (- (* x y) (* z t)))