Linear.V3:cross from linear-1.19.1.3

Percentage Accurate: 99.3% → 99.7%

Time: 3.6s

Alternatives: 4

Speedup: 1.0×

• 23.2% 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: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.0%

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

3. Taylor expanded in x around 0

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

   1. sub-neg N/A

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

   2. mul-1-neg N/A

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

   3. +-commutative N/A

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

   4. mul-1-neg N/A

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

   5. distribute-rgt-neg-in N/A

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

   6. mul-1-neg N/A

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

   7. lower-fma.f64 N/A

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

   8. mul-1-neg N/A

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

   9. lower-neg.f64 N/A

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

   10. lower-*.f64 99.2

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

5. Simplified 99.2%

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

Alternative 2: 74.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -t \cdot z\\ \mathbf{if}\;t \cdot z \leq -200000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \cdot z \leq 2 \cdot 10^{-103}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* t z))))
   (if (<= (* t z) -200000.0) t_1 (if (<= (* t z) 2e-103) (* x y) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = -(t * z);
	double tmp;
	if ((t * z) <= -200000.0) {
		tmp = t_1;
	} else if ((t * z) <= 2e-103) {
		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 = -(t * z)
    if ((t * z) <= (-200000.0d0)) then
        tmp = t_1
    else if ((t * z) <= 2d-103) 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 = -(t * z);
	double tmp;
	if ((t * z) <= -200000.0) {
		tmp = t_1;
	} else if ((t * z) <= 2e-103) {
		tmp = x * y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = -(t * z)
	tmp = 0
	if (t * z) <= -200000.0:
		tmp = t_1
	elif (t * z) <= 2e-103:
		tmp = x * y
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(-Float64(t * z))
	tmp = 0.0
	if (Float64(t * z) <= -200000.0)
		tmp = t_1;
	elseif (Float64(t * z) <= 2e-103)
		tmp = Float64(x * y);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = -(t * z);
	tmp = 0.0;
	if ((t * z) <= -200000.0)
		tmp = t_1;
	elseif ((t * z) <= 2e-103)
		tmp = x * y;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = (-N[(t * z), $MachinePrecision])}, If[LessEqual[N[(t * z), $MachinePrecision], -200000.0], t$95$1, If[LessEqual[N[(t * z), $MachinePrecision], 2e-103], N[(x * y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -2e5 or 1.99999999999999992e-103 < (*.f64 z t)

   1. Initial program 96.6%

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

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 77.4

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

   5. Simplified 77.4%

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

   if -2e5 < (*.f64 z t) < 1.99999999999999992e-103

   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. lower-*.f64 81.6

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

   5. Simplified 81.6%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot z \leq -200000:\\ \;\;\;\;-t \cdot z\\ \mathbf{elif}\;t \cdot z \leq 2 \cdot 10^{-103}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;-t \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot y - t \cdot z \end{array} \]

FPCore

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

C

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

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) - (t * z)
end function

Java

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

Python

def code(x, y, z, t):
	return (x * y) - (t * z)

Julia

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

MATLAB

function tmp = code(x, y, z, t)
	tmp = (x * y) - (t * z);
end

Wolfram

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

Derivation

1. Initial program 98.0%

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

3. Final simplification 98.0%

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

Alternative 4: 52.1% 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 98.0%

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

3. Taylor expanded in x around inf

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

   1. lower-*.f64 49.3

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

5. Simplified 49.3%

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

Reproduce

herbie shell --seed 2024215 
(FPCore (x y z t)
  :name "Linear.V3:cross from linear-1.19.1.3"
  :precision binary64
  (- (* x y) (* z t)))