Linear.V3:cross from linear-1.19.1.3

Percentage Accurate: 99.3% → 99.6%

Time: 5.3s

Alternatives: 4

Speedup: 1.0×

• 24.1% 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

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

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 99.6%

   \[x \cdot y - z \cdot t \]
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 100.0%

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

4. Applied egg-rr 100.0%

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

Alternative 2: 75.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1.2 \cdot 10^{-24}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \cdot y \leq 9.2 \cdot 10^{-23}:\\ \;\;\;\;0 - z \cdot t\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* x y) -1.2e-24)
   (* x y)
   (if (<= (* x y) 9.2e-23) (- 0.0 (* z t)) (* x y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * y) <= -1.2e-24) {
		tmp = x * y;
	} else if ((x * y) <= 9.2e-23) {
		tmp = 0.0 - (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) <= (-1.2d-24)) then
        tmp = x * y
    else if ((x * y) <= 9.2d-23) then
        tmp = 0.0d0 - (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) <= -1.2e-24) {
		tmp = x * y;
	} else if ((x * y) <= 9.2e-23) {
		tmp = 0.0 - (z * t);
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x * y) <= -1.2e-24:
		tmp = x * y
	elif (x * y) <= 9.2e-23:
		tmp = 0.0 - (z * t)
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x * y) <= -1.2e-24)
		tmp = Float64(x * y);
	elseif (Float64(x * y) <= 9.2e-23)
		tmp = Float64(0.0 - Float64(z * 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) <= -1.2e-24)
		tmp = x * y;
	elseif ((x * y) <= 9.2e-23)
		tmp = 0.0 - (z * t);
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x * y), $MachinePrecision], -1.2e-24], N[(x * y), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 9.2e-23], N[(0.0 - N[(z * t), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -1.1999999999999999e-24 or 9.2000000000000004e-23 < (*.f64 x y)

   1. Initial program 99.2%

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

   3. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 72.2%

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

   5. Simplified 72.2%

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

   if -1.1999999999999999e-24 < (*.f64 x y) < 9.2000000000000004e-23

   1. Initial program 100.0%

      \[x \cdot y - z \cdot t \]
   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 83.9%

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

   5. Simplified 83.9%

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

      1. cancel-sign-sub-inv N/A

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

      2. flip3-+ N/A

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

   7. Applied egg-rr 83.9%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1.2 \cdot 10^{-24}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \cdot y \leq 9.2 \cdot 10^{-23}:\\ \;\;\;\;0 - z \cdot t\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
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 99.6%

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

Alternative 4: 52.2% 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 99.6%

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

3. Taylor expanded in x around inf

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

   1. *-lowering-*.f64 48.1%

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

5. Simplified 48.1%

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

Reproduce

herbie shell --seed 2024163 
(FPCore (x y z t)
  :name "Linear.V3:cross from linear-1.19.1.3"
  :precision binary64
  (- (* x y) (* z t)))