Linear.V3:cross from linear-1.19.1.3

Percentage Accurate: 99.3% → 99.6%

Time: 3.1s

Alternatives: 4

Speedup: 1.0×

• 24.2% 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: 99.3% 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(z, Float64(-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. Add Preprocessing

3. Taylor expanded in x around 0 99.2%

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

   1. associate-*r* 99.2%

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

   2. neg-mul-1 99.2%

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

   3. *-commutative 99.2%

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

   4. fma-define 99.6%

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

5. Simplified 99.6%

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

6. Final simplification 99.6%

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

Alternative 2: 74.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2.15 \cdot 10^{+18} \lor \neg \left(x \cdot y \leq -1.12 \cdot 10^{-43} \lor \neg \left(x \cdot y \leq -7.5 \cdot 10^{-84}\right) \land x \cdot y \leq 6.5 \cdot 10^{-98}\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (* x y) -2.15e+18)
         (not
          (or (<= (* x y) -1.12e-43)
              (and (not (<= (* x y) -7.5e-84)) (<= (* x y) 6.5e-98)))))
   (* x y)
   (* z (- t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((x * y) <= -2.15e+18) || !(((x * y) <= -1.12e-43) || (!((x * y) <= -7.5e-84) && ((x * y) <= 6.5e-98)))) {
		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) <= (-2.15d+18)) .or. (.not. ((x * y) <= (-1.12d-43)) .or. (.not. ((x * y) <= (-7.5d-84))) .and. ((x * y) <= 6.5d-98))) 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) <= -2.15e+18) || !(((x * y) <= -1.12e-43) || (!((x * y) <= -7.5e-84) && ((x * y) <= 6.5e-98)))) {
		tmp = x * y;
	} else {
		tmp = z * -t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((x * y) <= -2.15e+18) or not (((x * y) <= -1.12e-43) or (not ((x * y) <= -7.5e-84) and ((x * y) <= 6.5e-98))):
		tmp = x * y
	else:
		tmp = z * -t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(x * y) <= -2.15e+18) || !((Float64(x * y) <= -1.12e-43) || (!(Float64(x * y) <= -7.5e-84) && (Float64(x * y) <= 6.5e-98))))
		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) <= -2.15e+18) || ~((((x * y) <= -1.12e-43) || (~(((x * y) <= -7.5e-84)) && ((x * y) <= 6.5e-98)))))
		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], -2.15e+18], N[Not[Or[LessEqual[N[(x * y), $MachinePrecision], -1.12e-43], And[N[Not[LessEqual[N[(x * y), $MachinePrecision], -7.5e-84]], $MachinePrecision], LessEqual[N[(x * y), $MachinePrecision], 6.5e-98]]]], $MachinePrecision]], N[(x * y), $MachinePrecision], N[(z * (-t)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -2.15e18 or -1.12e-43 < (*.f64 x y) < -7.50000000000000026e-84 or 6.50000000000000017e-98 < (*.f64 x y)

   1. Initial program 98.6%

      \[x \cdot y - z \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 77.6%

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

   if -2.15e18 < (*.f64 x y) < -1.12e-43 or -7.50000000000000026e-84 < (*.f64 x y) < 6.50000000000000017e-98

   1. Initial program 100.0%

      \[x \cdot y - z \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 87.2%

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

      1. associate-*r* 87.2%

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

      2. neg-mul-1 87.2%

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

      3. *-commutative 87.2%

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

   5. Simplified 87.2%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2.15 \cdot 10^{+18} \lor \neg \left(x \cdot y \leq -1.12 \cdot 10^{-43} \lor \neg \left(x \cdot y \leq -7.5 \cdot 10^{-84}\right) \land x \cdot y \leq 6.5 \cdot 10^{-98}\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.3% 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. Add Preprocessing

3. Final simplification 99.2%

   \[\leadsto x \cdot y - z \cdot t \]
4. Add Preprocessing

Alternative 4: 53.3% 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. Add Preprocessing

3. Taylor expanded in x around inf 51.2%

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

4. Final simplification 51.2%

   \[\leadsto x \cdot y \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024034 
(FPCore (x y z t)
  :name "Linear.V3:cross from linear-1.19.1.3"
  :precision binary64
  (- (* x y) (* z t)))