Linear.V3:cross from linear-1.19.1.3

Percentage Accurate: 99.2% → 99.6%

Time: 2.2s

Alternatives: 5

Speedup: 1.0×

• 23.5% 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: 99.6% 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(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. distribute-rgt-neg-out 99.2%

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

   3. +-commutative 99.2%

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

   4. distribute-rgt-neg-out 99.2%

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

   5. distribute-lft-neg-in 99.2%

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

   6. 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: 99.5% accurate, 0.1× speedup

Math

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

FPCore

(FPCore (x y z t) :precision binary64 (fma x y (* z (- t))))

C

double code(double x, double y, double z, double t) {
	return fma(x, y, (z * -t));
}

Julia

function code(x, y, z, t)
	return fma(x, y, Float64(z * Float64(-t)))
end

Wolfram

code[x_, y_, z_, t_] := N[(x * y + N[(z * (-t)), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.2%

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

   1. fma-neg 99.6%

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

   2. distribute-rgt-neg-in 99.6%

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

3. Simplified 99.6%

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

4. Final simplification 99.6%

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

Alternative 3: 68.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.4 \cdot 10^{+68}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-62}:\\ \;\;\;\;z \cdot \left(-t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -5.4e+68) (* x y) (if (<= x 1.1e-62) (* z (- t)) (* x y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -5.4e+68) {
		tmp = x * y;
	} else if (x <= 1.1e-62) {
		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 <= (-5.4d+68)) then
        tmp = x * y
    else if (x <= 1.1d-62) 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 <= -5.4e+68) {
		tmp = x * y;
	} else if (x <= 1.1e-62) {
		tmp = z * -t;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -5.4e+68:
		tmp = x * y
	elif x <= 1.1e-62:
		tmp = z * -t
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -5.4e+68)
		tmp = Float64(x * y);
	elseif (x <= 1.1e-62)
		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 <= -5.4e+68)
		tmp = x * y;
	elseif (x <= 1.1e-62)
		tmp = z * -t;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -5.4e+68], N[(x * y), $MachinePrecision], If[LessEqual[x, 1.1e-62], N[(z * (-t)), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -5.39999999999999982e68 or 1.10000000000000009e-62 < x

   1. Initial program 98.6%

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

   2. Taylor expanded in x around inf 69.2%

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

   if -5.39999999999999982e68 < x < 1.10000000000000009e-62

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 72.1%

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

      1. mul-1-neg 72.1%

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

      2. distribute-rgt-neg-in 72.1%

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

   4. Simplified 72.1%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.4 \cdot 10^{+68}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-62}:\\ \;\;\;\;z \cdot \left(-t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 4: 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]

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 5: 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 52.9%

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

3. Final simplification 52.9%

   \[\leadsto x \cdot y \]

Reproduce

herbie shell --seed 2023196 
(FPCore (x y z t)
  :name "Linear.V3:cross from linear-1.19.1.3"
  :precision binary64
  (- (* x y) (* z t)))