Linear.V2:$cdot from linear-1.19.1.3, A

Percentage Accurate: 99.2% → 99.6%

Time: 4.2s

Alternatives: 3

Speedup: 1.2×

• 22.8% 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, 1.2× 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, 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. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{z \cdot t + x \cdot y} \]

   2. accelerator-lowering-fma.f64 N/A

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

   3. *-lowering-*.f64 99.2

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

4. Applied egg-rr 99.2%

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

Alternative 2: 75.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{-23}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+43}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z t) -2e-23) (* z t) (if (<= (* z t) 5e+43) (* x y) (* z t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * t) <= -2e-23) {
		tmp = z * t;
	} else if ((z * t) <= 5e+43) {
		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 ((z * t) <= (-2d-23)) then
        tmp = z * t
    else if ((z * t) <= 5d+43) 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 ((z * t) <= -2e-23) {
		tmp = z * t;
	} else if ((z * t) <= 5e+43) {
		tmp = x * y;
	} else {
		tmp = z * t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z * t) <= -2e-23:
		tmp = z * t
	elif (z * t) <= 5e+43:
		tmp = x * y
	else:
		tmp = z * t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * t) <= -2e-23)
		tmp = Float64(z * t);
	elseif (Float64(z * t) <= 5e+43)
		tmp = Float64(x * y);
	else
		tmp = Float64(z * t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * t) <= -2e-23)
		tmp = z * t;
	elseif ((z * t) <= 5e+43)
		tmp = x * y;
	else
		tmp = z * t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * t), $MachinePrecision], -2e-23], N[(z * t), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 5e+43], N[(x * y), $MachinePrecision], N[(z * t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -1.99999999999999992e-23 or 5.0000000000000004e43 < (*.f64 z t)

   1. Initial program 98.3%

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

   3. Taylor expanded in x around 0

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

      1. +-rgt-identity N/A

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

      2. accelerator-lowering-fma.f64 83.0

         \[\leadsto \color{blue}{\mathsf{fma}\left(t, z, 0\right)} \]

   5. Simplified 83.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, z, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 83.0

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

   7. Applied egg-rr 83.0%

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

   if -1.99999999999999992e-23 < (*.f64 z t) < 5.0000000000000004e43

   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. +-rgt-identity N/A

         \[\leadsto \color{blue}{x \cdot y + 0} \]

      2. accelerator-lowering-fma.f64 77.1

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, 0\right)} \]

   5. Simplified 77.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 77.1

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

   7. Applied egg-rr 77.1%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{-23}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+43}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 52.4% 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

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

   1. +-rgt-identity N/A

      \[\leadsto \color{blue}{x \cdot y + 0} \]

   2. accelerator-lowering-fma.f64 51.3

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, 0\right)} \]

5. Simplified 51.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, 0\right)} \]
6. Step-by-step derivation

   1. +-rgt-identity N/A

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

   2. *-commutative N/A

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

   3. *-lowering-*.f64 51.3

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

7. Applied egg-rr 51.3%

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

8. Final simplification 51.3%

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

Reproduce

herbie shell --seed 2024195 
(FPCore (x y z t)
  :name "Linear.V2:$cdot from linear-1.19.1.3, A"
  :precision binary64
  (+ (* x y) (* z t)))