Linear.V3:cross from linear-1.19.1.3

Percentage Accurate: 99.2% → 99.6%

Time: 5.6s

Alternatives: 4

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

\[\begin{array}{l} \\ \mathsf{fma}\left(0 - z, t, x \cdot y\right) \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (fma (- 0.0 z) t (* x y)))

C

double code(double x, double y, double z, double t) {
	return fma((0.0 - z), t, (x * y));
}

Julia

function code(x, y, z, t)
	return fma(Float64(0.0 - z), t, Float64(x * y))
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(0.0 - z), $MachinePrecision] * t + N[(x * y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.4%

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. distribute-lft-neg-in N/A

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

   4. fma-define N/A

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

   5. fma-lowering-fma.f64 N/A

      \[\leadsto \mathsf{fma.f64}\left(\left(\mathsf{neg}\left(z\right)\right), \color{blue}{t}, \left(x \cdot y\right)\right) \]

   6. neg-sub0 N/A

      \[\leadsto \mathsf{fma.f64}\left(\left(0 - z\right), t, \left(x \cdot y\right)\right) \]

   7. --lowering--.f64 N/A

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, z\right), t, \left(x \cdot y\right)\right) \]

   8. *-lowering-*.f64 98.4%

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, z\right), t, \mathsf{*.f64}\left(x, y\right)\right) \]

4. Applied egg-rr 98.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0 - z, t, x \cdot y\right)} \]
5. Add Preprocessing

Alternative 2: 74.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 0 - z \cdot t\\ \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+79}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+47}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- 0.0 (* z t))))
   (if (<= (* z t) -2e+79) t_1 (if (<= (* z t) 2e+47) (* x y) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = 0.0 - (z * t);
	double tmp;
	if ((z * t) <= -2e+79) {
		tmp = t_1;
	} else if ((z * t) <= 2e+47) {
		tmp = x * y;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = 0.0d0 - (z * t)
    if ((z * t) <= (-2d+79)) then
        tmp = t_1
    else if ((z * t) <= 2d+47) then
        tmp = x * y
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = 0.0 - (z * t);
	double tmp;
	if ((z * t) <= -2e+79) {
		tmp = t_1;
	} else if ((z * t) <= 2e+47) {
		tmp = x * y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = 0.0 - (z * t)
	tmp = 0
	if (z * t) <= -2e+79:
		tmp = t_1
	elif (z * t) <= 2e+47:
		tmp = x * y
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(0.0 - Float64(z * t))
	tmp = 0.0
	if (Float64(z * t) <= -2e+79)
		tmp = t_1;
	elseif (Float64(z * t) <= 2e+47)
		tmp = Float64(x * y);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = 0.0 - (z * t);
	tmp = 0.0;
	if ((z * t) <= -2e+79)
		tmp = t_1;
	elseif ((z * t) <= 2e+47)
		tmp = x * y;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(0.0 - N[(z * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(z * t), $MachinePrecision], -2e+79], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], 2e+47], N[(x * y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -1.99999999999999993e79 or 2.0000000000000001e47 < (*.f64 z t)

   1. Initial program 96.1%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(t \cdot z\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{t \cdot z} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(t \cdot z\right)}\right) \]

      4. *-lowering-*.f64 88.5%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(t, \color{blue}{z}\right)\right) \]

   5. Simplified 88.5%

      \[\leadsto \color{blue}{0 - t \cdot z} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(t \cdot z\right) \]

      2. +-lft-identity N/A

         \[\leadsto \mathsf{neg}\left(\left(0 + t \cdot z\right)\right) \]

      3. flip3-+ N/A

         \[\leadsto \mathsf{neg}\left(\frac{{0}^{3} + {\left(t \cdot z\right)}^{3}}{0 \cdot 0 + \left(\left(t \cdot z\right) \cdot \left(t \cdot z\right) - 0 \cdot \left(t \cdot z\right)\right)}\right) \]

      4. distribute-neg-frac N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left({0}^{3} + {\left(t \cdot z\right)}^{3}\right)\right)}{\color{blue}{0 \cdot 0 + \left(\left(t \cdot z\right) \cdot \left(t \cdot z\right) - 0 \cdot \left(t \cdot z\right)\right)}} \]

   7. Applied egg-rr 88.5%

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

   if -1.99999999999999993e79 < (*.f64 z t) < 2.0000000000000001e47

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot y} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 81.2%

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{y}\right) \]

   5. Simplified 81.2%

      \[\leadsto \color{blue}{x \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+79}:\\ \;\;\;\;0 - z \cdot t\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+47}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;0 - z \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 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 98.4%

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

Alternative 4: 50.8% 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 98.4%

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

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{x \cdot y} \]
4. Step-by-step derivation

   1. *-lowering-*.f64 54.8%

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{y}\right) \]

5. Simplified 54.8%

   \[\leadsto \color{blue}{x \cdot y} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024158 
(FPCore (x y z t)
  :name "Linear.V3:cross from linear-1.19.1.3"
  :precision binary64
  (- (* x y) (* z t)))